Principles of Real and Complex Analysis

Su Hyeong Lee

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Table of Contents

1 Chapter 1 (The Real and Complex Number Systems) 3

2 Chapter 2 (Basic Topology) 6

3 Chapter 3 (Numerical Sequences and Series) 9

4 Chapter 4 (Continuity) 18

5 Chapter 5 (Differentiation) 24

6 Chapter 6 (The Riemann-Stieltjes Integral) 27

7 Chapter 7 (Sequences and Series of Functions) 32

8 Chapter 8 (Some Special Functions) 37

9 Chapter 9 (Functions of Several Variables) 42

10 Chapter 10 (Geometry and Topology in the Complex Plane) 43

11 Chapter 11 (Paths, Holomorphic Functions, and Complex Series) 47

12 Chapter 12 (Conformal Mappings and Multifunctions) 51

13 Chapter 13 (Integration and Cauchy’s Theorem) 56

14 Chapter 14 (Power Series Representations and Zeros of Holomorphic Func-

tions) 61

15 Chapter 15 (Modulus theorems, Meromorphic Functions, and Cauchy’s Residue

Theorem) 65

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1 Chapter 1 (The Real and Complex Number Systems)

Definition 1.1. Let S be a set. An order < on S is a relation, such that:

(i) If x, y ∈ S, exactly one of x < y, x = y, y < x holds.

(ii) If x, y, z ∈ S, x < y, y < z, then x < z.

An ordered set is a set in which an order is defined.

Theorem 1.2. An ordered set S has the least upper bound property if E ⊂ S, E 6= ∅, and E is

bounded above, then sup(E) exists in S. Every ordered set with the least upper bound property

also has the greatest lower bound property.

Definition 1.3. An ordered field F is an ordered set such that:

(i) x+ y < x+ z if x, y, z ∈ F and y < z

(ii) xy > 0 if x ∈ F , y ∈ F , x > 0, y > 0

Remark. Note that if x > 0, x is called positive, and similarly for negative. All the familiar

(calculation) rules for working with inequalities apply in every ordered field. This is proposition

1.18 on page 8, which also contains statements like, ‘a non-zero element squared is positive’

and ‘0 < x < y =⇒ 0 < 1/y < 1/x’.

Theorem 1.4 (Existence Theorem). Exists an ordered field R which has the least upper bound

property. Moreover, R contains Q as a subfield. The elements of R(:= R) are called real

numbers.

Remark. With this theorem, you can prove ‘Archimedian property of R’, and ‘denseness of Q

in R’. Here, denseness is stated: If x, y ∈ R, x < y, then ∃p ∈ Q s.t. x < p < y. It is easy to

prove that this is equivalent to the statement ‘Q is dense in R’, where the definition of dense

is given in topological terms (as in M2PM5).

Theorem 1.5. For every real x > 0 and integer n > 0, there is one and only one positive real

y such that yn = x. Let n√x := x1/n := y.

Remark. How is x1/a defined where a is not an integer?

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Definition 1.6. A complex number is an ordered pair (a, b) of real numbers. ‘Ordered’ means

(a, b) and (b, a) are distinct if a 6= b. Define addition and multiplication conventionally, and

this becomes a field. Call this C.

Definition 1.7. We preserve the already-defined order in R, and extend R to include +∞ and

−∞. This is the extended real number system, and it does not form a field. You know the

convention for the arithmetic using ∞ already. One thing to note is that by definition, the

order (the ‘<’ symbol) is defined to be ∀x ∈ R, −∞ < x < +∞.

Theorem 1.8 (Schwartz Inequality). Let ai, bi ∈ C, for i = 1, . . . , n. Then,∥∥∥∥∥∥n∑j=1

ajbj

∥∥∥∥∥∥2

≤

n∑j=1

∥∥aj∥∥2

n∑j=1

∥∥bj∥∥2

The equality condition for this inequality is: let aj = rj exp(iθj), bj = sj exp(iωj), rj, sj ∈ R≥0.

Then for every i, j s.t. i < j, risj = rjsi and (either risirjsj = 0 or θi − ωi − θj + ωj = 2nπ).

Remark. Note carefully the norms in the Schwartz Inequality, because replacing norms with

brackets will give an incorrect statement. Annoyingly, for x ∈ C, x2 6= x · x. This is despite

v2 := v · v, for v ∈ Rk (for k ≥ 3, I presume).

Definition 1.9. For k ∈ Z+, let Rk be the set of all ordered k-tuples x = (x1, . . . , xk). These

are called vectors, and addition and multiplication are defined conventionally. This defines a

vector field for all such k.

Proposition 1.10. The steps to constructing the reals by Dedekind cuts are:

(1) Suppose we already have Q. We define a set R. The members of R are certain subsets of

Q, called cuts. A cut is, by definition, any set α ⊂ Q that satisfies:

(i) α 6= ∅, α 6= Q.

(ii) If p ∈ α, q ∈ Q, q < p, then q ∈ α.

(iii) If p ∈ α, then ∃r ∈ α s.t. p < r.

(2) Letters of the greek alphabet α, β, γ, . . . denote cuts, and letters of the english alphabet

p, q, r, . . . denote rational numbers. Define α < β to mean α is a proper subset of β (and α = β

to mean α, β are equal). This is an order for R.

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(3) The ordered set R has the least upper bound property.

(4) If α, β ∈ R, define α+β to be the set of all sums r+s, where r ∈ α, s ∈ β. Define 0∗ := Q<0.

0∗ is a cut, and R satisfies the field axioms with respect to addition (operator +).

(Clever step) Fix α ∈ R. Let β := ∀p ∈ Q s.t. ∃r > 0 s.t. − p− r /∈ α. In other words, some

rational number smaller than −p fails to be in α. Then, β ∈ R and α + β = 0∗. −α := β.

(5) We can now prove that if α, β, γ ∈ R and β < γ, then α + β < α + γ, and also that

α > 0∗ ⇐⇒ −α < 0∗.

(6) The product of negative rationals are positive, so this makes defining multiplication more

bothersome. So we firstly restrict ourselves to R+ := ∀α ∈ R s.t. α > 0∗. If α, β ∈ R+,

define αβ := ∀p ∈ Q s.t. p ≤ rs for some choice of r ∈ α, s ∈ β, r > 0, s > 0. Define

1∗ := ∀q ∈ Q s.t. q < 1. Note, in particular, that α, β > 0∗ =⇒ αβ > 0∗.

(7) Complete the definition of multiplication by setting α0∗ = 0∗α = 0∗, and:

αβ :=

(−α)(−β) if α < 0∗, β < 0∗

−[(−α)β] if α < 0∗, β > 0∗

−[α(−β)] if α > 0∗, β < 0∗

We can now finish the proof that R is an ordered field with the least upper bound property.

(8) Associate with each r ∈ Q the set r∗ := ∀p ∈ Q s.t. p < r. Then r∗ ∈ R, and it satisfies

properties (i) r∗ + s∗ = (r + s)∗, (ii) r∗s∗ = (rs)∗, (iii) r∗ < s∗ ⇐⇒ r < s.

(9) The above step shows that replacing rational numbers r by their ‘rational cuts’ r∗ ∈ R

preserves sums, products, and order. Thus, the ordered field Q is isomorphic to the ordered

field Q∗, whose elements are the rational cuts. This allows us to regard Q as a subfield of R. It

is also a fact (no proof given) that any two ordered fields with the least upper bound property

are isomorphic.

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2 Chapter 2 (Basic Topology)

Remark. Prove that set A is infinite if it is bijective to one of its proper subsets. Easy. Then

does it hold that A is infinite only if it is bijective to one of its proper subsets?

Definition 2.1. Let ai, bi ∈ R s.t. ai < bi, for i = 1, . . . , k, k ∈ Z+. Then, the set of all

x = (x1, . . . , xk) ∈ Rk s.t. ai ≤ xi ≤ bi is called a k-cell. So a 1-cell is a closed line, a 2-cell a

closed rectangle, and so on. It is easily proved that k-cells are convex. E ⊂ Rk is convex if for

any two elements, (informally) the ‘line segment’ connecting the two is a subset of Rk.

Remark. From now on, I won’t write ‘informally’. Should be fairly clear within context.

Definition 2.2. In this book, a neighbourhood around p ∈ X, X a metric space, is defined to

be an open set around it (commonly an open ball). A point p ∈ E is a limit point of E if every

neighbourhood around p contains a point in E\p. If p ∈ E and p is not a limit point of E,

then p is called an isolated point of E.

Definition 2.3. E is perfect if E is closed and every point in E is a limit point of E. E is

dense in metric space X if every point in X is either a limit point or an element of E.

Remark. In this chapter, we establish results encountered in the ‘Introduction to Metric Spaces

and Topology’ module, but using slightly different definitions which are based on the language

of R instead of Topological Spaces. This chapter shows how topological theorems derived using

more general topological definitions (of ‘open’, ‘closed’, etc.,) are totally consistent with the

‘more specific’ definitions presented in Real Analysis. I’m finally satisfied; the proof of this

observation had not been explicitly remarked on in lectures, only implicitly used.

Remark. Let X be a metric space. Let K ⊂ Y ⊂ X. Then K can be open relative to Y but

not X. So when we say set A is open or closed, we should identify which set A is open relative

to. However, compactness behaves better.

Theorem 2.4. Let X be a metric space. Let K ⊂ Y ⊂ X. Then K is compact relative to X

⇐⇒ K is compact relative to Y .

Remark. Thus, although it does not make sense to talk about open/closed spaces without

mentioning the metric space to which the space is embedded to, it does make sense to talk

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about compact spaces. This is because it can be directly inferred from the theorem above that

K compact in X ⇐⇒ K compact in Y ⇐⇒ K compact in K (= K is compact).

Theorem 2.5. Compact subsets of Hausdorff spaces are closed.

Theorem 2.6. Closed subsets of compact sets are compact. Therefore, if F is closed and K is

compact, F ∩K is compact.

Theorem 2.7. Ka is a collection of compact subsets of a metric space X s.t. the intersection

of every finite subcollection of Ka is nonempty. Then, ∩Ka is nonempty. (Indexing set can

be uncountable)

Theorem 2.8. Ka is a collection of compact sets of a metric space X s.t. Kn+1 ⊂ Kn, for

n ∈ Z+. Then ∩∞1 Ka is nonempty.

Remark. From now on, due to context, whenever ‘compact set’ is mentioned, we are talking

about a metric space, as compactness in this book was defined on a metric space.

Theorem 2.9. If E is an infinite subset of a compact set K (K a metric space), then E has

a limit point in K. (This means K contains a limit point of E.)

Remark. The intuition is: K is compact, hence ‘small’. E is an infinite subset of a ‘small’ set

K, so this means that elements of E are very closely packed, and are also very close to elements

of K\E. So E has a limit point in K.

Theorem 2.10. Let In be a sequence of intervals in R (In = [an, bn]) s.t. In+1 ⊂ In, n ∈ Z+.

Then, ∩∞1 In is not empty.

Theorem 2.11. Let k ∈ Z+. If Ik is a sequence of k-cells such that In+1 ⊂ In, for n ∈ Z+,

then ∩∞1 In is not empty.

Theorem 2.12. Every k-cell is compact. (Note to self: remember that k-cells in this book are

defined in Rk, equipped with the Euclidean Metric)

Theorem 2.13. Let E ⊂ Rk. Then the following are equivalent:

(a) E is closed and bounded.

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(b) E is compact.

(c) Every infinite subset of E has a limit point in E.

Remark. This concisely details a lot of lemmas presented in the Real Analysis module. The

equivalence of (a) and (b) above are called ‘Heine-Borel theorem’. (b) and (c) are equivalent

in any metric space, but (a) does not in general imply (b) or (c) in a metric space.

Theorem 2.14. Every bounded infinite subset of Rk has a limit point in Rk.

Theorem 2.15. Let P be a nonempty perfect set in Rk. Then P is uncountable. Notably,

every interval [a, b] is uncountable, and thus R. (Proof is hard to motivate.)

Theorem 2.16. Let P be the Cantor set. Then, P is compact, nonempty, and perfect. Fur-

thermore, no segment(

3k+13m

, 3k+23m

)where k,m ∈ Z+ contains a point of P .

Theorem 2.17. Let Vi be a family of compact subsets of Rk, for any indexing set I. Then,

∩Vi is compact.

Proof. Since each Vi is compact, due to Heine-Borel, each Vi is closed and bounded. Since any

one of Vi is bounded, ∩Vi is bounded. Since each Vi is closed, ∩Vi is closed. Therefore, ∩Vi is

closed and bounded, hence compact.

Definition 2.18. (D1) Let A,B ⊂ X, X a metric space. A and B are separated if A ∩B = ∅

and A ∩B = ∅. E ⊂ X is connected if E is not the union of two nonempty separated sets.

Remark. In lectures, the definition given was: (D2) ‘(Wikipedia) A connected space is a topo-

logical space that cannot be represented as the union of two or more disjoint non-empty open

subsets’. (D1) and (D2) are not equivalent; (D1) contains more information than (D2). Also,

(D1) =⇒ (D2) but (D2) does not imply (D1), even when limiting ourselves to metric spaces.

See Question sheet for more details.

Theorem 2.19. E ⊂ R is connected ⇐⇒ (If x, y ∈ E, x < z < y for some z ∈ R, then

z ∈ E.)

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3 Chapter 3 (Numerical Sequences and Series)

Remark. The results about convergence are being stated in (slightly more) general terms of

metric spaces. The Algebra of Limits theorem has been proven for complex sequences. Finally!

Theorem 3.1 (Algebra of Limits, for vectors). Suppose xn, yn are sequences in Rk, βn is

a sequence in R. xn −→ x, yn −→ y, βn −→ β. Then, limn→∞(xn+yn) = x+y, limn→∞ xn·yn = x·y,

limn→∞ βnxn = βx.

Remark. Use that xn −→ x iff each coordinate sequence in xn converges to each coordinate in x,

and use Algebra of Limits for sequences. Done.

Theorem 3.2. pn in a metric space X converges to p ⇐⇒ Every subsequence of pn

converges to p.

Theorem 3.3. The following hold:

(i) If pn is a sequence in a compact metric space X, then some subsequence of pn converges

to a point of X. (Note: can’t use H-B)

(ii) Every bounded sequence in Rk contains a convergent subsequence.

Theorem 3.4. The subsequential limits of a sequence pn in a metric space X form a closed

subset of X.

Theorem 3.5. E ⊂ X, (X, d). E closed ⇐⇒ all limit points of E are in E. Furthermore,

any one point is closed in a metric space. Proof easy.

Definition 3.6. A sequence pn in a metric space X is a Cauchy sequence if ∀ε, ∃n0 ∈ Z>0

s.t. d(pn, pm) < ε for all m,n ≥ n0.

Theorem 3.7. pn is a sequence in metric space X. EN := pN , pN+1, . . . . Then, pn is a

Cauchy sequence ⇐⇒ limN→∞ diam(EN) = 0.

Definition 3.8. Note that if E ⊂ X, (X, d), E0 := The set of all limit points of E, then the

closure of E, E := E ∪ E0.

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Theorem 3.9. The following hold:

(i) E ⊂ X, X a metric space. Then, diam(E) = diam(E).

(ii) Kn is a sequence of compact sets in X s.t. Kn+1 ⊂ Kn, for n ∈ Z>0, limn→∞ diam(Kn) = 0.

Then, ∩∞1 Kn consists of exactly one point.

Theorem 3.10. The following hold:

(i) In any metric space X, every convergent sequence is a Cauchy sequence.

(ii) X is a compact metric space and pn is a Cauchy sequence in X. Then pn converges

to some point of X.

(iii) In Rk, every Cauchy sequence converges.

Remark. A metric space where every Cauchy sequence converges called complete. Thus, all

compact metric spaces and all Euclidean spaces are complete. Also, every closed subset of a

complete metric space is complete.

Definition 3.11. sn is a sequence in R. Let E be the set of numbers x ∈ R∗ (extended real

number system), such that snk → x for some subsequence snk. (Thus, E possibly contains

the numbers ∞,−∞.) Define s∗ := sup(E), s∗ = inf(E). The numbers s∗, s∗ are called the

upper and lower limits of sn, and we use notation lim supn→∞(sn) = s∗, lim infn→∞(sn) = s∗.

Remark. Note that E, defined as above, can not be empty. For if sn is bounded, it will have

a limit point, and if sn is unbounded, it will have a subsequence that diverges to ∞ or −∞.

Theorem 3.12. sn, E are defined as above. Then:

(i) s∗ ∈ E

(ii) If x > s∗, there is an integer N s.t. n ≥ N =⇒ sn < x.

Moreover, s∗ is the only number with the properties (i) and (ii). An analogous result holds for

s∗

Remark. For a real-valued sn, limn→∞ sn = s ⇐⇒ lim supn→∞(sn) = lim infn→∞(sn) = s.

Theorem 3.13. (Proof easy when using immediately aforementioned theorem) If sn ≤ tn for

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n ≥ N , for fixed N , then:

lim infn→∞

(sn) ≤ lim infn→∞

(tn)

lim supn→∞

(sn) ≤ lim supn→∞

(tn)

Remark. Recall the lemma in M1P1; let sn, xn be real sequences. If 0 ≤ xn ≤ sn for n ≥ N ,

where N is fixed, and if sn → 0, then xn → 0.

Theorem 3.14. (Recall from Chapter 1) For every real x > 0 and integer n > 0, there is one

and only one positive real y such that yn = x. Let n√x := x1/n := y. (Note: y may not be

unique in R. It is only unique in R>0.)

Proposition 3.15. (Question 6, Chapter 1, page 22) Let b ∈ R>1.

(a) Let m, p ∈ Z and n, q ∈ Z>0, r = mn

= pq. Then, (bm)

1n = (bp)

1q , and therefore it makes

sense to define br = (bm)1n . (Question: Then what about (b

√2)

1√2 ? Answer: Wait for it!)

(b) Prove that br+s = brbs if r, s ∈ Q.

(c) If x ∈ R, define B(x) := bt : t ∈ Q, t ≤ x. Prove br = supB(r) when r ∈ Q. Thus, it

makes sense to now define bx := supB(x), ∀x ∈ R.

(d) Prove that for x, y ∈ R, bx+y = bxby.

(e) Here are some questions which are not included in the text, but I thought would be inter-

esting to think about: For c, d ∈ R, prove that (bc)d = bcd and ( 1bc

)d = 1bcd

. Prove that when

p ∈ R>0, c > d ≥ 0 =⇒ cp > dp. For c1 > d1, prove bc1 > bd1 . Prove for x ∈ R that

a1 > b1 > 0 =⇒ ax1 > bx1 . axbx = (ab)x. What about the case b < 0? What about case

b ∈ [0, 1]?

(Remark: I have written my proof to this proposition in the ‘Question sheet’, because it is too

long.)

Proposition 3.16. (Question 7, Chapter 1, page 22) Fix b ∈ R>1, y ∈ R>0. Then, there is a

unique x ∈ R s.t. bx = y. (This x is called a logarithm of y to the base b) Follow these steps:

(a) For n ∈ Z>0, bn − 1 ≥ n(b− 1).

(b) Hence b− 1 ≥ n(b1n − 1).

(c) If t > 1 and n > (b− 1)/(t− 1), then b1n < t

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(d) If w ∈ R is s.t. bw < y, then bw+ 1n < y for sufficiently large n.

(e) If bw > y, then bw−1n > y for sufficiently large n.

(f) Let A be the set of all w s.t. bw < y. Then, x = supA satisfies bx = y.

(g) Prove that this x is unique.

My question: Then what about b < 1? (Remark: I have written my proof to this proposition in

the ‘Question sheet’, because it is too long.) In particular, I think the method I used to prove

(g) is interesting, because there is a sort of asymmetry in the proof. I actually checked to see

if I had used all the given conditions in the question after writing my solution because it felt

intuitively strange.

Theorem 3.17. The following hold:

(a) If p > 0, then limn→∞1np

= 0

(b) If p > 0, then limn→∞n√p = 1

(c) limn→∞n√n = 1

(d) If p > 0, α ∈ R, then limn→∞nα

(1+p)n= 0

(e) If |x| < 1, then limn→∞ xn = 0

Remark. The proofs to the above theorem is standard, but still interesting. But I wonder how

to come up with some of them, as I can’t see the motivation behind them.

Remark. For the remainder of this chapter, all sequences and series under consideration are

complex, unless explicitly stated otherwise. Some theorems can be extended to series with

terms in Rk, which will be remarked on (by me). In the case of Rk, replace all | · | by || · ||.

Remark. For a sequence an in whatever space,∑∞

n=1 an is called a series. We define the

numbers sn :=∑n

k=1 ak to be the partial sums of the series. Sometimes, for convenience of

notation, we consider a series to be∑∞

n=0 an or∑∞

n=1 an, and write it as∑an.

Theorem 3.18. Let an be a sequence in Rk. The Cauchy criterion can be restated as:∑an

converges ⇐⇒ ∀ε > 0, ∃N ∈ Z>0 s.t. m ≥ n ≥ N =⇒∥∥∑m

k=n ak∥∥ ≤ ε.

Theorem 3.19. Let an be a sequence in Rk. If∑an converges, then limn→∞ an = 0.

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Theorem 3.20. A series of non-negative real terms converges if and only if its partial sums

form a bounded sequence.

Theorem 3.21 (General Comparison Test). The following hold:

(a) Let an be a sequence in Rk and cn a sequence in R. If ‖an‖ ≤ cn, for n ≥ N0, where

N0 ∈ Z fixed, and if∑cn converges, then

∑an converges absolutely.

(b) Let an be a sequence of non-negative real terms. If an ≥ dn ≥ 0 for n ≥ N0, where N0 ∈ Z

fixed, and if∑dn diverges, then

∑an diverges.

Theorem 3.22. Suppose a1 ≥ a2 ≥ a3 ≥ · · · ≥ 0. Then the series∑∞

n=1 an converges if and

only if the series∑∞

k=0 2ka2k = a1 + 2a2 + 4a4 + 8a8 + . . . converges.

Remark. A feature of this theorem is that a ‘thin’ subsequence of an determines the con-

vergence or divergence of∑an. Also, whenever inequalities (<,≤ etc) are being used we are

talking about elements of R, (I presume) because of ‘It is also a fact (no proof given) that any

two ordered fields with the least upper bound property are isomorphic’.

Theorem 3.23.∑

1np

converges if p > 1 and diverges if p ≤ 1.

Remark. You can use the above theorem to prove this for case p > 0!

Theorem 3.24. If p > 1,∑∞

n=21

n(logn)pconverges. If p ≤ 1, the series diverges. (You can

prove this using the 2 immediately aforementioned theorems)

Definition 3.25. e :=∑∞

n=01n!

. This series converges, so the definition makes sense.

Theorem 3.26. limn→∞(1 + 1

n

)n= e, and e is irrational.

Remark. At first glance, it is somewhat surprising to use inequalities to prove that some number

is irrational!

Theorem 3.27. (Root Test) Let an be a sequence in Rk. Let α = lim supn→∞n√‖an‖. Then,

for a given∑an:

(a) if α < 1,∑an converges absolutely

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(b) if α > 1,∑an diverges

(c) if α = 1, the test gives no information

Remark. One might wonder if the statement of the Root Test presented here is more powerful

(I mean ‘weaker’) than the one given during M1P1, where α := limn→∞n√‖an‖. This is the

case. Suppose an = nn if 3|n, and an = 0 otherwise. Then n√‖an‖ does not converge, hence

limn→∞n√‖an‖ does not exist. However, lim supn→∞

n√‖an‖ exists, and is∞. With this version

of the Root Test, we can conclude∑an diverges, but we cannot with the M1P1 version.

Theorem 3.28. (Half-Recalled theorem) sn is a sequence in R. Let E be the set of numbers

x ∈ R∗ (extended real number system), such that snk → x for some subsequence snk. (Thus,

E possibly contains the numbers ∞,−∞) Then:

(i) s∗ ∈ E

(ii) If x > s∗, there is an integer N s.t. n ≥ N =⇒ sn < x.

Theorem 3.29. (Ratio Test) Let an be a sequence in Rk. Then, the series∑an:

(a) converges absolutely if lim supn→∞‖an+1‖‖an‖ < 1

(b) diverges if ‖an+1‖‖an‖ ≥ 1 for all n ≥ n0, where n0 ∈ Z is fixed.

Remark. lim supn→∞‖an+1‖‖an‖ ≥ 1 is not much help in proving that condition (b) is satisfied/not

satisfied. (Condition (b) is not equivalent to lim supn→∞‖an+1‖‖an‖ ≥ 1) So once you calculate

lim supn→∞‖an+1‖‖an‖ < 1, you can use the ratio test, but when lim supn→∞

‖an+1‖‖an‖ ≥ 1, you need to

do additional working.

Remark. One might wonder, as with the Root Test, if the statement of the Ratio Test presented

here is weaker than the one given during M1P1, where α := limn→∞‖an+1‖‖an‖ . This is the case. For

example, let an be defined recursively by a1 = 1, an+1

an= 1

2for n even, and an+1

an= 1

3for n odd.

Then lim supn→∞an+1

an= 1

2, lim infn→∞

an+1

an= 1

3. Since lim supn→∞

an+1

an6= lim infn→∞

an+1

an,

limn→∞an+1

andoes not exist, and we cannot use the M1P1 version. However, lim supn→∞

an+1

an<

1, therefore we can see with the weaker version of the Ratio Test that∑an will converge.

Remark. Whenever the ratio test shows convergence, the root test does too. Whenever the root

test is inconclusive, the ratio test is too. Therefore, the root test has wider scope of application.

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This is illustrated by the below theorem.

Theorem 3.30. For any sequence cn of R>0 (so cn might be ‖an‖, an ∈ Rk):

lim infn→∞

cn+1

cn≤ lim inf

n→∞n√cn

lim supn→∞

n√cn ≤ lim sup

n→∞

cn+1

cn

Definition 3.31. Given a sequence cn of complex numbers, the series∑∞

n=0 cnzn is called a

power series.

Theorem 3.32. Given the power series∑∞

n=0 cnzn, (cn a sequence of complex numbers) let

α = lim supn→∞n√‖cn‖, R = 1

α. (If α = 0, R = ∞ and if α = ∞, R = 0) Then

∑∞n=0 cnz

n

converges if ‖z‖ < R, and diverges if ‖z‖ > R. R is called the radius of convergence of∑∞n=0 cnz

n.

Remark. You can do this sort of test with the ratio test as well, not just with the root test.

Write out statement, then prove. Also, note that case R = 0 is not a counterexample to the

theorem. There exist no z ∈ C s.t. ‖z‖ < 0, and the ‘converges if ‖z‖ < R’ condition cannot

fail.

Remark. The statement would be something like: given the power series∑∞

n=0 cnzn, cn a

sequence of complex numbers, let α = limn→∞‖cn+1‖‖cn‖ , R = 1

α. (If α = 0, R = ∞ and if

α =∞, R = 0) Then∑∞

n=0 cnzn converges if ‖z‖ < R, and diverges if ‖z‖ > R. R is called the

radius of convergence of∑∞

n=0 cnzn. Proof is just the M1P1 version of the ratio test applied to∑∞

n=0 cnzn. I have been unable to prove this statement for the case when α = limn→∞

‖cn+1‖‖cn‖ is

replaced by α = lim supn→∞‖cn+1‖‖cn‖ .

Theorem 3.33. Given two sequences an, bn in C, let An =∑n

k=0 ak for n ≥ 0. For

n = −1, let A−1 = 0. Then, if 0 ≤ p ≤ q,∑q

n=p anbn =(∑q−1

n=pAn(bn − bn+1))

+Aqbq−Ap−1bp.

Theorem 3.34. Let an be a sequence in Rk, bn a sequence in R. Suppose:

(a) the partial sums An of∑an form a bounded sequence

(b) b0 ≥ b1 ≥ b2 ≥ . . .

(c) limn→∞ bn = 0.

15

Then,∑anbn converges.

Theorem 3.35 (Alternating series test). For a sequence cn in C, suppose:

(a) ‖c1‖ ≥‖c2‖ ≥‖c3‖ ≥ . . .

(b) c2m−1 ≥ 0, c2m ≤ 0, for m ∈ Z≥1

(c) limn→∞ cn = 0

Then,∑cn converges.

Theorem 3.36. For a sequence cn in R, suppose the radius of convergence of∑cnz

n is 1,

c0 ≥ c1 ≥ c2 ≥ . . . , and limn→∞ cn = 0. Then∑cnz

n converges at every point on the circle

‖z‖ = 1, except possibly at z = 1.

Remark. Theorems 3.35, 3.36 can be proved using theorem 3.34.

Theorem 3.37. Let an be a sequence in Rk. If∑an converges absolutely, then

∑an con-

verges.

Remark. Prove that power series converge absolutely in the interior of the circle of convergence.

(Just use root test) Furthermore, it is easy to add and multiply absolutely convergent series,

but for non-absolutely, but still convergent series, things become more complicated.

Theorem 3.38. Let an, bn be sequences in Rk. If∑an = A,

∑bn = B, then

∑an+bn =

A+B, and∑can = cA, for any fixed c ∈ R.

Remark. The series may be non-absolutely convergent. However, notice that we are not rear-

ranging any terms of the original series as we add. (We add using partial sums, as that is how

the value of series is defined)

Definition 3.39. Let an, bn be sequences in C, for n ∈ Z≥0. We define cn :=∑n

k=0 akbn−k,

and call∑cn the product of the two given series.

Remark. The product of two convergent series may diverge.

Theorem 3.40. Let an, bn be sequences in C, for n ∈ Z≥0. Suppose:

16

(a)∑∞

n=0 an converges absolutely

(b)∑∞

n=0 an = A

(c)∑∞

n=0 bn = B

(d) cn =∑n

k=0 akbn−k

Remark. Therefore, the product of two convergent series converges, to the value that you would

expect it to, if at least one of the two series converges absolutely. Another question to ask is:

If such∑cn as above converge, where

∑∞n=0 an need not converge absolutely, then would it

converge to∑cn = AB? The answer is given by the following theorem.

Theorem 3.41. For n ∈ Z≥0, let an, bn, cn be sequences in C that converge to A, B,

C. Then if cn =∑n

k=0 akbn−k, then C = AB.

Theorem 3.42. Let k(n) : Z>0 → Z>0 be a bijection. For a sequence an, define bn = ak(n).

We say∑bn is a rearrangement of

∑an.

Theorem 3.43. Let an be a sequence in R. The series∑an converges, but not absolutely.

Suppose −∞ ≤ α ≤ β ≤ ∞. Then there exists a rearrangement∑bn with partial sums sn s.t.

lim infn→∞ sn = α, lim supn→∞ sn = β.

Theorem 3.44. If∑an is a series of elements of Rk which converges absolutely, then every

rearrangement of∑an converge, and to the same value.

17

4 Chapter 4 (Continuity)

Definition 4.1. (X, dx), (Y, dy), E ⊂ X, f : E → Y , p is a limit point of E but need not

be in E, and q ∈ Y . Then, we write limx→p f(x) = q if ∀ε > 0,∃δ > 0 s.t. if x ∈ E and

0 < dX(x, p) < δ, then dy(f(x), q) < ε.

Theorem 4.2. Letting the notation be the same as definition 4.1, limx→p f(x) = q ⇐⇒

limn→∞ f(pn) = q for every sequence pn in E s.t. pn 6= p, limn→∞ pn = p.

Theorem 4.3. Letting the notation be as in definition 4.1, if f has a limit at p, the limit is

unique.

Theorem 4.4. Suppose E ⊂ X, X a metric space. p ∈ X is a limit point of E, and f, g :

E → C s.t. limx→p f(x) = A, limx→p g(x) = B. Then,

(a) limx→p(f + g)(x) = A+B

(b) limx→p(fg)(x) = AB

(c) limx→p(fg)(x) = A

B, if B 6= 0.

Theorem 4.5. Suppose E ⊂ X, X a metric space. p ∈ X is a limit point of E, and f, g :

E → Rk s.t. limx→p f(x) = A, limx→p g(x) = B. Then,

(a) limx→p(f + g)(x) = A+B

(b) limx→p(f · g)(x) = A ·B

Remark. If you think of the limit in terms of sequences, the above two theorems are immediate.

Definition 4.6. X and Y are metric spaces, E ⊂ X, p ∈ E, f : E → Y . f is continuous at p

if ∀ε > 0, ∃δ > 0 s.t. (∀x ∈ E s.t. dX(x, p) < δ), dY (f(x), f(p)) < ε. By expanding definitions,

note that f is continuous at p ⇐⇒ limx→p f(x) = f(p). (I sometimes interpret this as: ‘f is

continuous ⇐⇒ you can shove the limit inside f , i.e. limx→p f(x) = f(limx→p x) = f(p)’)

Remark. Note that f has to be defined at the point p in order to be continuous at p. If p is an

isolated point of E, then every function f which has domain E is continuous at p.

Theorem 4.7. X, Y , Z are metric spaces, E ⊂ X, f : E → Y , g : f(E)→ Z, and h : E → Z

18

s.t. h(x) = g(f(x)), for x ∈ E. If f is continuous at p ∈ E and g is continuous at f(p), then

h is continuous at p.

Theorem 4.8. X, Y are metric spaces. f : X → Y is continuous on X if and only if f−1(V )

is open in X for every open set V in Y .

Theorem 4.9. X, Y are metric spaces. f : X → Y is continuous on X if and only if f−1(V )

is closed in X for every closed set V in Y .

Theorem 4.10. f, g : X → C, X a metric space. Then, f + g,fg, and fg

are continuous on

X. (For fg, we must assume that g(x) 6= 0, ∀x ∈ X.)

Remark. Proof follows very easily from things we’ve done before. An application of theorem

4.10 would be: because the coordinate function is continuous, from repeated application of the

theorem above, we deduce that every polynomial P (x) =∑cn1...nkx

n11 . . . xnkk , where x ∈ Rk and

ni ∈ Z≥0, is continuous. Furthermore, note that the norm function defined on Rk is continuous.

Remark. f : X → Y , X, Y a metric space. Note again that when p is an isolated point of

X, limx→p f(x) is not defined. Therefore when proving theorem 4.10 using theorem 4.4, it is

necessary to split cases into when p ∈ X is a limit point of X and when it is an isolated point

of X.

Theorem 4.11. The following hold:

(a) f1, . . . , fk : X → R, X a metric space. f : X → Rk is defined by f(x) = (f1(x), . . . , fk(x)),

∀x ∈ X. Then f is continuous ⇐⇒ each f1, . . . , fk is continuous.

(b) If f, g : X → Rk are continuous, then f + g, f · g are continuous on X.

Remark. Continuity of a function is defined on a subset E (which is the domain of the function)

of a metric space X. But elements of X\E plays absolutely no role in determining whether

a function is continuous or not. (This isn’t really the case when considering limits of functions,

as the point in consideration need only be a limit point of E). Therefore, when considering

continuity, from now on we discard X\E. (We lose nothing of interest by doing so) Therefore,

we from now on talk only about continuous mappings of one metric space into another, rather

than of mappings of subsets.

19

Definition 4.12. f : E → Rk, E a metric space. f is bounded if ∃M ∈ R s.t.∥∥f(x)

∥∥ ≤ M ,

∀x ∈ E.

Theorem 4.13. f : X → Y is continuous, where X is a compact metric space, and Y a metric

space. Then f(X) is compact.

Remark. f : X → Y is continuous, where X, Y are metric spaces. For E ⊂ Y , f(f−1(E)) ⊂ E

and for E ⊂ X, E ⊂ f−1(f(E))

Remark. f : X → Rk is continuous, where X is a compact metric space. Then f(X) is compact,

thus closed and bounded, and thus bounded.

Theorem 4.14. f : X → R continuous, X a compact metric space. Let M = supp∈X f(p),

m = infp∈X f(p). Then, ∃p, q ∈ X s.t. f(p) = M , f(q) = m.

Theorem 4.15. f : X → Y is a continuous bijection, where X is a compact metric space, and

Y is a metric space. Then f−1 is continuous.

Definition 4.16. X, Y metric space, f : X → Y . f is uniformly continuous on X if ∀ε > 0,

∃δ > 0 s.t. dY (f(p), f(q)) < ε for all p, q ∈ X for which dX(p, q) < δ.

Theorem 4.17. f : X → Y is continuous, where X is a compact metric space, and Y is a

metric space. Then, f is uniformly continuous on X.

Theorem 4.18. Let E be a noncompact set in R. Then,

(a) there exists a continuous function on E which is not bounded

(b) there exists a continuous and bounded function on E which has no maximum

If E is also bounded, then in addition to (a) and (b) holding,

(c) there exists a continuous function on E which is not uniformly continuous

Remark. Therefore, we conclude that the assumption of compactness is essential in the theorems

above. We prove this theorem by providing examples. Different examples are given for (a), (b)

for the cases E bounded, and E unbounded.

20

Theorem 4.19. f : X → Y continuous, where X, Y are metric spaces. If E ⊂ X is connected,

then f(E) is connected.

Remark. We proved that the definition of connectedness given in this text contains more infor-

mation than and in some way ‘implies’ the one given in M2PM5. (So if a set is connected in

the definition of this text, it is also connected in the general definition. But it does not work

the other way around.) Note that the proofs given in the text are valid in the general sense, as

well.

Theorem 4.20. f : [a, b]→ R is continuous. If f(a) < c < f(b) (c ∈ R), then ∃x ∈ (a, b) s.t.

f(x) = c.

Definition 4.21. Let f be defined on (a, b), a segment of R. Consider any x s.t. a ≤ x < b.

We write f(x+) = q if, for all sequences tn in (x, b) s.t. tn → x, f(tn)→ q as n→∞.

Definition 4.22. Let f be defined on (a, b), a segment of R. Consider any x s.t. a < x ≤ b.

We write f(x−) = q if, for all sequences tn in (a, x) s.t. tn → x, f(tn)→ q as n→∞.

Remark. For any x ∈ (a, b), limt→x f(t) exists ⇐⇒ f(x+) = f(x−) = limt→x f(t).

Definition 4.23. Let f be defined on (a, b), a segment of R. If f is discontinuous at a point

x, and if f(x+) and f(x−) exist, then f is said to have a discontinuity of the first kind (or a

simple discontinuity) at x. Otherwise, the discontinuity is said to be of the second kind.

Definition 4.24. Let f : (a, b)→ R. f is monotonically increasing on (a, b) if a < x < y <

b =⇒ f(x) ≤ f(y). The definition of a monotonically decreasing function is obtained by

reversing the last inequality. By ‘monotonic functions’, we mean both monotonically increasing

and decreasing functions.

Theorem 4.25. Let f : (a, b) → R be monotonically increasing on (a, b). Then, f(x+) and

f(x−) exist at every point x ∈ (a, b). More precisely,

supa<t<x

f(t) = f(x−) ≤ f(x) ≤ f(x+) = infx<t<b

f(t)

Furthermore, if a < x < y < b, then f(x+) ≤ f(y−). Analogous results hold for monotonically

decreasing functions.

21

Remark. Monotonic functions have no discontinuities of the second kind.

Remark. From now on, when I say ‘f is monotonic on (a, b)’, I mean that f : (a, b)→ R and f

is monotonic.

Theorem 4.26. f is monotonic on (a, b). Then the set of points of (a, b) where f is discon-

tinuous is at most countable.

Remark. Notably, the discontinuities of a monotonic function need not be isolated. In fact,

for any countable subset E of (a, b), which may even be dense, we can construct a function f ,

monotonic on (a, b), discontinuous at every point of E, and at no other point of (a, b). A nice

construction is given; in summary, let all points of E be arranged in any sequence xn, for

n ∈ Z>0. Let cn be a sequence of positive numbers for which the series∑cn converges. For

any x ∈ (a, b), define the function f as follows: Given x ∈ (a, b), sum cn over indices n for

which xn < x. Since∑cn converges absolutely, note that the order in which we sum is not

important. This function f is monotonically increasing, discontinuous at every point of E, and

continuous at every other point of (a, b). In fact, f(xn+)− f(xn−) = cn.

Remark. To operate in the extended real number system, we generalize our definition of limits

(Definition 4.1) by reformulating it in terms of neighborhoods. For any x, δ ∈ R, define a

neighbourhood of x to be any segment (x− δ, x+ δ).

Definition 4.27. For any c ∈ R, the set of x ∈ R s.t. x > c is called a neighborhood of +∞,

and is written (c,+∞). Similarly, the set (−∞, c) is a neighborhood of −∞. (Personally,

I call it an open neighbourhood, but all my courses seem to mean an open neighbourhood

whenever they say neighbourhood. So I will adapt to it.)

Definition 4.28. Let f : E → R, E ⊂ R. Let A, x ∈ R∗, the extended number system. We

say f(t) → A as t → x if for every neighbourhood U of A, there exists a neighbourhood V in

R of x s.t. V ∩ E is not empty and f(t) ∈ U for ∀t ∈ V ∩ E, t 6= x.

Remark. This is consistent with Definition 4.1 when A, x ∈ R. Think of the definition of a limit

given in topological spaces.

Theorem 4.29. f, g : E → R, E ⊂ R. Suppose f(t) → A and g(t) → B as t → x, where

22

A,B, x ∈ R∗. Then, (We must assume that the RHS of (b), (c), (d) are defined, though)

(a) f(t) = A′ =⇒ A = A′

(b) (f + g)(t)→ A+B

(c) (fg)(t)→ AB

(d) (fg)(t)→ A

B,

Remark. Note that ∞−∞, 0 · ∞, ∞∞ , A0

are not defined, under the convention stated in this

textbook.

23

5 Chapter 5 (Differentiation)

Important Remark: In this chapter, we only examine real functions defined on intervals or

segments, except in the final section. This is because genuine differences appear when we go

from real functions to vector-valued functions. The latter is examined in chapter 9.

Definition 5.1. f : [a, b]→ Rk. For x ∈ [a, b], define (where a < t < b, t 6= x)

φ(t) =f(t)− f(x)

t− x

We define f ′(x) = limt→x φ(t), provided the limit exists. f ′ is called the derivative of f .

Remark. For f : [a, b]→ C, f ′ is defined by one sided limits at the endpoints, if the limit exists.

For f : (a, b) → C, f ′(a) and f ′(b) are not defined, while f ′ for x ∈ (a, b) is defined the same

way as in definition 5.1.

Remark. For f : [a, b]→ Rk, let f1, . . . , fk be the component functions of f = (f1, . . . , fk). Then,

f is differentiable at a point x if and only if each of the functions f1, . . . , fk are differentiable at

x. If f ′ exists, f ′ = (f ′1, . . . , f′k). In particular, when f is a complex function, f is differentiable

⇐⇒ the two functions defined by the real and imaginary parts of f are both differentiable.

The proofs to these are easy.

Theorem 5.2. f : [a, b] → Rk or f : [a, b] → C. If f is differentiable at x ∈ [a, b], then f is

continuous at x.

Remark. Personally, I am hesitant to exclude writing C in this proposition because × is not

preserved into · via the standard isomorphism between R2 and C. But this remark is highly

irrelevant, and in the proof that I gave R2 and C are completely interchangeable.

Theorem 5.3. Suppose f, g : [a, b]→ C are differentiable at point x ∈ [a, b]. Then, f + g, fg,

fg

are differentiable at x, and:

(a) (f + g)′(x) = f ′(x) + g′(x)

(b) (fg)′(x) = f ′(x)g(x) + f(x)g′(x)

(c)(fg

)′(x) = g(x)f ′(x)−g′(x)f(x)

g2(x)(assuming in (c) that g(x) 6= 0)

Remark. When f, g : [a, b]→ Rk, (a) holds. If we modify fg to f · g in (b), it also holds.

24

Theorem 5.4. f : [a, b] → R is continuous, f ′(x) exists at some point x ∈ [a, b], g : I → R

where f([a, b]) ⊂ I, and g is differentiable at point f(x). If, for a ≤ t ≤ b, h(t) = g(f(t)), then

h is differentiable at x, and h′(x) = g′(f(x))f ′(x).

Definition 5.5. Let X be a metric space. f : X → R. f has a local maximum at p ∈ X if

∃δ > 0 s.t. q ∈ X, d(p, q) < δ =⇒ f(q) ≤ f(p). A local minimum is defined similarly.

Theorem 5.6. f : [a, b]→ R. If f has a local maximum at x ∈ (a, b), and if f ′(x) exists, then

f ′(x) = 0. The analogous statement for local minima is also true.

Theorem 5.7 (Generalized Mean Value Theorem). f, g : [a, b] → R are continuous, and are

differentiable in (a, b). Then, ∃x ∈ (a, b) s.t. (f(b)− f(a))g′(x) = (g(b)− g(a))f ′(x). Note that

differentiability at the end points (defined as one-sided limits in this book) is not required.

Theorem 5.8 (Mean Value Theorem). f : [a, b]→ R is continuous, and also differentiable in

(a, b). Then, ∃x ∈ (a, b) s.t. f(b)− f(a) = (b− a)f ′(x).

Remark. To prove Theorem 5.8, let g(x) = x in Theorem 5.7. To see that the Mean Value

Theorem fails for complex functions, an example would be f : R→ C s.t. f(x) = exp (ix).

Theorem 5.9. f : (a, b)→ R is differentiable.

(a) ∀x ∈ (a, b), f ′(x) ≥ 0 =⇒ f is monotonically increasing

(b) ∀x ∈ (a, b), f ′(x) = 0 =⇒ f is constant

(c) ∀x ∈ (a, b), f ′(x) ≤ 0 =⇒ f is monotonically decreasing

Theorem 5.10. f : [a, b] → R is differentiable, and f ′(a) < λ < f ′(b). Then, ∃x ∈ (a, b) s.t.

f ′(x) = λ. A similar result holds when f ′(a) > f ′(b).

Theorem 5.11. f : [a, b]→ R is differentiable, then f ′ cannot have any simple discontinuities

on [a, b].

Theorem 5.12 (L’Hospital’s Rule). f, g : (a, b) → R are differentiable, g′(x) 6= 0 for ∀x ∈

(a, b), for −∞ ≤ a < b ≤ +∞. Suppose f ′(x)g′(x)→ A as x → a. If f(x), g(x) → 0 as x → a, or

if g(x)→ +∞ as x→ a, then f(x)g(x)→ A as x→ a. An analogous statement is true if x→ b or

25

if g(x)→ −∞. Note that we use the extended limit concept, defined in definition 4.28.

Remark. This rule fails for complex functions. For example, f, g : (0, 1)→ C s.t. f(x) = x and

g(x) = x+ x2 exp ( ix2

)

Theorem 5.13. f : [a, b] → R, n ∈ Z>0, f (n−1) is continuous on [a, b], f (n)(t) exists for

∀t ∈ (a, b). Let α, β be distinct points of [a, b], and define P (t) =∑n−1

k=0f (k)(α)k!

(t − α)k. Then,

∃x ∈ (α, β) (or (β, α)) s.t. f(β) = P (β) + f (n)(x)n!

(β − α)n.

Remark. Let E1 ⊂ E ⊂ E0 ⊂ Y , Y a metric space (so its subsets are all metric spaces). In order

for f (n) : E1→ Y to exist at a point x, f (n−1) : E → Y must exist in a neighbourhood of x in E

(or a one-sided neighbourhood, if x is an end-point of interval [a, b] ⊂ R) and be differentiable.

This is because by definition ∀ε > 0, ∃δ > 0 s.t. x, t ∈ E and 0 < ‖t− x‖ < δ =⇒∥∥∥f (n−1)(t)−f (n−1)(x)t−x

∥∥∥ < ε. f (n−1) must be defined in the δ−neighbourhood in E of x for the

definition to hold. Similarly, f (n−2) : E0→ Y must be differentiable in that δ−neighbourhood

(in E0, as E ⊂ E0).

Theorem 5.14. f : [a, b]→ Rk is continuous, and f is differentiable in (a, b). Then, ∃x ∈ (a, b)

s.t.∥∥f(b)− f(a)

∥∥ ≤ (b− a)∥∥f ′(x)

∥∥.

26

6 Chapter 6 (The Riemann-Stieltjes Integral)

Definition 6.1. By a partition P of [a, b], we define P = x0, . . . , xn for finite n s.t.

a = x0 ≤ x1 ≤ · · · ≤ xn−1 ≤ xn = b. Define ∆xi := xi − xi−1 for i = 1, . . . , n. For a

bounded function f : [a, b] → R, we define Mi := sup f(x), mi = inf f(x) for x ∈ [xi−1, xi].

Then, U(P, f) :=∑n

i=1Mi∆xi and L(P, f) :=∑n

i=1mi∆xi. Finally,∫ bafdx := inf U(P, f) and∫ b

afdx := supL(P, f), where the infimum and supremum are taken over all partitions.

Remark.∫ bafdx and

∫ bafdx are called the upper and lower Riemann integrals of f , respectively.

If∫ bafdx =

∫ bafdx, we write f ∈ <, where < denotes the set of Riemann-Integrable functions,

and say that f is Riemann-Integrable on [a, b].∫ baf(x)dx :=

∫ bafdx :=

∫ bafdx =

∫ bafdx. If f is

bounded, then L(P, f), U(P, f) are bounded, hence∫ bafdx and

∫ bafdx exist (can be defined).

Definition 6.2. α : [a, b] → R is monotonically increasing. Since α(a) and α(b) are finite, α

is bounded on [a, b]. For a given partition P , ∆αi := α(xi) − α(xi−1) ≥ 0. For an bounded

f : [a, b] → R, define Mi := sup f(x), mi = inf f(x) for x ∈ [xi−1, xi]. Then, U(P, f, α) :=∑ni=1 Mi∆αi and L(P, f, α) :=

∑ni=1mi∆αi. Finally,

∫ bafdα := inf U(P, f, α) and

∫ bafdα :=

supL(P, f, α), where the infimum and supremum are taken over all partitions.

Remark. If∫ bafdα =

∫ bafdα, we denote the common value by

∫ baf(x)dα(x) :=

∫ bafdα. If∫ b

afdα is defined, we write f ∈ <(α), and say that f is integrable with respect to α. Note that

α need not even be continuous.

Remark. From this point onward, in this chapter, f will be assumed to be real and

bounded, defined on [a, b], and α is monotonically increasing, defined on [a, b]. When

f is not real and bounded, etc., I will explicitly state: ‘This proposition deviates

from the conventions of this chapter’. If that statement is not present, then f is real and

bounded, etc. When there is no ambiguity,∫

is written in place of∫ ba. (Note: this convention

seems different than in other applied courses.)

Definition 6.3. For partitions P, P ∗, P1, P2, we say that P ∗ is a refinement of P if P ⊂ P ∗.

Given P1, P2, we define their common refinement P ∗ := P1 ∪ P2.

Remark. I won’t say ‘For partitions P ∗, P ’ every time. It should be inferable and fairly obvious

within context.

27

Theorem 6.4. If P ∗ is a refinement of P , then L(P, f, α) ≤ L(P ∗, f, α) ≤ U(P ∗, f, α) ≤

U(P, f, α).

Theorem 6.5.∫ bafdα ≥

∫ bafdα.

Theorem 6.6. f is real and bounded on [a, b]. f ∈ <(α) on [a, b] ⇐⇒ ∀ε > 0, ∃ partition P

s.t. U(P, f, α)− L(P, f, α) < ε.

Remark. It is convenient to note that if a, b, c, d, ε ∈ R, a ≤ b ≤ c ≤ d, d − a < ε, then

‖b− c‖ < ε.

Theorem 6.7. f is real and bounded on [a, b]. The following hold:

(a) If U(P, f, α)−L(P, f, α) < ε holds for some P and some ε, then U(P ∗, f, α)−L(P ∗, f, α) < ε

holds, with the same ε, for every refinement P ∗ of P .

(b) If U(P, f, α)− L(P, f, α) < ε holds for P = x0. . . . , xn and if si, ti are arbitrary points in

[xi−1, xi], then∑n

i=1

∥∥f(si)− f(ti)∥∥∆αi < ε.

(c) If f ∈ <(α), U(P, f, α) − L(P, f, α) < ε holds for P = x0. . . . , xn, and if ti ∈ [xi−1, xi],

then∥∥∥∑n

i=1 f(ti)∆αi −∫ bafdα

∥∥∥ < ε.

Theorem 6.8. If f : [a, b]→ R is continuous, then f ∈ <(α) on [a, b].

Theorem 6.9. f : [a, b]→ R is monotonic, and α : [a, b]→ R is continuous, then f ∈ <(α).

Theorem 6.10. f : [a, b]→ R is bounded on [a, b] and has finitely many points of discontinuity

on [a, b]. α is continuous at every point at which f is discontinuous. Then, f ∈ <(α).

Theorem 6.11. f : [a, b]→ R, f ∈ <(α) on [a, b], m ≤ f ≤M . φ : [m,M ]→ R is continuous,

and h(x) = φ(f(x)) on [a, b]. Then, h ∈ <(α) on [a, b].

Theorem 6.12 (Properties of the Integral). The following hold:

(a) If f1, f2 ∈ <(α) on [a, b], then f1 + f2 ∈ <(α), cf ∈ <(α) for every constant c ∈ R,∫ ba(f1 + f2)dα =

∫ baf1dα +

∫ baf2dα, and

∫ bacfdα = c

∫ bafdα.

(b) If f1(x) ≤ f2(x) on [a, b], then∫ baf1dα ≤

∫ baf2dα.

28

(c) If f ∈ <(α) on [a, b] and a < c < b, then f ∈ <(α) on [a, c] and on [c, b], and∫ cafdα +∫ b

cfdα =

∫ bafdα.

(d) If f ∈ <(α) on [a, b] and∥∥f(x)

∥∥ ≤M on [a, b], then∥∥∥∫ ba fdα∥∥∥ ≤M [α(b)− α(a)].

(e) If f ∈ <(α1), f ∈ <(α2), then f ∈ <(α1 + α2) and∫ bafd(α1 + α2) =

∫ bafdα1 +

∫ bafdα2. If

f ∈ <(α) and c ∈ R+ is a constant, then f ∈ <(cα) and∫ bafd(cα) = c

∫ bafdα.

Theorem 6.13. f, g ∈ <(α) on [a, b]. Then,

(a) fg ∈ <(α)

(b) ‖f‖ ∈ <(α) and∥∥∥∫ ba fdα∥∥∥ ≤ ∫ ba ‖f‖ dα.

Definition 6.14. The unit step function I is defined by I(x) = 0 (x ≤ 0) and I(x) = 1

(x > 0).

Theorem 6.15. a < s < b, f is bounded on [a, b], f is continuous at s, and α(x) = I(x− s),

then∫ bafdα = f(s).

Theorem 6.16. cn ≥ 0, n ∈ Z>0.∑cn converges, sn is a sequence of distinct points in

(a, b), and α(x) =∑∞

n=1 cnI(x − sn). Let f be real, bounded, and continuous on [a, b]. Then,∫ bafdα =

∑∞n=1 cnf(sn).

Theorem 6.17. Assume α increases monotonically and α′ ∈ <(α) on [a, b]. f is a bounded

real function on [a, b]. Then, f ∈ <(α) ⇐⇒ fα′ ∈ <. Furthermore in this case,∫ bafdα =∫ b

af(x)α′(x)dx.

Theorem 6.18 (Change of variable). φ is a strictly increasing continuous function that maps

[A,B] onto [a, b]. (So φ(A) = a, φ(B) = b.) α is monotonically increasing on [a, b] and

f ∈ <(α) on [a, b]. Define β and g on [A,B] by β(y) := α(φ(y)), g(y) := f(φ(y)). Then,

g ∈ <(β) and∫ BAgdβ =

∫ bafdα.

Remark. The intuitive picture for theorem 6.18 would be: Notice that f ∈ <(α) already exists

on [a, b]. As we integrate g along [A,B],∫ BAgdβ traces in the space of [a, b], or behaves the

same way as, integrating f along [a, b], i.e.∫ bafdα.

Theorem 6.19. f ∈ < on [a, b]. For a ≤ x ≤ b, let F (x) :=∫ xaf(t)dt. Then F is continuous

29

on [a, b]. And furthermore, if f is continuous at point x0 ∈ [a, b], then F is differentiable at x0

and F ′(x0) = f(x0).

Theorem 6.20 (Fundamental Theorem of Calculus). f ∈ < on [a, b] and if there is a differ-

entiable function F on [a, b] s.t. F ′ = f , then∫ baf(x)dx = F (b)− F (a).

Theorem 6.21 (Integration by Parts). F,G are differentiable functions on [a, b], F ′ = f ∈ <,

G′ = g ∈ <. Then∫ baF (x)g(x)dx = F (b)G(b)− F (a)G(a)−

∫ baf(x)G(x)dx.

Definition 6.22. f1, . . . , fk : [a, b] → R, and f := (f1, . . . , fk) : [a, b] → Rk. If α : [a, b] → R

is monotonically increasing, we say f ∈ <(α) if and only if fj ∈ <(α) for j = 1, . . . , k. If

f ∈ <(α), we define∫ bafdα := (

∫ baf1dα, . . . ,

∫ bafkdα).

Theorem 6.23 (Properties of the Integral). This proposition deviates from the conventions

of this chapter. f, f1, f2 : [a, b] → Rk, and α : [a, b] → R is monotonically increasing. The

following hold:

(a) If f1, f2 ∈ <(α) on [a, b], then f1 + f2 ∈ <(α), cf ∈ <(α) for every constant c ∈ R,∫ ba(f1 + f2)dα =

∫ baf1dα +

∫ baf2dα, and

∫ bacfdα = c

∫ bafdα.

(c) If f ∈ <(α) on [a, b] and a < c < b, then f ∈ <(α) on [a, c] and on [c, b], and∫ cafdα +∫ b

cfdα =

∫ bafdα.

(e) If f ∈ <(α1), f ∈ <(α2), then f ∈ <(α1 + α2) and∫ bafd(α1 + α2) =

∫ bafdα1 +

∫ bafdα2. If

f ∈ <(α) and c ∈ R+ is a constant, then f ∈ <(cα) and∫ bafd(cα) = c

∫ bafdα.

Theorem 6.24. This proposition deviates from the conventions of this chapter. α increases

monotonically and α′ ∈ <(α) on [a, b]. f : [a, b]→ Rk bounded. Then, f ∈ <(α) ⇐⇒ fα′ ∈ <.

Furthermore in this case,∫ bafdα =

∫ baf(x)α′(x)dx.

Theorem 6.25. This proposition deviates from the conventions of this chapter. f(: [a, b] →

Rk) ∈ < on [a, b]. For a ≤ x ≤ b, let F (x) :=∫ xaf(t)dt. Then F is continuous on [a, b].

And furthermore, if f is continuous at point x0 ∈ [a, b], then F is differentiable at x0 and

F ′(x0) = f(x0).

Theorem 6.26 (Fundamental Theorem of Calculus for vector-valued functions). This propo-

sition deviates from the conventions of this chapter. f(: [a, b]→ Rk) ∈ < on [a, b] and if there

30

is a differentiable function F (: [a, b]→ Rk) on [a, b] s.t. F ′ = f , then∫ baf(x)dx = F (b)−F (a).

Remark. A simpler way to type this would be: This proposition deviates from the conventions

of this chapter. f, F : [a, b]→ Rk, f ∈ < on [a, b], F ′ = f , then∫ baf(t)dt = F (b)− F (a).

Theorem 6.27. This proposition deviates from the conventions of this chapter. f : [a, b]→ Rk,

f ∈ <(α), then ‖f‖ ∈ <(α) and∥∥∥∫ ba fdα∥∥∥ ≤ ∫ ba ‖f‖ dα.

Definition 6.28. This definition deviates from the conventions of this chapter. A continuous

mapping γ : [a, b]→ Rk is called a curve in Rk. We may express this as ‘γ is a curve on [a, b]’.

If γ is injective, γ is called an arc. If γ(a) = γ(b), γ is a closed curve. Note that a curve is

defined to be a mapping, and not a point set.

Definition 6.29. This definition deviates from the conventions of this chapter. For each

partition P = x0, . . . , xn of [a, b], and for each curve γ on [a, b], define the following:

Λ(P, γ) :=∑n

i=1

∥∥γ(xi)− γ(xi−1)∥∥. Hence, Λ(P, γ) is the length of the polygonal path with

vertices γ(x0), γ(x1), . . . , γ(xn), in this order. Define the length of γ as Λ(γ) := sup Λ(P, γ),

where the supremum is taken over all partitions of [a, b]. If Λ(γ) <∞, γ is rectifiable.

Theorem 6.30. This proposition deviates from the conventions of this chapter. If γ′ is con-

tinuous on [a, b], then γ is rectifiable, and Λ(γ) =∫ ba

∥∥γ′(t)∥∥ dt.Remark. Thus, for a continuously differentiable curve γ, Λ(γ) is given by a Riemann integral.

31

7 Chapter 7 (Sequences and Series of Functions)

Many results presented in this chapter extend without difficulty to more general settings of

vector-valued functions, or mappings into general metric spaces. However, we stay within the

specific setting of complex valued functions to put emphasis of some key points. (From the

future (after a lot of confusion): For more clarity, all functions fn defining sequences of functions

fn are assumed in the text to be mapped to C.)

Definition 7.1. fn for n ∈ Z>0 is a sequence of functions defined on a set E (definition

does not state range), and suppose that for every arbitrarily fixed x ∈ E, the sequence fn(x)

converges with respect to n. Define f(x) := limn→∞ fn(x), x ∈ E. We say that fn converges

on E and that f is the limit, or limit function, of f . If f(x) = limn→∞ fn(x), ∀x ∈ E holds,

we say that fn converges to f pointwise on E.

Definition 7.2. fn for n ∈ Z>0 is a sequence of functions defined on a set E (definition

does not state range), and suppose that for every arbitrarily fixed x ∈ E, the series∑∞

n=1 fn(x)

converges. Define f :=∑∞

n=1 fn(x), for ∀x ∈ E. Then f is called the sum of the series∑∞n=1 fn(x).

Definition 7.3. fn for n ∈ Z>0 is a sequence of functions defined on a set E (definition does

not state range). We say fn converges uniformly on E to a function f if ∀ε > 0, ∃N ∈ Z

s.t. n ≥ N =⇒∥∥fn(x)− f(x)

∥∥ ≤ ε for ∀x ∈ E.

Definition 7.4. fn for n ∈ Z>0 is a sequence of functions defined on a set E (definition does

not state range). We say the series∑∞

n=1 fn(x) converges uniformly on E if the sequence of

partial sums sn, defined by sn(x) :=∑n

i=1 fi(x) converges uniformly on E.

Theorem 7.5 (Cauchy criterion for uniform convergence). Sequence (fn : E → Y ) for

n ∈ Z>0, where Y is a complete metric space. Then fn converges uniformly on E ⇐⇒

∀ε > 0, ∃N ∈ Z>0 s.t. m,n ≥ N and x ∈ E =⇒∥∥fn(x)− fm(x)

∥∥ ≤ ε.

Theorem 7.6. fn for n ∈ Z>0 is a sequence of functions defined on a set E (definition does

not state range). Suppose limn→∞ fn(x) = f(x), for ∀x ∈ E. Let Mn = supx∈E∥∥fn(x)− f(x)

∥∥.

Then, fn → f uniformly on E ⇐⇒ Mn → 0 as n→∞.

32

Theorem 7.7 (Weierstrass test). Sequence (fn : E → Y ) for n ∈ Z>0, where Y is a complete

metric space, and suppose∥∥fn(x)

∥∥ ≤ Mn for ∀x ∈ E, n ∈ Z>0. Then∑∞

n=1 fn(x) converges

uniformly on E if∑∞

n=1 Mn converges.

Theorem 7.8. (fn : E → Y ), n ∈ Z>0, is a sequence of functions, E a metric space, Y a

complete metric space, and fn → f uniformly on E. Let x be a limit point of E, and suppose that

limt→x fn(t) = An, for n ∈ Z>0, exists. Then An converges, and limt→x f(t) = limn→∞An.

In other words, limt→x limn→∞ fn(t) = limn→∞ limt→x fn(t).

Theorem 7.9. (fn : E → Y ), n ∈ Z>0, is a sequence of continuous functions, E a metric

space, Y a complete metric space. If fn → f uniformly on E, then f is continuous on E.

Remark. The converse is not true. A sequence of continuous functions may converge to a

continuous function, although the convergence is not uniform. A ‘partial’ converse is theorem

7.10.

Theorem 7.10. K is a compact metric space, and

(a) (fn : K → R) is a sequence of continuous functions

(b) fn converges pointwise to a continuous function f on K

(c) fn(x) ≥ fn+1(x) for ∀x ∈ K, n ∈ Z>0.

Then, fn → f uniformly on K.

Definition 7.11. If X is a metric space, C (X) will denote the set of all complex-valued, contin-

uous, and bounded functions with domain X. Associate with each f ∈ C (X) the supremum

norm, ‖f‖sup = supx∈X∥∥f(x)

∥∥. Define the distance/metric between f ∈ C (X) and g ∈ C (X)

to be ‖f − g‖sup. Then, C (X) is a metric space.

Remark. Theorem 7.6 can now be rephrased: Let E be a set. A sequence (fn : E → C)

converges to f in C (X) ⇐⇒ fn → f uniformly on X. (Converges to f in C (X) = with

respect to the metric of C (X))

Theorem 7.12. The distance metric defined in definition 7.11 makes C (X) into a complete

metric space.

33

Theorem 7.13. α is monotonically increasing on [a, b]. fn gets mapped to Rk, fn ∈ <(α)

on [a, b] for n ∈ Z>0, and suppose fn → f uniformly on [a, b]. Then f ∈ <(α) on [a, b],

and∫ bafdα = limn→∞

∫ bafndα. (The existence of the limit is a part of the conclusion of the

theorem.)

Theorem 7.14. If fn gets mapped to Rk, fn ∈ <(α) on [a, b] and f(x) =∑∞

n=1 fn(x), x ∈

[a, b], is the series converging uniformly on [a, b], then∫ bafdα = limk→∞

(∑kn=1

∫ bafndα

)=∑∞

n=1

∫ bafndα. In other words, the series may be integrated term by term.

Theorem 7.15. fn is a sequence of functions mapped to R, differentiable on [a, b]. Assume

∃x0 ∈ [a, b] s.t. fn(x0) converges with respect to n. If f ′n converges uniformly on [a, b], then

fn converges uniformly on [a, b] to a function f , and f ′(x) = limn→∞ f′n(x), x ∈ [a, b].

Theorem 7.16. There exists f : R→ R which is continuous and nowhere differentiable.

Definition 7.17. Let fn be a sequence of functions defined on a set E. fn is pointwise

bounded if for any arbitrarily fixed x0 ∈ E, fn(x0) is a bounded sequence. In other words,

∃φ : E → R s.t. φ is finite-valued and∥∥fn(x)

∥∥ < φ(x), for ∀x ∈ E, n ∈ Z>0.

Definition 7.18. Let fn be a sequence of functions defined on a set E. fn is uniformly

bounded on E if ∃M ∈ R s.t.∥∥fn(x)

∥∥ < M , for ∀x ∈ E, n ∈ Z>0.

Remark. Even if fn is a uniformly bounded sequence of continuous functions defined on a

compact set E, there need not exist a subsequence which converges pointwise on E. Even if gn

is a (pointwise) convergent sequence, it need not contain a uniformly convergent subsequence.

Definition 7.19. A family F of complex functions f defined on E ⊂ X, X a metric space, is

said to be equicontinuous on E if for ∀ε > 0, ∃δ > 0 s.t. dX(x, y) < δ, x ∈ E, y ∈ E, f ∈ F

=⇒∥∥f(x)− f(y)

∥∥ < ε. It is clear that every member of an equicontinuous family is uniformly

continuous.

Remark. There is a close relationship between equicontinuity and the uniform convergence of

sequences of continuous functions. This is illustrated in threorems 7.21, 7.22.

Theorem 7.20. If fn is a pointwise bounded sequence of complex functions on a countable

34

set E, then fn has a subsequence fnk s.t. fnk(x) converges for every x ∈ E.

Theorem 7.21. If K is a compact metric space, fn ∈ C (K) for n ∈ Z>0, and fn converges

uniformly on K, then fn is equicontinuous on K.

Theorem 7.22. If K is compact, fn ∈ C (K) for n ∈ Z>0, and if fn is pointwise bounded

and equicontinuous on K, then:

(a) fn is uniformly bounded on K

(b) fn contains a uniformly convergent subsequence.

Theorem 7.23 (Stone-Weierstrass Theorem). f : [a, b] → C continuous. Then, exists a

sequence of polynomials Pn : [a, b] → C s.t. limn→∞ Pn(x) = f(x) uniformly on [a, b]. If f is

real, then the Pn may be taken to be a sequence of real polynomials.

Theorem 7.24. For every interval [−a, a] there is a sequence of real polynomials Pn s.t.

Pn(0) = 0 and limn→∞ Pn(x) =‖x‖ uniformly on [−a, a].

Definition 7.25. A family A of complex functions defined on a set E is said to be an algebra

if for all f, g ∈ A and for all c ∈ C, (i) f + g ∈ A , (ii) fg ∈ A , (iii) cf ∈ A hold. In other

words, A is closed under addition, multiplication, and scalar multiplication. We also consider

algebras of real functions, and in this case (iii) is required to hold only for all c ∈ R.

Definition 7.26. If A has the property fn ∈ A for n ∈ Z>0, fn → f uniformly on E =⇒

f ∈ A , then A is said to be uniformly closed. Let B be the set of all functions which are

limits of uniformly convergent sequences of members of A . Then B is called the uniform

closure of A .

Theorem 7.27. B is the uniform closure of an algebra A of bounded functions. Then B is a

uniformly closed algebra.

Definition 7.28. A is a family of functions on a set E. A is said to separate points on E

if for every pair of distinct points x1, x2 ∈ E, there exists a function f ∈ A s.t. f(x1) 6= f(x2).

If for each x ∈ E there exists a function g ∈ A s.t. g(x) 6= 0, A vanishes at no point of E.

Remark. The algebra of all polynomials in one variable clearly has these properties on R. The

35

algebra of all even polynomials on R do not, as f(x) = f(−x), for x ∈ R.

Theorem 7.29. Suppose A is an algebra of functions on set E, A separates points on E,

A vanishes at no point of E. Suppose x1, x2 are distinct points of E, and c1, c2 are complex

constants (they are real if A is a real algebra). Then A contains a function f s.t. f(x1) = c1,

f(x2) = c2.

Theorem 7.30. Let A be an algebra of real continuous functions having compact set K as

domain. If A separates points on K and A vanishes at no point of K, then the uniform closure

B of A consists of all real continuous functions on K.

Remark. Theorem 7.30 does not hold for complex algebras. But it holds if an extra condition

is imposed on A , namely that A is self-adjoint.

Definition 7.31. An algebra A is self-adjoint if for every f ∈ A , its complex conjugate

f ∈ A . f is defined by f(x) = f(x).

Theorem 7.32. Suppose A is a self-adjoint algebra of complex continuous functions on a

compact set K, A separates points on K, A vanishes at no point of K. Then the uniform

closure B of A consists of all complex continuous functions on K. In other words, A is dense

in C (K).

Remark. Based on the fact that C (X) is a metric space for metric space X, and what we have

proved about algebra A and its uniform closure B, I think it’s safe to say B = A . This is

because the closure of A is the union of A and its limit points. Note that A (⊂ C (X)) is

also a metric space. To be a limit point in A ⊂ C (X) is to have points in A that converge

to it, with respect to the distance function (metric) of C (X), the supremum norm. We have

proved that convergence in C (X) with respect to the supremum norm is equivalent to uniform

convergence on X. Furthermore, any sequence of elements of A that converge with respect to

the metric of C (X) converges to a limit point of A , which is an element of A . It is clear that

any element of A has a sequence of points in A that converge to it (the constant sequence of

itself). Therefore, B = A .

36

8 Chapter 8 (Some Special Functions)

Definition 8.1. Functions which are represented by power series, f(x) =∑∞

n=0 cn(x−a)n, are

called analytic functions. If f converges for ‖x− a‖ < R, for some R > 0 (R may be ∞), f

is said to be expanded in a power series about the point x = a. As a matter of convenience, we

very often take a = 0 without saying so explicitly, and without any loss of generality.

Remark. In this chapter, we restrict ourselves to real values of x, i.e. f(x) =∑∞

n=0 cn(x− a)n

s.t. f : R→ R.

Theorem 8.2. Suppose the series∑∞

n=0 cnxn converges for ‖x‖ < R, and define f(x) =∑∞

n=0 cnxn, for‖x‖ < R. Then f(x) converges uniformly on [−R+ ε, R− ε], for ε ∈ (0, R]. The

function f is continuous and differentiable in (−R,R), and f ′(x) =∑∞

n=0 ncnxn−1 for‖x‖ < R.

Remark. Under the hypotheses of Theorem 8.2, f has derivatives of all orders in (−R,R), which

are given by f (k)(x) =∑∞

n=k n(n − 1) . . . (n − k + 1)cnxn−k. In particular, f (k)(0) = k!ck, for

k ∈ Z≥0. (f (0) = f) This shows that the coefficients of the power series development of f are

determined by the values of f and its derivatives at a single point. If the coefficients are given,

the values of the derivatives of f at the center of the interval of convergence can be read off

immediately from the power series. This shows that if two power series converge to the same

function in (−R,R), then the two series must be identical, i.e. must have the same coefficients.

Remark. Let the assumptions be as in Theorem 8.2. If the series∑∞

n=0 cnxn converges at an

endpoint, say at x = R, then f is continuous not only in (−R,R), but also at x = R. This

follows from Abel’s theorem.

Theorem 8.3 (Abel’s theorem). Suppose∑∞

n=0 cnRn converges. Let f(x) =

∑∞n=0 cnx

n for

x ∈ (−R,R), R ∈ R>0. Then, limx→R− f(x) =∑∞

n=0 cnRn. (The limit is from the left)

Theorem 8.4. Given a double sequence aij ∈ Rk, i, j ∈ Z>0, suppose∑∞

j=1

∥∥aij∥∥ = bi, for

i ∈ Z>0, and∑∞

i=1 bi converges. Then,∑∞

i=1

∑∞j=1 aij =

∑∞j=1

∑∞i=1 aij.

Theorem 8.5 (Taylor’s theorem). Suppose the series f(x) =∑∞

n=0 cnxn converges for‖x‖ < R.

If a ∈ (−R,R), then f can be expanded in a power series about the point x = a which converges

in ‖x− a‖ < R−‖a‖ and f(x) =∑∞

n=0f (n)(a)n!

(x− a)n, for ‖x− a‖ < R−‖a‖.

37

Theorem 8.6. Suppose the series∑anx

n and∑bnx

n converge in the segment S = (−R,R) ⊂

R. Let E be the set of all x ∈ S at which∑∞

n=0 anxn =

∑∞n=0 bnx

n. If E has a limit point in

S, then an = bn for n ∈ Z≥0, and hence∑∞

n=0 anxn =

∑∞n=0 bnx

n holds for all x ∈ S.

Definition 8.7. Define for ∀z ∈ C, E(z) :=∑∞

n=0zn

n!. It is shown in the text that E(x) = ex

for ∀x ∈ R, where e is as defined in definition 3.25.

Theorem 8.8. For ex : R→ R,

(a) ex is continuous and differentiable for all x ∈ R

(b) (ex)′ = ex

(c) ex is a strictly increasing function of x and ex > 0

(d) ex+y = exey

(e) ex → +∞ as x→ +∞, ex → 0 as x→ −∞

(f) limx→+∞ xne−x = 0, n ∈ Z>0

Remark. Pages 180-182. A nice section about defining log functions.

Definition 8.9. Define C(x) := 12[E(ix) + E(−ix)], S(x) := 1

2i[E(ix) − E(−ix)]. We can

show these coincide with cos(·) and sin(·) functions, whose definitions usually are based on

‘geometric considerations’. It is possible to prove that there exists smallest positive x0 ∈ R>0

s.t. C(x0) = 0. Define the number π := 2x0. From this, we deduce usual properties of E(·).

Theorem 8.10. The following hold:

(a) The function E is periodic, with period 2πi

(b) The functions C and S are periodic, with period 2π

(c) t ∈ (0, 2π) =⇒ E(it) 6= 1

(d) z ∈ C s.t. ‖z‖ = 1, then there exists a unique t ∈ [0, 2π) s.t. E(it) = z.

Remark. We can now prove that the complex field is algebraically complete, i.e. every noncon-

stant polynomial with complex coefficients has a complex root.

38

Theorem 8.11. a0, . . . , an ∈ C, n ∈ Z≥1, an 6= 0, P (z) =∑n

k=0 akzk. Then P (z) = 0 for some

z ∈ C.

Definition 8.12. Let x ∈ R, N ∈ Z≥1, and a0, . . . , aN , b1, . . . , bN ∈ C. A trigonometric

polynomial is a finite sum of the form f(x) = a0 +∑N

n=1(an cos(nx) + bn sin(nx)). This can

be alternately written f(x) =∑N

n=−N cneinx, c−N , . . . , cN ∈ C. It is clear that f(x + 2π) = f .

Note again that x ∈ R.

Definition 8.13. In agreement with definition 8.12, a trigonometric series is a series of the

form∑∞−∞ cne

inx, x ∈ R. The Nth partial sum of this series is defined to be∑N

n=−N cneinx. If

f is an integrable function on [−π, π], the numbers cm := 12π

∫ π−π f(x)e−imxdx, m ∈ Z are called

the Fourier coefficiencts of f . The series∑∞−∞ cne

inx, x ∈ R formed by these coefficients is

called the Fourier series of f .

Definition 8.14. Let φn : [a, b] → C, n,m ∈ Z>0 be a sequence of functions such that for

n 6= m,∫ baφn(x)φm(x)dx = 0. Then φn is said to be an orthogonal system of functions

on [a, b]. If, in addition,∫ ba

∥∥φn(x)∥∥2dx = 1 for ∀n ∈ Z>0, φn is said to be orthonormal.

If φn is orthonormal on [a, b], define cn =∫ baf(t)φn(t)dt, n ∈ Z>0, and call cn the nth

Fourier coefficient of f relative to φn. We write f ∼∑∞

n=1 cnφn(x) and call this series the

Fourier series of f (relative to φn). The ∼ symbol used does not imply anything about

the convergence of the series. It just means that the coefficients are given by cn defined above.

Remark. From this point onward, f ∈ < is assumed, although this hypothesis can be

weakened.

Theorem 8.15. φn is orthonormal on [a, b]. Let sn(x) =∑n

m=1 cmφm(x) be the nth partial

sum of the Fourier series of complex function f , and suppose tn(x) =∑n

m=1 γmφm(x), for

complex numbers γm. Then∫ ba‖f − sn‖2 dx ≤

∫ ba‖f − tn‖2 dx, and equality holds if and only if

γm = cm, m = 1, . . . , n.

Remark. Among all functions tn, sn gives the best possible mean square approximation to f .

Theorem 8.16. φn is orthonormal on [a, b], and f ∼∑∞

n=1 cnφn(x). Then,∑∞

n=1‖cn‖2 ≤∫ b

a

∥∥f(x)∥∥2dx. In particular, limn→∞ cn = 0.

39

Remark. From now on in this chapter, f satisfies f(x+ 2π) = f(x) , f ∈ < on [−π, π] (and

hence on every bounded interval), The Fourier series of f is∑∞−∞ cne

inx, x ∈ R, where the

Fourier coefficients are given by cm := 12π

∫ π−π f(x)e−imxdx. sN(x) = sN(f ;x) =

∑Nn=−N cne

inx

is the Nth partial sum of the Fourier series of f .

Remark. It can be proven 12π

∫ π−π

∥∥sN(x)∥∥2dx =

∑N−N‖cn‖

2 ≤ 12π

∫ π−π

∥∥f(x)∥∥2dx.

Definition 8.17. Define the Dirichlet kernel DN(x) =∑N−N e

inx =sin (N+ 1

2)x

sin (x2

). We have that

sN(f ;x) = 12π

∫ π−π f(t)DN(x− t)dt = 1

2π

∫ π−π f(x− t)DN(t)dt.

Theorem 8.18. If, for some x ∈ R, there are constants δ > 0 and M < ∞ such that∥∥f(x+ t)− f(x)∥∥ ≤ M‖t‖ for ∀t ∈ (−δ, δ), then limN→∞ sN(f ;x) = f(x). This is a theo-

rem about pointwise convergence of Fourier series.

Theorem 8.19. If f(x) = 0 for ∀x ∈ J , where J is some open interval of the real line, then

limN→∞ sN(f ;x) = 0 (pointwise) for every arbitrarily fixed x ∈ J .

Remark. Here is another formulation of theorem 8.19: If f(t) = g(t) for all t in some neigh-

bourhood of x, then sN(f ;x) − sN(g;x) = sN(f − g;x) → 0 as N → ∞. This is called the

localization theorem. For an arbitrarily fixed x, the behaviour of the sequence sN(f ;x),

as far as convergence is concerned, depends only on the values of f in some arbitrarily small

neighbourhood of x. Two different Fourier series may have the same behaviour in one interval

and behave in entirely different ways in other intervals. This is a difference between Fourier

series and power series.

Theorem 8.20. If f is continuous s.t. f(x+2π) = f(x) and ε > 0, then there is a trigonometric

polynomial P s.t.∥∥P (x)− f(x)

∥∥ < ε for ∀x ∈ R.

Theorem 8.21 (Parseval’s theorem). Suppose f, g ∈ < and f(x+2π) = f(x), g(x+2π) = g(x).

Let f(x) ∼∑∞−∞ cne

inx, g(x) ∼∑∞−∞ γne

inx. Then, limN→∞1

2π

∫ π−π

∥∥f(x)− sN(f ;x)∥∥2dx = 0,

12π

∫ π−π f(x)g(x)dx =

∑∞−∞ cnγn, 1

2π

∫ π−π

∥∥f(x)∥∥2dx =

∑∞−∞‖cn‖

2.

Definition 8.22 (Gamma Function). For 0 < x < ∞, Γ(x) :=∫∞

0tx−1e−tdt. The integral

converges for these x.

40

Theorem 8.23. The following hold:

(a) The functional equation Γ(x+ 1) = xΓ(x) holds if 0 < x <∞.

(b) Γ(n+ 1) = n! for n ∈ Z≥1.

(c) log Γ is convex on (0,∞).

These properties characterize Γ completely.

Theorem 8.24. If f : (0,∞)→ R is a positive function s.t.

(a) f(x+ 1) = xf(x)

(b) f(1) = 1

(c) log f is convex,

Then f(x) = Γ(x).

Theorem 8.25. If x, y ∈ R>0, then B(x, y) :=∫ 1

0tx−1(1− t)y−1dt = Γ(x)Γ(y)

Γ(x+y). B(x, y) is called

the beta function.

Remark. We can now deduce Γ(x) = 2x−1√π

Γ(x2)Γ(x+1

2).

Theorem 8.26 (Stirling’s formula). This is a simple approximation for Γ(x + 1) when x is

large (hence for n! when n is large). The formula is limx→∞Γ(x+1)

(xe

)x√

2πx= 1.

41

9 Chapter 9 (Functions of Several Variables)

Definition 9.1. A mapping A : X → Y , X, Y vector spaces, is said to be a linear transfor-

mation if A(x1 + x2) = Ax1 + Ax2, A(cx) = cAx for all x, x1, x2 ∈ X and all scalars c. One

often writes Ax instead of A(x) if A is linear.

Definition 9.2. Linear transformations of a vector space X into X are often called linear

operators on X. If A is a linear operator on X which is a bijection, we say that A is invertible.

We can define an operator A−1 by requiring A−1(Ax) = x, for ∀x ∈ X. Such A−1 is linear, and

also satisfies A(A−1x) = x, for ∀x ∈ X.

Theorem 9.3. A linear operator A on a finite-dimensional vector space X is an injection if

and only if A(X) = X. (The range of A is all of X)

Theorem 9.4. A linear operator A on a finite-dimensional vector space X is an injection if

and only if it is a surjection.

Definition 9.5. L(X, Y ) is the set of all linear transformations of the vector space X into the

vector space Y . Instead of writing L(X,X), write L(X). For A ∈ L(Rn,Rm), define the norm

‖A‖ of A to be the sup of all numbers‖Ax‖, where x ranges over all vectors in Rn s.t. ‖x‖ ≤ 1.

Note that the supremum of ‖Ax‖ as above is in fact a maximum of ‖Ax‖, and has to occur at

‖x‖ = 1. Furthermore,‖Ax‖ ≤‖A‖‖x‖ for ∀x ∈ Rn. Also, if λ is s.t. ‖Ax‖ ≤ λ‖x‖ for ∀x ∈ Rn,

then ‖A‖ ≤ λ.

Theorem 9.6. The following hold:

(a) If A ∈ L(Rn,Rm), then ‖A‖ <∞ and A is a uniformly continuous mapping of Rn into Rm.

(b) If A,B ∈ L(Rn,Rm), and c is a scalar, then ‖A+B‖ ≤‖A‖+‖B‖ and ‖cA‖ =‖c‖‖A‖.

(c) If A ∈ L(Rn,Rm) and B ∈ L(Rm,Rk), then ‖BA‖ ≤‖B‖‖A‖.

Remark. The norm ‖·‖ acts as a metric for the space L(Rn,Rm).

Theorem 9.7. Let Ω be the set of all invertible linear operators on Rn. Then the following

hold:

(a) If A ∈ Ω, B ∈ L(Rn) and ‖B − A‖∥∥A−1

∥∥ < 1, then B ∈ Ω.

42

(b) Ω is an open subset of L(Rn) and the mapping A→ A−1 is continuous on Ω. Note that this

mapping is a 1− 1 mapping of Ω onto Ω, hence is a bijection, which is its own inverse.

Definition 9.8. Suppose E is an open set in Rn, f : E → Rm, and x ∈ E. If ∃A ∈ L(Rn,Rm)

s.t. limh∈Rn→0‖f(x+h)−f(x)−Ah‖

h= 0, then we say that f is differentiable at x, and we write

f ′(x) = A. If f is differentiable at every x ∈ E, we say that f is differentiable in E. It can be

proven that A, if it exists, is unique.

Theorem 9.9 (Chain rule). Suppose E is an open set in Rn, f : E → Rm, f is differentiable

at x0 ∈ E, g maps an open set containing f(E) into Rk, and g is differentiable at f(x0). Then

the mapping F : E → Rk defined by F (x) = g(f(x)) is differentiable at x0, and F ′(x0) =

g′(f(x0))f ′(x0).

Theorem 9.10 (Inverse function theorem). Suppose f is a C ′−mapping of an open set E ⊂ Rn

into Rn, f ′(a) is invertible for some a ∈ E, and b = f(a). Then,

(a) There exist open sets U, V ⊂ Rn such that a ∈ U , b ∈ V , f is an injection on U , and

f(U) = V (so f is a bijection locally)

(b) If g is the inverse of f , which exists by (a), defined in V by g(f(x)) = x (x ∈ U), then

g ∈ C ′(V ).

Remark. The inverse function theorem states, roughly speaking, that a continuously differen-

tiable mapping f is invertible in a neighbourhood of any point x at which the linear transfor-

mation f ′(x) is invertible.

Remark. The rest of the chapter concerns the implicit function theorem, the rank theorem, and

derivatives of higher order.

10 Chapter 10 (Geometry and Topology in the Complex

Plane)

Definition 10.1. For a set B, the characteristic function of B, χB, is given by χB(x) = 1

if x ∈ B and χB(x) = 0 otherwise. Note that z, w ∈ C, zw = 0 =⇒ z = 0 or w = 0.

Proposition 10.2. ∀z, w ∈ C, the following hold(s):

43

(a) |Re(z)| ≤ |z|, |Im(z)| ≤ |z|

(b) |z + w| ≤ |z|+ |w|

(c) |z + w| ≥ ||z| − |w||

Proposition 10.3. α, β ∈ C, α 6= β. λ ∈ R>0\1. Then, the equation |z−α||z−β| = λ represents a

circle.

Proposition 10.4. Suppose α, β ∈ C, α 6= β. Then the equation for a circular arc that passes

through α, β is for fixed µ (and z 6= α, β):

arg(z − α)− arg(z − β) ≡ arg

(z − αz − β

)≡ µ or − (π − µ) (mod 2π)

Here, arg(z) := ∀θ s.t. z = |z|eiθ.

Remark. Given fixed µ, α, and β, the two arcs, which form a circle with points α, β missing, are

unique. You can use geometry to prove this. You can also prove the proposition above using

geometry.

Definition 10.5. The open disc centre a ∈ C and radius r > 0 is defined to be D(a; r) :=

z ∈ C : |z − a| < r. The closed disc, for same conditions D(a; r) := z ∈ C : |z − a| ≤ r.

The punctured disc with same conditions D′(a; r) := z ∈ C : 0 < |z − a| < r.

Definition 10.6. Any set of form z ∈ C : s < |z − a| < r (0 ≥ s < r), or z ∈ C : s <

|z − a| (0 ≥ s) is called an (open) annulus. The open upper half-plane Π+ := z ∈ C :

Im(z) > 0. the closed upper half-plane, and other half planes are defined similarly. Π− is the

open lower half-plane.

Definition 10.7. Sα,β := z ∈ C : 0 6= z = |z|riθ ∈ C with α < θ < β, (α < β), is called a

sector.

Definition 10.8. Embed C by identifying x+ iy with (x, y, 0). Σ := (x, y, u) ∈ R3 : x2 + y2 +

(u− 12)2 = 1

4 is the Riemann sphere. Define the extended complex plane C := C∪∞.

Remark. In the stereographic projection, note that ‘all roads lead to∞’. −∞ is nowhere to be

found on the Riemann sphere.

44

Definition 10.9. The following are conventions of arithmetic when working with C:

(A) a±∞ = ±∞+ a =∞, ∀a ∈ C (B) a\∞ = 0, ∀a ∈ C

(C) a · ∞ =∞ · a =∞, ∀a ∈ C\0 (D) a\0 =∞, ∀a ∈ C\0

(E) ∞+∞ =∞ ·∞ =∞ =∞

Note that ∞−∞ and ∞\∞ are not defined. We are allowed to divide a complex number a

by 0, as long as a 6= 0.

Proposition 10.10. Prove the following:

(a) Prove that any line on C is stereographically projected onto the Riemann sphere(Σ) to be

a circle that goes through the North Pole in R3. Prove that any circle in Σ that goes through

the North Pole is stereographically projected onto a line on C. Therefore, the stereographical

projection is a line in C ⇐⇒ the original projected circle in Σ goes through north pole.

(b) Prove that any circle on C is stereographically projected to a circle in R3 in Σ. Prove that

any circle in Σ that does not go through the north pole is stereographically projected onto C

to be a circle.

The ‘intuitive explanation’ for this has been written in the Question Sheet.

Proposition 10.11. α, β ∈ C, α 6= β. λ ∈ R>0. Consider the circline given by equation

|z−α||z−β| = λ in C. The points α, β are known as inverse points with respect to the circline, and

have a geometrical significance.

Theorem 10.12. α, β ∈ C are inverse points with respect to the circle |z − a| = r ⇐⇒

(α− a)(β − a) = r2.

Remark. Note that always one of α, β is inside the circle and the other outside. Furthermore,

regard α = a, β =∞ as a pair of inverse points for |z − a| = r.

Definition 10.13. We define a ‘disc’ in C centered on ∞ by D(∞; r) := z ∈ C : |z| >

r ∪ ∞, (r > 0). We define S ⊂ C to be open if, ∀r ∈ S, ∃r > 0 s.t. D(z; r) ⊂ S.

Remark. Due to significant time constraints (i.e. time left in UROP) I will only write down the

essentials.

45

Remark. Let a = z0, . . . , zn = b ∈ C. We call [z0, z1] ∪ · · · ∪ [zn−1, zn] the polygonal route

from a to b. Case a = b is allowed. A non-empty S ⊂ C is polygonally connected if, given

any two points a, b ∈ S, there exists a polygonal route from a to b lying solely in S.

Definition 10.14. A region is a non-empty open connected subset of C.

Theorem 10.15. Let G be a non-empty open subset of C. Then, G is a region ⇐⇒ G is

polygonally connected. In particular, any non-empty open convex set is a region.

Remark. The same proof works if:

(a) We only allow polygonal routes whose line segments are all horizontal or vertical.

(b) We only allow polygonal routes some of whose line segments are circular arcs, or all of

whose line segments are circular arcs.

One may guess from this that there are other conventions defining polygonal routes.

Theorem 10.16. The following hold:

(a) zn is a complex sequence. Then zn converges if and only if the real sequences Re(zn)

and Im(zn) both converge. In addition, zn → a =⇒ ‖zn‖ →‖a‖, zn → a.

(b) Let f : S → C and f = u + iv. Then for any a ∈ S, the closure of S, limz→a exists if

and only if both limz→a u and limz→a v exist. Then f(z) → w implies that u(z) → Re(w) and

v(z)→ Im(w). In addition,∥∥f(z)

∥∥→‖w‖ and f(z)→ w.

(c) Let f : S → C and f = u+ iv. Then f is continuous at a ∈ S (on S) if and only if u, v are

continuous at a ∈ S (on S). In addition, continuity of f(z) implies continuity of∥∥f(z)

∥∥, for

z ∈ S.

Theorem 10.17 (Bolzano-Weierstrass Theorem). Any infinite compact subset S of C has a

limit point in S.

46

11 Chapter 11 (Paths, Holomorphic Functions, and Com-

plex Series)

Definition 11.1. A curve is a continuous map. Denote the image of a curve γ : [a, b]→ C by

γ∗. γ is said to lie in a set S if γ∗ ⊂ S. Note that γ∗ is compact in C. Such curve γ is said

to be smooth if the function γ is continuously differentiable on [a, b], the derivatives at a, b

being one-sided limits. A path is the join of finitely many smooth curves, which need not be

differentiable as a whole.

Definition 11.2. A circline path is a path which is the join of finitely many paths of (a) line

segments or (b) circular arcs (arcs of a circle). A contour is a simple closed circline path. If,

for contour γ(t), γ moves anticlockwise around any point ‘inside’ it as t increases, we say it to

be positively oriented.

Remark. The image of a contour consists of finitely many line segments and circular arcs and

does not cross itself. Define γ(a; r)(t) := a + reit, t ∈ [0, 2π], which denotes the circle centre a

and radius r, and Γr(t) := reit, t ∈ [0, π].

Theorem 11.3. γ a path lying in open set G. Then ∃ constant m > 0 s.t. D(z;m) ⊂ G for

∀z ∈ γ∗.

Remark. If γ is a path lying in an open set G, then there exists an infinite chain of discs covering

γ∗, due to theorem 11.3.

Theorem 11.4 (Covering Theorem). G is an open set and γ : [a, b]→ C is a path s.t. γ∗ ⊂ G.

Then, ∃m > 0 which is a constant, and finitely many open discs D0, . . . , DN s.t.

(i) For k = 0, 1, . . . , N , Dk = D(γ(tk);m), where α = t0 < t1 < · · · < tN = β

(ii) For k = 0, 1, . . . , N − 1, Dk ∩Dk+1 6= ∅

(iii) For k = 0, 1, . . . , N − 1, γ([tk, tk+1]) ⊂ Dk

(iv) γ∗ ⊂ ∪Nk=0Dk ⊂ G

The disc DN is not needed for the covering, and put in only for notational convenience later

on. When γ is closed, D0 = DN . Furthermore, I think it’s safe to say that for k = 1, . . . , N ,

γ([tk−1, tk]) ⊂ Dk, as m is constant and due to (iii).

47

Remark. If γ is a path lying in an open set G, then γ∗ can be covered by a finite chain of open

sets contained in G, each overlapping the next.

Theorem 11.5. By refining the proof of theorem 11.4, we get for the notation used in 11.4

that there ∃α > 0 s.t. the open strip S := z ∈ C : ∃w ∈ γ∗ with ‖z − w‖ < α is such that

γ∗ ⊂ S ⊂ G.

Theorem 11.6 (Jordan curve theorem, restricted to contour). γ a contour. Then the comple-

ment of γ∗ is of form I(γ) ∪ O(γ), where I(γ) and O(γ) are disjoint connected open sets, and

I(γ) is bounded (the inside of γ) and O(γ) is unbounded (the outside of γ).

Remark. This asserts that a simple closed path has an ‘inside’ and ‘outside’. It is true that for

contour γ, γ∗ = ∂I(γ) = ∂O(γ) and that γ∗ ∪ I(γ), γ∗ ∪O(γ) are closed.

Theorem 11.7 (Triangulation of a polygon). γ is a polygonal contour in C and z1, . . . , zn,

n > 3, are vertices of γ∗. Then it is possible to insert n − 3 line segments [zj, zk] such that

I(γ) is divided into n − 2 triangles (i.e. triangular contours). Each of the inserted segments,

excluding its endpoints, lies in I(γ).

Theorem 11.8. A complex function f : G→ C for G ⊂ C open is differentiable at z ∈ G if

limh→0f(z+h)−f(z)

hexists. The limit, when it exists, is denoted by f ′(z).

Remark. Note the role of G being open. No matter which direction h approaches 0, f(z+h)−f(z)h

must tend to the same limiting value. Otherwise, the limit does not exist.

Theorem 11.9 (Cauchy Riemann equations (C-R equations)). For open G ⊂ C, let f : G→ C

be differentiable at z = x + iy ∈ G. (Deduce common sense notations) Let f(z) = u(x, y) +

iv(x, y). Then u, v have first-order partial derivatives at (x, y) and satisfy ux = vy, uy = −vx.

Theorem 11.10 (Partial converse to Cauchy Riemann equations). For open G ⊂ C, let f :

G→ C, f(z) = u(x, y) + iv(x, y), z = x + iy ∈ G. Assume u, v satisfy the C-R equations and

that the first order partial derivatives are continuous. Then, f ′(z) exists on G.

Definition 11.11. A complex valued function f which is differentiable at every point of open

set G is said to be holomorphic in G. The set of functions holomorphic in G is denoted H(G).

48

f is holomorphic at a point a ∈ C if there exists r > 0 s.t. f is defined and holomorphic in

D(a; r).

Theorem 11.12. The following hold:

(a) Let f, g be holomorphic in G and λ ∈ C. Then λf , f + g, fg are holomorphic in G and the

usual differentiation rules apply. For ∀z ∈ G, (λf)′(z) = λf ′(z), (f + g)′(z) = f ′(z) + g′(z),

(fg)′(z) = f ′(z)g(z) + f(z)g′(z).

(b) f is holomorphic in G and g is holomorphic in an open set containing f(G). Then g f is

holomorphic in G, and for ∀z ∈ G, (g f)′(z) = g′(f(z))f ′(z).

(c) f is holomorphic in G and f(z) 6= 0, ∀z ∈ G. Then 1f

is holomorphic in G and for any

z ∈ G, ( 1f)′(z) = − f ′(z)

(f(z))2.

Theorem 11.13. If f is holomorphic in G, then f is continuous on G. If S is a compact

subset of G, then f is bounded on S.

Theorem 11.14. Suppose f is holomorphic in a region G. Then any of the following conditions

force f to be a constant in G:

(a) f ′(z) = 0, ∀z ∈ G

(b) ‖f‖ constant in G

(c) f(z) is real for ∀z ∈ G.

Remark. A non-constant real-valued function in an open disc or in a region cannot be holomor-

phic, such as ‖f‖ , Re(f), Im(f). (Technically the last one is an imaginary function) Contrast

this with continuity, where the three aforementioned functions are continuous.

Theorem 11.15. an is a sequence in C. If∑an converge, then an → 0, n → ∞, and ∃M

s.t. ‖an‖ ≤M , ∀n. Algebra of limits hold for complex sequences. If a complex series converges

absolutely, then it converges. The comparison test, the ratio test, and the root test are tests for

absolute convergence.

Remark. The facts presented in theorem 11.15 have already been studied in ‘Principles of

Mathematical Analysis’, by Rudin.

49

Remark. Note the conditions for being able to rearrange the terms of a power series, studied

in ‘Principles of Mathematical Analysis’. This book does not seem to remark too deeply on it.

Definition 11.16 (Power series and radius of convergence). A series of form∑cn(z − a)n,

where a ∈ C and cn ∈ C (n ∈ Z≥0) is said to be a power series. We often assume without loss

of generality that a = 0. The radius of convergence (ROC) of power series∑cn(z − a)n is

defined to be R := sup‖z − a‖ :∑∥∥cn(z − a)n

∥∥ converges.

Remark. The following lemma implies that every power series f(z) =∑cn(z − a)n with ROC

R > 0 has a ‘disc of convergence’ D(a;R). We prove later that f(z) is holomorphic in this disc.

The series diverges for ‖z − a‖ > R. Any behaviour is possible on ‖z − a‖ = R.

Theorem 11.17. f(z) =∑cn(z − a)n is a power series with ROC R > 0. Then,

(a)∑cn(z − a)n converges absolutely for ∀z ∈ C s.t. ‖z − a‖ < R

(b)∑cn(z − a)n fails to converge for any z ∈ C s.t. ‖z − a‖ > R.

Theorem 11.18 (Differentiation theorem for power series). Let∑cn(z−a)n be a power series

with ROC R > 0, and define f in D(a;R) by f(z) =∑cn(z − a)n. Then the following

statements are true:

(a)∑ncn(z − a)n−1 has a ROC R

(b) f is continuous in D(a;R)

(c) f is holomorphic in D(a;R), and f ′ is given by term-by-term differentiation, f ′(z) =∑ncn(z − a)n−1, for (‖z − a‖ < R)

(d) f has derivatives of all orders in D(a;R). Furthermore, f (n)(a) = n!cn, for n ∈ Z≥0.

Theorem 11.19.∑cn(z − a)n and

∑ncn(z − a)n−1 have same ROC.

50

12 Chapter 12 (Conformal Mappings and Multifunctions)

Theorem 12.1. ddz

cos z = − sin z, ddz

sin z = cos z, ddz

cosh z = sinh z, ddz

sinh z = cosh z

Theorem 12.2 (Osborn’s rules). cos iz = cosh z, sin iz = i sinh z

Theorem 12.3. Exactly the same addition/subtraction formulae hold for complex trigonomet-

ric and hyperbolic functions as their real counterparts. (e.g. cos(z + w) = cos(z) cos(w) −

sin(z) sin(w))

Remark. cos, sin are unbounded in C. It is an important fact that ez = 1 ⇐⇒ z = 2kπi,

k ∈ Z, and ez = −1 ⇐⇒ z = (2k + 1)πi, k ∈ Z.

Theorem 12.4. For the trigonometric and hyperbolic functions we have, for k ∈ Z:

(a) cos z = 0 ⇐⇒ z = (k + 12)π

(b) sin z = 0 ⇐⇒ z = kπ

(c) cosh z = 0 ⇐⇒ z = (k + 12)πi

(d) sinh z = 0 ⇐⇒ z = kπi

Note that sin, cos behave the same way as in the real case. In (c), (d), the situation is rather

different than in the real case.

Remark. In the complex plane, cos, sin are periodic with period 2π, and cosh, sinh are periodic

with period 2πi.

Remark. For z ∈ C, the argument of z, [arg z], is a set of elements of R. The brackets are

there to emphasize that the argument is defined as an infinite set of numbers, and not a single

number.

Remark. In this book, log has base e. We define for z ∈ C/0, [log z] := log‖z‖ + iθ : θ ∈

[arg z]. log is the inverse of the exponential function.

Definition 12.5. For z ∈ C/0 and α ∈ C, define [zα] := eα(log z). Only when α is an

integer does [zα] not produce multiple values.

51

Remark. Complex powers must be treated carefully. For x ∈ R>0, α, β ∈ R, xαxβ = xα+β

can be shown to have a complex analogue, which the values of the multifunctions have to be

appropriately selected. But for x1, x2 ∈ R>0, α ∈ R, xα1xα2 = (x1x2)α has no universally valid

complex generalization.

Remark. Now, we investigate how to extract holomorphic functions from [log z] and [zα], where

α /∈ Z.

Remark. In chapter 7, a very ‘economic’ explanation of branch cut, branch point are given.

Remark. Cut C at (−π, π], meaning to cut along non-positive real axis (−∞, 0]. In the cut

plane Cπ := C/(−∞, 0], define fk(z) := log‖z‖+ i(θ+ 2kπ) for z ∈ Cπ, θ ∈ (−π, π], θ ∈ [arg z]

for k ∈ Z. Then, Imfk and Refk are continuous at any point not on the cut, so fk is continuous

in Cπ. But it is discontinuous on the cut (−∞, 0]. It is possible to prove using 11.10 that fk is

holomorphic in Cπ, and with some extra working, that f ′k(z) = 1z

on Cπ. We call functions fk

the holomorphic branches of the logarithm.

Remark. n ∈ Z/−1, 0, 1, and consider the multifunction [z1n ] = ‖z‖

1n e

iθn : 0 6= z =‖z‖ eiθ.

We will restrict θ ∈ [0, 2π), thus we cut the axis at (−∞, 0]. Define gk(z) := e2kπin r

1n e

iθn for

(0 6= z = reiθ). Then the functions gk are 1-valued and s.t. (gk(z))n = z, for any point in Cπ.

Furthermore, [z1n ] = gk(z) : 0 ≤ k ≤ n−1. It turns out that gk is holomorphic (and therefore

continuous) in Cπ, and gk is called a holomorphic branch of the n’th root. Similarly, we can

think of holomorphic branches of [zα], for arbitrary α, although there may be infinitely many

holomorphic branches.

Remark. In order to consider nice holomorphic functions, we cut the plane (take some elements

out of the plane) so that the multifunctions, confined to a single branch, are holomorphic on

the cut plane, so that we can apply all sorts of theorems.

Remark. From now on in this chapter, we let arg z denote any choice from the set [arg z].

Theorem 12.6 (Conformality theorem). Suppose f is holomorphic in an open set G, and γ1,

γ2 are paths, with parameter interval [0, 1], in G meeting at ζ = γ1(0) = γ2(0) with f ′(ζ) 6= 0.

Then f preserves angles between γ1, γ2 at point ζ into f(γ1), f(γ2) at point f(ζ).

52

Definition 12.7. A complex function is conformal in an open set G ⊂ C (or G ⊂ C) if

f ∈ H(G) and f ′(z) 6= 0, for every z ∈ G. It is conformal at a point ζ if it is conformal in

some D(ζ; r), r ∈ R>0.

Remark. The conformality theorem shows that a conformal mapping preserves the magnitude

of and ‘sense of angles’ between paths. In some sense, orientation is preserved.

Remark. Consider the Mobius Transformation (MT) f(z) = az+bcz+d

, where (ad− bc 6= 0). Then,

f is conformal in C/−dc, for c 6= 0.

Remark. We normally regard MT as a mapping from C to itself, so we want to extend our

conformality definition to such a mapping. Let an MT f(z) = az+bcz+d

, where (ad− bc 6= 0). If for

ζ ∈ C, f(ζ) =∞, then we call f(z) conformal at ζ if g(z1) = 1f(z1)

is conformal at z1 = ζ. We

say that f is conformal at ∞ is f(z2) = f( 1z2

) is conformal at z2 = 0.

Remark. Consider the Mobius Transformation (MT) f(z) = az+bcz+d

, where (ad− bc 6= 0). Then,

f is conformal in C. In particular, an MT has non-zero derivative at every point of C.

Remark. The original definition of conformality uses the definition of holomorphic functions,

and the conformality theorem uses the notion of an angle. The conformality theorem has not

been proven using the the extended definition of conformal mappings in MT’s. Therefore it is

unclear if the conformality theorem can be directly applied without any modification. Alas, it

is worth noting that ‘angles’ are preserved ‘slightly differently’ when mapping intersection of

circles to two lines, in that finite angles in the intersection of the circles become non-definable

as circles become parallel lines. But in general, the conformality theorem can be applied.

Remark. The map f(z) = zn, for n ∈ Z≥2, is conformal except at 0. Note that f is a single-

valued function, without the need to take holomorphic branches.

Remark. Let α ∈ R>0, and consider zα =‖z‖α eiαθ, for θ ∈ (0, 2π). Then zα is conformal in the

plane cut along [0,∞). The position of the cut may be altered with respect to the region

that we consider. For example, to find a conformal mapping from C/(−∞, 0] into the open

right half-plane, we would cut the plane along (−∞, 0] and take the branch z12 := ‖z‖

12 e

iθ2 ,

where z = ‖z‖ eiθ, θ ∈ (−π, π). (Due to the position of the cut, the value of θ is restricted to

(−π, π)) This branch is holomorphic, and conformal in C/(−∞, 0].

53

Remark. In the rest of Chapter 8, the usage of functions to map one region of C to another

well-behaved region of C are explained. It is a very nice section.

Theorem 12.8. There is no way to impose a restriction which selects θ(z) ∈ [arg z], ∀z ∈

C/0, s.t. θ is a continuous function of z. (This tells us that there can be no continuous loga-

rithm in C/0 as if there were, its imaginary part, an argument function, would be continuous

too.)

Definition 12.9. Take a multifunction [w(z)] such that w(z) is a non-empty subset of C for

each z in the domain of w. a ∈ C is a branch point of [w(z)] if, for sufficiently small r > 0, it

is not possible to choose f(z) ∈ [w(z)] s.t. f(z) is a continuous function on γ(a; r)∗ (the image

of the circle centre a and radius r).

Remark. Suppose we are given a multifunction [w(z)]. We want to select a function f(z) ∈

[w(z)], for each z in as large a domain as possible, so that f is holomorphic. (In particular, f

has to be continuous.)

Remark. For a branch point a of a multifunction, we introduce new polar variables (r, θ), where

for arbitrary z, z − a = reiθ. Multibranches are explained (but not defined) via example.

Definition 12.10. Let γ be a positively oriented closed contour with parameter interval [a, b].

We say z performs a circuit round γ if we allow z = γ(t) to vary with t continuously

increasing from the start value a to the final value b. It is a fact that as z, where z − a = reiθ,

performs a circuit round γ, (i) if a ∈ I(γ), then θ increases by 2π and (ii) if a ∈ O(γ), then θ

returns to its initial value.

Definition 12.11. Suppose we have a complete set of multibranches Fλλ∈Λ for the multi-

function [w(z)]. (The meaning of this statement is explained in page 109) Here, Λ is some finite

or infinite indexing set. Suppose a1, . . . , an ∈ C are the branch points of w(z), and let (rk, θk)

be the polar variables relative to the point ak for variable z, defined as (almost directly) before.

Let γ be a contour passing through none of a1, . . . , an. For k = 1, . . . , n, let Θk := θk + 2π if

ak ∈ I(γ), and Θk := θk if ak ∈ O(γ).

We write Fλ −→γFµ for µ ∈ Λ if Fµ(rk, θk) = Fλ(rk,Θk) for k = 1, . . . , n. We shall say that γ is

an admissible contour for [w(z)] if Fλ −→γFλ for all λ ∈ Λ. Otherwise, it is inadmissible.

54

Remark. The rest of this chapter focuses on choosing multifunctions nicely. For example, for a

multifunction [w(z)], by considering circuits round contours which exclude the various branch

points, suppose we have found which contours are admissible and which are inadmissible. We

now want to restrict the movement of z such that inadmissible contours are outlawed. We do

this by cutting the plane, which z is not allowed to cross. We do not remove the points of a cut

from the plane, but we do think of a cut as having two edges. An explanation of a holomorphic

branch is given.

55

13 Chapter 13 (Integration and Cauchy’s Theorem)

Definition 13.1. A complex function h is piecewise continuous on a compact interval [α, β]

in R if there exist points α = t0 < t1 < · · · < tn = β and continuous functions hk on [tk, tk+1]

s.t. h(t) = hk(t) for t ∈ (tk, tk+1), for k = 0, . . . , n− 1. h need not be defined at all or some of

the points tk.

Definition 13.2. Let γ be a path with parameter interval [α, β]. There exist points α = t0 <

t1 < · · · < tn = β s.t. γ, restricted to each [tk, tk+1], coincides with a continuously differentiable

function on [tk, tk+1]. At intermediate points tk, γ′ might not exist. Let f : γ∗ → C be

continuous. Then, define∫γf(z)dz :=

∫ βαf(γ(t))γ′(t)dt. We call this the integral of f along

γ, or the integral of f round γ if γ is closed.

Theorem 13.3 (Integrals along paths). Suppose γ is a path with parameter interval [α, β] and

that f : γ∗ → C is continuous. Then the following hold:

(a) Reversal.∫−γ f(z)dz = −

∫γf(z)dz

(b) Joining. Let γ1, γ2 join to form γ. Then,∫γf(z)dz =

∫γ1f(z)dz +

∫γ2f(z)dz.

(c) Reparametrization. Let γ be another path, with parameter interval [α, β]. Let ψ be a

strictly increasing bijection mapping [α, β] to [α, β] which has a positive continuous derivative.

Suppose γ = γ ψ. Then,∫γf(z)dz =

∫γf(z)dz. So the value of

∫γf(z)dz only depends on the

image γ∗ and the direction that γ is traced. The speed(s) at which γ is traced bears no influence

on the value of the integral.

Theorem 13.4 (Integrals along a join of paths). Suppose γ is a path with parameter inter-

val [α, β] and is the join of paths γ1, γ2, . . . , γn, and that f : γ∗ → C is continuous. Then,∫γf(z)dz =

∑nk=1

∫γkf(z)dz.

Theorem 13.5 (Fundamental theorem of calculus). γ is a path with parameter interval [α, β],

and F : G → C s.t. G is an open set and γ∗ ⊂ G. Suppose F ′(z) exists and is continuous at

each point of γ∗. Then,∫γF ′(z)dz = F (γ(β))− F (γ(α)).

Remark. The fundamental theorem of calculus, in complex analysis, is an interim result which

is superseded by Cauchy’s theorem and its consequences. That is not the case in real analysis.

56

Theorem 13.6 (Estimation theorem). γ is a path with parameter interval [α, β] and f : γ∗ → C

is continuous. Then,∥∥∥∫γ f(z)dz

∥∥∥ ≤ ∫ βα

∥∥f(γ(t))γ′(t)∥∥ dt. In particular, if

∥∥f(z)∥∥ ≤ M for

∀z ∈ γ∗, then∥∥∥∫γ f(z)dz

∥∥∥ ≤ M ×∫ βα

∥∥γ′(t)∥∥ dt. Note that∫ βα

∥∥γ′(t)∥∥ dt is defined to be the

length of γ.

Definition 13.7. Cauchy’s theorem may fail for a function f which is not holomorphic inside

of or on the contour. We say that f is holomorphic inside and on a contour γ if f ∈ H(G)

for some open set G s.t. γ∗ ∪ I(γ) ⊂ G.

Theorem 13.8 (Cauchy’s theorem for a triangle). f is holomorphic on an open set G which

contains a triangle (i.e. triangular contour) γ and I(γ). Then,∫γf(z)dz = 0.

Theorem 13.9 (Indefinite integral theorem 1). Recall that for a, z ∈ C, [a, z] is the line

segment linking a, z. Let f be a continuous complex-valued function on a convex region G

s.t.∫γf(z)dz = 0 for any triangle γ in G. Let a be an arbitrary fixed point of G. Then,

F (z) :=∫

[a,z]f(w)dw is holomorphic in G, with F ′ = f .

Theorem 13.10. Let G be a convex region and let f ∈ H(G). Then there exists F ∈ H(G)

s.t. F ′ = f .

Theorem 13.11 (Cauchy’s theorem for a contour). f is holomorphic inside and on a closed

contour γ. Then,∫γf(z)dz = 0.

Theorem 13.12 (Deformation theorem 1). The following hold:

(a) Suppose γ is a positively oriented contour and that D(a; r) ⊂ I(γ). Let f be holomorphic

inside and on γ except possibly at a. Then∫γf(z)dz =

∫γ(a;r)

f(z)dz.

(b) Suppose γ1, γ2 are positively oriented contours such that γ2 lies inside γ1, that is, γ∗2∪I(γ2) ⊂

I(γ1). Let f be holomorphic inside and on γ1. Then,∫γ1f(z)dz =

∫γ2f(z)dz.

(c) Suppose γ1, γ2 are circline paths with a common initial point and common final point. Let

γ := γ1 ∪ (−γ2), and suppose that γ is simple. Let f be holomorphic inside and on γ. Then∫γ1f(z)dz =

∫γ2f(z)dz.

Theorem 13.13 (Logarithm in a convex region). Suppose that G is a convex region not con-

taining 0. Then there exists a function f = logG ∈ H(G) s.t. ef(z) = z for all ∀z ∈ G and

57

f(z) − f(a) =∫γ

1wdw for ∀a, z ∈ G, where γ is any path in G with endpoints a and z. The

function f is uniquely determined up to the addition of an integer multiple of 2πi. Furthermore,

for each z ∈ G, logG z = log‖z‖+ iθ(z), where θ(z) ∈ [arg z] and θ(z) is a continuous function

in G.

Definition 13.14. G ⊂ C is a non-empty open set in C and let γ, γ be closed paths in G.

We say γ can be obtained from γ by an elementary deformation if there exist open convex

subsets G0, G1, . . . , GN−1 of G s.t. γ can be expressed as the join of paths γ1, γ2, . . . , γN−1 and

γ as the join of paths γ1, γ2, . . . , γN−1 in such a way that, for k = 0, . . . , N − 1, γk and γk lie

in Gk and have common initial and final points. Furthermore, closed paths γ′1, γ′2 in G are

said to be to be homotopic in G if γ′2 can be obtained from γ′1 by a finite number of elementary

deformations.

Remark. In Metric Spaces and Topology, homotopy was defined, at first sight, slightly differ-

ently. Let the notation be as above. Intuitively, I will explain this way: we can think of path

γ′1 as rubber band positioned over γ′∗. Then, path γ′2 is homotopic to γ′1 if the rubber band

can be slid and stretched to coincide with γ′2∗, correctly oriented, without ever moving outside

G. This is non-trivially equivalent to the definition of homotopy presented here.

Definition 13.15. A path γ lying in a set G is said to be null if γ∗ = a for some a ∈ G. A

region G is simply connected if every closed path in G is homotopic to a null path in G.

Theorem 13.16 (Deformation theorem 2). Suppose f is holomorphic in an open set G and

that γ, γ are homotopic closed paths in G. Then,∫γf(z)dz =

∫γf(z)dz.

Theorem 13.17 (Cauchy’s theorem 2). Suppose f is holomorphic in a simply connected region

G. Then,∫γf(z)dz = 0 for every closed path γ in G.

Theorem 13.18 (Antiderivative theorem 2). Let G be a simply connected region and let f ∈

H(G). Then there exists F ∈ H(G) s.t. F ′ = f .

Theorem 13.19 (Logarithm in a simply connected region). Suppose that G is a simply con-

nected region not containing 0. Then there exists a function f = logG ∈ H(G) s.t. ef(z) = z for

all ∀z ∈ G and f(z) − f(a) =∫γ

1wdw for ∀a, z ∈ G, where γ is any path in G with endpoints

a and z. The function f is uniquely determined up to the addition of an integer multiple of

58

2πi. Furthermore, for each z ∈ G, logG z = log‖z‖ + iθ(z), where θ(z) ∈ [arg z] and θ(z) is a

continuous function in G.

Definition 13.20. Let γ be a closed path and let w /∈ γ∗. Define the index, or winding

number, n(γ, w) of γ around w by n(γ, w) := 12πi

∫γ

1z−wdz.

Remark. A contour, or more generally a simple closed path, is positively oriented if n(γ, w) =

1 for any w ∈ I(γ). For any closed path γ, we clearly have n(−γ, w) = −n(γ, w). Because

a contour is simple, it cannot wind around the same point more than once. The idea of the

winding number is that n(γ, w) should measure the number of times γ winds around w, taking

orientation into account. This is justified in the next theorem. For w /∈ γ∗, let γw(t) = γ(t)−w.

Then n(γ, w) = n(γw, 0). We may, therefore, without loss of generality, take w = 0 in the next

theorem.

Theorem 13.21 (Properties of the winding number). Let γ be a closed path with parameter

interval [α, β] and let 0 /∈ γ∗. Then,

(i) n(γ, 0) is an integer, where 2πin(γ, 0) =∫γz−1dz

(ii) There exists a continuous function µ : [α, β] → R, unique up to an integer multiple of 2π,

such that (a) 2πn(γ, 0) = µ(β)− µ(α) and (b) µ(t) ∈ [arg(γ(t))] for all ∀t ∈ [α, β].

Theorem 13.22 (Continuous selection of argument). We call the function µ in theorem 13.21

a continuous selection of argument along γ. Note that µ is required to vary continuously

with t, rather than with z = γ(t). When µ(γ, 0) 6= 0, we cannot find a continuous argument

function of z on γ∗ since no choice from [arg z] at z = γ(α) = γ(β) is compatible with continuity.

Theorem 13.23 (Cauchy’s theorem). Let G be a region and let f ∈ H(G). Then∫γf(z)dz = 0

for any closed path γ in G s.t. n(γ, w) = 0 for ∀w /∈ G.

Theorem 13.24. Let G be a region. Then the following are equivalent:

(i) G is simply connected

(ii) n(γ, w) = 0 for all closed paths in G and for all w /∈ G

(iii)∫γf(z)dz = 0 for all closed paths γ in G and all f ∈ H(G)

(iv) Each f ∈ H(G) has an antiderivative (i.e. f = F ′ for some F ∈ H(G))

59

(v) Given any f ∈ H(G) with f : G → C\0, there exists a holomorphic logarithm of f , i.e.

there exists g ∈ H(G) s.t. eg = f .

60

14 Chapter 14 (Power Series Representations and Zeros

of Holomorphic Functions)

Remark. In Cauchy’s theorem, the orientation of the contour did not need to be specified. In

Cauchy’s integral formula, all subsequent results giving formulae for integrals (whose values

are in general non-zero), the contour is taken to be positively oriented, without explicitly

mentioning it in every single theorem.

Theorem 14.1 (Cauchy’s integral formula). Let f be holomorphic inside and on a positively

oriented contour γ. Then, if a is inside γ (i.e. a ∈ I(γ)), then f(a) = 12πi

∫γf(w)w−adw.

Theorem 14.2 (Liouville’s theorem). Let f be holomorphic and bounded in the complex plane

C. Then, f is constant. (So a bounded, entire function is constant)

Theorem 14.3 (Fundamental theorem of algebra). Let p(z) be a non-constant polynomial

with constant coefficients. Then there exists ζ ∈ C s.t. p(ζ) = 0. Consequently, a complex

polynomial with degree n > 1 has n roots (not necessarily distinct) in C.

Theorem 14.4 (The existence of derivatives). Suppose that f is holomorphic in an open set

G. Then,

(i) f ′ ∈ H(G)

(ii) f has derivatives of all orders in G.

Theorem 14.5 (Morera’s theorem). Suppose f is continuous on an open set G, and∫γf(z)dz =

0 for all triangles γ in G. Then f ∈ H(G).

Theorem 14.6 (Cauchy’s formula for derivatives). Let f be holomorphic inside and on a

positively oriented contour γ and let a ∈ I(γ). Then f (n) exists for all n ∈ Z≥1, and f (n)(a) =

n!2πi

∫γ

f(w)(w−a)n+1dw.

Theorem 14.7 (Interchange theorem (simple form)). Let γ be a path, and u0, u1, . . . be con-

tinuous on γ∗, and assume that∑∞

k=0 uk(z) converges pointwise to U(z), for ∀z ∈ γ∗. Assume

that there exist constants Mk s.t. Mk converges and∥∥uk(z)

∥∥ ≤ Mk for ∀z ∈ γ∗. Then,∑∞k=0

∫γuk(z)dz = (

∫γU(z)dz :=

∫γ

∑∞k=0 uk(z)dz).

61

Theorem 14.8 (Coefficients in a power series). For a ∈ C, let f(z) =∑∞

k=0 ck(z − a)k, where

the power series has ROC R > 0. Then, cn = 12πi

∫γ(a;r)

f(z)zn+1 for r ∈ [0, R), n ∈ Z≥0.

Theorem 14.9 (Taylor’s theorem). Let f ∈ H(D(a;R)), for R > 0. Then there exist unique

constants cn s.t. f(z) =∑∞

n=0 cn(z − a)n, for ∀z ∈ D(a;R). The constant cn is given by

cn = 12πi

∫γ

f(w)(w−a)n+1dw = f (n)(a)

n!. Here, γ is a circle γ(a;R) for some r ∈ (0, R), or is any

positively oriented simple closed path homotopic in D′(a;R) := D(a;R)\a to γ(a;R).

Remark. There is a nice section on page 164 for finding the Taylor expansion for a holomorphic

branch of the logarithm away from the branch cut in the complex plane.

Theorem 14.10. Let f be holomorphic in C, with taylor expansion f(z) =∑∞

n=0 cnzn valid

for all z ∈ C. Suppose there exist positive constants M,K and k ∈ Z+ s.t.∥∥f(z)

∥∥ ≤ M‖z‖k,

for ‖z‖ ≥ K. Then, f is a polynomial of degree at most k.

Remark. Taylor’s theorem shows that every holomorphic function in an open set G is analytic,

meaning that it is locally representable by power series.

Theorem 14.11 (Multiplication theorem for power series). Suppose f(z) =∑∞

n=0 anzn and

g(z) =∑∞

n=0 bnzn are complex power series with ROC R1 and R2, respectively. Let h(z) =∑∞

n=0 cnzn, where cn =

∑nr=0 arbn−r. Then

∑∞n=0 cnz

n has ROC at least R := minR1, R2,

and h(z) = f(z)g(z) for ‖z‖ < R.

Definition 14.12. Suppose fn is a sequence of complex-valued functions defined on some

set S. We write fn → f if fn converges pointwise on S to f , and fnu−→ f if fn converges

uniformly on S to f .

Definition 14.13. Suppose uk is a sequence of complex-valued functions defined on some

set S. Given∑∞

k=1 uk, let fn := u1 + · · · + un. We say∑∞

k=1 uk converges uniformly on a

set S if fn converges uniformly on S.

Theorem 14.14 (Weierstrass’ M-test). The series∑uk converges uniformly on S if there

exist real numbers Mk s.t. ∀k, ‖uk‖ ≤Mk for ∀z ∈ S and∑Mk converges.

Remark. Note the Weierstrass’ M-test being used in in theorem 14.7.

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Theorem 14.15 (Interchange theorem for uniformly convergent sequences and series). Let γ

be a path with parameter interval [α, β].

(i) Let fn be a sequence of continuous complex-valued functions that converges uniformly on

γ∗ to a (necessarily continuous) function f . Then,

limn→∞

∫γ

fn(z)dz =

∫γ

f(z)dz

(:=

∫γ

limn→∞

fn(z)dz

).

(ii) Let uk be a sequence of continuous complex-valued functions such that∑uk(z) converges

uniformly on γ∗. (Take fn :=∑n

k=1 uk, fnu−→ f .) Then,∫

γ

∞∑k=0

uk(z)dz =

∫γ

limn→∞

fn(z)dz = limn→∞

∫γ

fn(z)dz = limn→∞

n∑k=1

∫γ

uk(z)dz =∞∑k=1

∫γ

uk(z)dz.

Definition 14.16 (Zeros and their orders). Suppose f is holomorphic at a, i.e. f ∈ H(D(a; r))

for some r ∈ R>0. The point a is said to be a zero of f if f(a) = 0. We say that the zero a of

f is of order m if 0 = f(a) = f ′(a) = · · · = f (m−1)(a) and f (m)(a) 6= 0. Zeros of orders 1, 2, . . .

are called simple, double, . . . . For convenience, we use the convention that f has a zero of

order 0 at a if f is holomorphic at a and f(a) 6= 0.

Theorem 14.17 (Characterization theorem for zeros of order m). Let f ∈ H(D(a;R)) and

suppose that f has Taylor expansion f(z) =∑∞

n=0 cn(z − a)n in D(a;R). Then the following

are equivalent:

(i) 0 = f(a) = f ′(a) = · · · = f (m−1)(a) and f (m)(a) 6= 0

(ii) f(z) =∑∞

n=m cn(z − a)n, where cm 6= 0

(iii) f(z) = (z − a)mg(z), where g ∈ H(D(a;R)) and g(a) 6= 0

(iv) There exists a non-zero constant C ∈ C s.t. limz→a(z − a)−mf(z) exists and equals C.

Theorem 14.18. If f, g have zeros of order m,n ≥ 0 at a, then fg has a zero of order m+ n

at a.

Theorem 14.19 (Identity theorem). Let G be a region and suppose f ∈ H(G). Assume the

set Z(f) of zeros of f has a limit point in G. Then, f is identically zero in G. (Therefore, the

zeros of a non-constant holomorphic function are isolated.)

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Theorem 14.20 (Uniqueness theorem). Suppose G is a region, and f, g belong to H(G), and

that f(z) = g(z) for ∀z ∈ S, where S has a limit point in G. Then f ≡ g in G.

Theorem 14.21 (Counting zeros). Let f be holomorphic inside and on a positively oriented

contour γ. Let f be non-zero on γ and have N zeros inside γ. Then, 12πi

∫γf ′(z)f(z)

dz = N . (N is

the order of all the zeros summed, i.e. a zero of order m is counted m times.)

Theorem 14.22 (Rouche’s theorem). Let f, g be holomorphic inside and on a contour γ and

suppose that∥∥f(z)

∥∥ >∥∥g(z)∥∥ on γ∗. Then f and f + g have the same number of zeros inside

γ.

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15 Chapter 15 (Modulus theorems, Meromorphic Func-

tions, and Cauchy’s Residue Theorem)

Theorem 15.1 (Local maximum modulus theorem). Suppose that f ∈ H(D(a;R)) and that∥∥f(z)∥∥ ≤∥∥f(a)

∥∥ for all z ∈ D(a;R). Then f is constant on D(a;R).

Theorem 15.2 (Maximum modulus theorem). Let G be a bounded region and let f be holo-

morphic in G and continuous on the closure G of G. Then ‖f‖ attains its maximum on the

boundary ∂G := G\G.

Theorem 15.3 (Schwartz lemma). Suppose that f is holomorphic in D(0;R), that f(0) = 0,

and that∥∥f(z)

∥∥ ≤ M in D(0;R). Then,∥∥f(z)

∥∥ ≤ MR‖z‖, for ‖z‖ ≤ R. If equality occurs for

some z s.t. ‖z‖ < R, then there exists a real constant λ s.t. f(z) = Mzeiλ/R for x ∈ D(0;R).

Theorem 15.4. Suppose that G is an open set and that f ∈ H(G). Let a ∈ G.

(i) Assume that G is a region and that f is non-constant. Then, in some D′(a; r), the function

f − f(a) is never zero.

(ii) Let f be an injection. Then f ′ cannot be identically zero and hence can only have isolated

zeros.

(iii) Choose r s.t. D(a; r) ⊂ G and suppose that f − f(a) is non-zero on γ∗, where γ = γ(a; r).

Let m := inf∥∥f(z)− f(a)

∥∥ : z ∈ γ∗. Then, (a) m > 0 and (b) for each w ∈ D(f(a);m), the

functions f − f(a) and f −w have the same number of zeros, counted according to multiplicity.

Theorem 15.5. Suppose that f is a holomorphic injection in an open set G. Then, f is

conformal in G.

Theorem 15.6 (Open mapping theorem). Suppose that f is holomorphic and non-constant in

an open set G. Then f(G) is open.

Theorem 15.7 (Inverse function theorem). Let G be an open set and let f be holomorphic and

one-to-one in G. Then f−1 is holomorphic in f(G).

Theorem 15.8 (Conformality of invertible maps). Let G be a region and let f ∈ H(G).

Suppose that f maps G one-to-one and onto (i.e. ’bijectively’) the region G := f(G), so there

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exists a well-defined inverse map f−1 : G→ G. Then we have

(i) f is conformal

(ii) f−1 is conformal

There is a partial converse: if f is conformal in a region G, then f is locally one-to-one in G.

Theorem 15.9 (Riemann mapping theorem). Let G be a simply connected region with G 6= C.

Then there exists a one-to-one conformal mapping f from G onto D(0; 1) with f−1 : D(0; 1)→

G also conformal.

Definition 15.10. By definition, a series∑∞

n=−∞ an converges to s1 + s2 if∑∞

n=0 an converges

to s1 and∑∞

n=1 a−n converges to s2.

Theorem 15.11 (Laurent’s theorem). Let A = z ∈ C : R <‖z − a‖ < S, (0 ≤ R < S ≤ ∞)

and let f ∈ H(A). Then, f(z) =∑∞

n=−∞ cn(z − a)n, for ∀z ∈ A, where cn = 12πi

∫γ

f(w)(w−a)n+1dw

with γ = γ(a; r) (R < r < S) or γ any closed path in A homotopic to γ(a; r).

Theorem 15.12 (Uniqueness of the Laurent expansion). Let f ∈ H(A), where A is the annulus

z ∈ C : R < ‖z − a‖ < S (0 ≤ R < S ≤ ∞) and suppose that f(z) =∑∞

n=−∞ dn(z − a)n,

(∀z ∈ A). Then dn = cn for ∀n ∈ Z, where cn is as in theorem 15.11.

Theorem 15.13 (Estimating Laurent coefficients). Suppose that f is holomorphic in an an-

nulus z ∈ C : R < ‖z‖ < S. Let f have a Laurent expansion f(z) =∑∞

n=−∞ cnzn, where

2πicn =∫γ(0;r)

f(w)w−n−1dw (R < r < S). Then, ‖cn‖ ≤ r−n sup∥∥f(z)

∥∥ :‖z‖ = r.

Definition 15.14 (Singularities). We say a is a regular point of f if f is holomorphic at a.

A point a is a singularity of f if a is a limit point of regular points which is not itself regular.

If a is a singularity of f and f is holomorphic in some punctured disc D′(a; r), r > 0, then a is

an isolated singularity. Otherwise a is a non-isolated (essential) singularity.

Remark. Suppose f has an isolated singularity at a. Then f is holomorphic in some annulus

z : 0 < ‖z − a‖ < r and has there a unique Laurent expansion f(z) =∑∞

n=−∞ cn(z − a)n.

We may write f(z) =∑−1

n=−∞ cn(z − a)n +∑∞

n=0 cn(z − a)n. The second term on the RHS is

holomorphic in D(a; r) and is in no way responsible for the singularity. This is caused by the

first sum,∑−1

n=−∞ cn(z− a)n, which is known as the principal part of the Laurent expansion.

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Definition 15.15. We classify isolated singularities according to the behaviour of the co-

efficients cn for n < 0. The feasibility of such classification relies on both the existence and

uniqueness of the Laurent expansion. The point a is said to be:

(i) a removable singularity if cn = 0 for ∀n < 0

(ii) a pole of order m (m ≥ 1) if c−m 6= 0 and cn = 0 for ∀n < −m

(iii) an isolated essential singularity if there does not exist m s.t. cn = 0 for all n < −m.

Poles of orders 1, 2, 3, . . . are called simple, double, triple, . . .

Theorem 15.16 (Characterization theorem for poles of order m). Let f ∈ H(D′(a; r)). Then

f has a pole of order m at a if and only if limz→a(z−a)mf(z) = D, where D is a finite non-zero

constant.

Theorem 15.17 (Poles and Zeros). Suppose that f is holomorphic in some open disc D(a; r).

Then f has a zero of order m if and only if 1f

has a pole of order m at a.

Theorem 15.18 (Cancellation and coalescence of zeros and poles). Suppose that f has a pole

of order m at a.

(i) Suppose that g ∈ H(D(a; r)) for some r > 0. Then at point a the function fg has (a) a pole

of order m if g(a) 6= 0, (b) a pole of order m− n if g has a zero of order n < m at a, and (c)

a removable singularity if g has a zero of order n ≥ m at a.

(ii) Suppose that g has a pole of order n at a. Then fg has a pole of order m+ n at a.

Remark. Page 204, section 17.15 explains where a ‘removable’ singularity gets its name.

Theorem 15.19 (Behaviour near a non-removable isolated singularity). Let f have an isolated

singularity at a and have Laurent expansion f(z) =∑∞

n=−∞ cn(z − a)n (0 <‖z − a‖ < r).

(i) If f has a pole at a, f(z)→∞ as z → a.

(ii) Suppose that f has an isolated essential singularity at a. Let w be any complex number.

Then there exists a sequence an s.t. an → a and f(an) → w. This is the Casorati-

Weierstrass theorem.

Remark (Extending our notions). Let us extend our investigation of singularities to the extended

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complex plane C. Consider a function f defined on some set z ∈ C : ‖z‖ > r but not

necessarily at point ∞. Define f(w) := f( 1w

), for ∀w ∈ D′(0; 1r), and let f(0) = f(∞) if f(∞)

is defined. If f(w) has a pole of order m at w = 0, then f(z) has a pole of order m at z =∞.

Definition 15.20 (Meromorphic function). Let G be an open subset of C or C. A complex

valued function which is holomorphic in G except possibly for poles is said to be meromorphic

in G.

Theorem 15.21 (Meromorphic functions in C). The following hold:

(i) Let f be holomorphic in C. Then f is constant.

(ii) Let f be meromorphic in C. Then f is a rational function.

Lemma 15.22 (Integration round a pole). Let f be holomorphic inside and on a positively

oriented contour γ except at the point a inside γ, where it has a pole of order m. Let f(z) =∑∞n=−m cn(z − a)n be the unique Laurent expansion of f about a. Then

∫γf(z)dz = 2πic−1.

Definition 15.23 (Residue). Suppose that f ∈ H(D′(a; r)) and that f has a pole at a. The

residue of f at a is the unique coefficient c−1 of (z− a)−1 in the Laurent expansion of f about

a, and is denoted resf(z); a.

Theorem 15.24 (Cauchy’s residue theorem). Let f be holomorphic inside and on a positively

oriented contour except for a finite number of poles, a1, . . . , aN inside γ. Then∫γf(z)dz =

2πi∑N

k=1 resf(z); ak.

Remark. The rest of this chapter focuses on the calculation of residues.

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