化工應用數學

授課教師： 郭修伯

Lecture 5 Solution by series (skip)Complex algebra

Infinite series

• They can be accepted as solutions if they are convergent.

– As n, SnS (some finite number), the series is “convergent”.

– As n, Sn ±, the series is “divergent”.

– In other cases, the series is “oscillatory”.

nn uuuuS ...321

Properties of infinite series

• If a series contains only positive real numbers or zero, it must be either convergent or divergent.

• If a series is convergent, then un 0, as n .• If a series is absolutely convergent, then it is also

convergent.– If the series is convergent,

it is absolutely convergent.

||...|||||| 321 nuuuu

Power Series

• A power series about x0 is:

• Not every differential equation can be solved using power series method. This method is valid if the coefficient functions in the

differential equation are analytical at a point • Taylor series:

– Maclaurin series (about zero)

• Frobenius series:

0

0 )(n

nn xxa

)(!

10

)( xfk

a kk

0

0 )(n

rnn xxcy

Important topics in “series”

• Method of Frobenius– The differential equation

• Bessel’s equation– The equation arises so frequently in practical

problems that the series solutions have been standardized and tabulated.

0)()(2

22 yxG

dx

dyxxF

dx

ydx

0n

rnn xaycan be solved by putting

Special Functions

• Bessel’s equation of order

– occurs in studies of radiation of energy and in other contexts, particularly those in cylindrical coordinates

– Solutions of Bessel’s equation• when 2 is not an integer

• when 2 is an integer

– when = n + 0.5– when = n + 0.5

0)( 222 yxyxyx

0

22 )1(!2

)1()(

n

nn

n

xnn

xJ )()()( 21 xJcxJcxy

)()()( 5.025.01 xJcxJcxy nn

Complex algebra

)sin(cos iryixz r

x

y

zwq

zwq

zwp

zwp

zwzw

/

/

Properties :

De Moivre’s theorem:

For all rational values of n,

nini n sincos)sin(cos

Note: is not included!

xix

xxxi

xx

xixxixe

n

yyyye

ix

ny

sincos

...!5!3

...!4!2

1

...!4!3!2

1

...!

...!3!2

1

5342

432

32

)sin(cos yiyee xiyx

i

eex

eex

ixix

ixix

2sin

2cos

Hyperbolic functions

xix

xi

ix

ixix

xiee

iix

xee

ix

xx

xx

tanhcosh

sinh

cos

sintan

sinh2

sin

cosh2

cos

Complex numbersandTrigonometric-exponential identities

Derivatives of a complex variable

Consider the complex variable to be a continuous function,and let and .Then the partial derivative of w w.r.t. x, is:

)(zfw ivuw iyxz

x

vi

x

u

x

w

ordz

df

x

z

dz

df

x

w

x

vi

x

u

dz

df

Similarily, the partial derivative of w w.r.t. y, is:

y

vi

y

u

y

w

or dz

dfi

y

z

dz

df

y

w

y

vi

y

u

dz

dfi

x

v

y

uand

y

v

x

u

Cauchy-Riemann conditions

They must be satisfied for the derivative of a complex number to have any meaning.

Analytic functions

• A function of the complex variable is called an analytic or regular function within a region R, if all points z0 in the region satisfies the following conditions:– It is single valued in the region R.– It has a unique finite value.

– It has a unique finite derivative at z0 which satisfies the Cauchy-Riemann conditions

• Only analytic functions can be utilised in pure and applied mathematics.

)(zfw iyxz

If w = z3, show that the function satisfies the Cauchy-Riemann conditions and state the region wherein the function is analytic.

322333 33)( iyxyyixxiyxzw

)3()3( 3223 yyxixyxw ivuw

xyy

u

yxx

u

6

33 22

xyx

v

yxy

v

6

33 22

x

v

y

uand

y

v

x

u

Cauchy-Riemann conditions

Satisfy!

Also, for all finite values of z, w is finite.Hence the function w = z3 is analytic in any region of finite size.(Note, w is not analytic when z = .)

If w = z-1, show that the function satisfies the Cauchy-Riemann conditions and state the region wherein the function is analytic.

221 )(

))((

)(1

yx

iyx

iyxiyx

iyx

iyxzw

)()(2222 yx

yi

yx

xw

ivuw

222

222

22

)(

2

)(

yx

xy

y

u

yx

xy

x

u

x

v

y

uand

y

v

x

u

Cauchy-Riemann conditions

Satisfy!Except from the origin

For all finite values of z, except of 0, w is finite.Hence the function w = z-1 is analytic everywhere in the z planewith except of the one point z = 0.

222

222

22

)(

2

)(

yx

xy

x

v

yx

xy

y

v

?

2

1

xx

u

At the origin, y = 0x

u1

)()(2222 yx

yi

yx

xivuw

2

1

yy

v

At the origin, x = 0y

v1

As x tends to zero through either positive or negative values, it tends to negative infinity.

As y tends to zero through either positive or negative values, it tends to positive infinity.

y

v

x

u

Consider half of the Cauchy-Riemann condition , which is not satisfied at the origin.

Although the other half of the condition is satisfied, i.e. 0

x

v

y

u

Singularities

• We have seen that the function w = z3 is analytic everywhere except at z = whilst the function w = z-1 is analytic everywhere except at z = 0.

• In fact, NO function except a constant is analytic throughout the complex plane, and every function except of a complex variable has one or more points in the z plane where it ceases to be analytic.

• These points are called “singularities”.

Types of singularities

• Three types of singularities exist:– Poles or unessential singularities

• “single-valued” functions

– Essential singularities• “single-valued” functions

– Branch points• “multivalued” functions

Poles or unessential singularities

• A pole is a point in the complex plane at which the value of a function becomes infinite.

• For example, w = z-1 is infinite at z = 0, and we say that the function w = z-1 has a pole at the origin.

• A pole has an “order”:– The pole in w = z-1 is first order.

– The pole in w = z-2 is second order.

The order of a pole

If w = f(z) becomes infinite at the point z = a, we define:

)()()( zfazzg n where n is an integer.

If it is possible to find a finite value of n which makes g(z) analytic at z = a,then, the pole of f(z) has been “removed” in forming g(z).The order of the pole is defined as the minimum integer value of n for whichg(z) is analytic at z = a.

比如： 在原點為 pole, (a=0) z

w1

)(1

)( zgz

z n 則

n 最小需大於 1 ，使得 w 在原點的 pole 消失。

Order = 1

什麼意思呢？

6.34.2 )(

1

azzw

在 0 和 a 各有一個 pole ，則 w在 0 這個 pole 的 order 為 3在 a 這個 pole 的 order 為 4

Essential singularities

• Certain functions of complex variables have an infinite number of terms which all approach infinity as the complex variable approaches a specific value. These could be thought of as poles of infinite order, but as the singularity cannot be removed by multiplying the function by a finite factor, they cannot be poles.

• This type of sigularity is called an essential singularity and is portrayed by functions which can be expanded in a descending power series of the variable.

• Example: e1/z has an essential sigularity at z = 0.

Essential singularities can be distinguished from poles by the fact thatthey cannot be removed by multiplying by a factor of finite value.

Example:..

!

1...

!2

111

22/1

nznzzew infinite at the origin

We try to remove the singularity of the function at the origin by multiplying zp

..!

...!2

21

n

zzzzwz

nppppp

It consists of a finite number of positive powers of z, followed by an infinite number of negative powers of z.

All terms are positive

wzzAs p,0

It is impossible to find a finite value of p which will remove the singularity in e1/z at the origin.The singularity is “essential”.

Branch points

• The singularities described above arise from the non-analytic behaviour of single-valued functions.

• However, multi-valued functions frequently arise in the solution of engineering problems.

• For example:

2

1

zw irez i

erw 2

1

2

1

z w

For any value of z represented by a point on the circumference of the circle in the zplane, there will be two corresponding values of w represented by points in the w plane.

ivuw

2

1

zw irez i

erw 2

1

2

1

2

1sin

2

1

2

1cos

2

1

ru

rr

u

sincos iei

2

1cos2

1

ru 2

1sin2

1

rv

2

1cos

2

1

2

1sin

2

1

rv

rr

v

and

u

rr

vand

v

rr

u 11

Cauchy-Riemann conditions in polar coordinates

when 0 2

A given range, where the functionis single valued: the “branch”

The particular value of z at whichthe function becomes infinite or zerois called the “branch point”.

The origin is the branch point here.

• A function is only multi-valued around closed contours which enclose the branch point.

• It is only necessary to eliminate such contours and the function will become single valued.– The simplest way of doing this is to erect a barrier from the

branch point to infinity and not allow any curve to cross the barrier.

– The function becomes single valued and analytic for all permitted curves.

Branch point

Barrier - branch cut

• The barrier must start from the branch point but it can go to infinity in any direction in the z plane, and may be either curved or straight.

• In most normal applications, the barrier is drawn along the negative real axis.– The branch is termed the “principle branch”.– The barrier is termed the “branch cut”.– For the example given in the previous slide, the region,

the barrier confines the function to the region in which the argument of z is within the range - < < .

The successive values of a complex variable z can be representedby a curve in the complex plane, and the function w = f (z) will haveparticular value at each point on this curve.

Integration of functions of complex variables

• The integral of f(z) with respect to z is the sum of the product fM(z)z along the curve in the complex plane:

CMz

dzzfzzf )()(lim0

where fM(z) is the mean value of f(z) in the length z of the curve;and C specifies the curve in the z plane along which the integrationis performed.

iyxzandzfivuw )(

CC

CC

udyvdxivdyudx

idydxivudzzf

)()(

))(()(

When w and z are both real (i.e. v = y = 0):

Cudx

This is the form that we have learnt about integration; actually, this is only a special case of a contour integration along the real axis.

Cauchy’s theorem

• If any function is analytic within and upon a closed contour, the integral taken around the contour is zero.

0)( C dzzf

If KLMN represents a closed curve and there are no singularities of f(z)within or upon the contour, the value of the integral of f(z) around the contour is:

CCC

udyvdxivdyudxdzzf )()()(

Since the curve is closed, each integral on the right-hand side can berestated as a surface integral using Stokes’ theorem:

AC

AC

dxdyy

v

x

uudyvdx

dxdyy

u

x

vvdyudx

)(

)(

But for an analytic function, each integral on the right-hand side iszero according to the Cauchy-Riemann conditions

0)( C dzzf

Stokes’ theorem

AC

dxdyy

P

x

QQdyPdx )(

Integral of f(z) between two points

• The value of an integral of f(z) between two points in the complex plane is independent of the path of integration, provided that the function is analytic everywhere within the region containing all of the paths.

P

Q

Show that the value of z2 dz between z = 0 and z = 8 + 6i is the same whether the integration is carried out along the path AB or around thepath ACDB.

A

C

B

D

The path of AB is given by the equation:

xy4

3 222

16

247)

4

3()( x

iixxiyxz

idxdxdz4

3

3

936352

4

34

16

24768

0

28

0

2 idxx

iidzz

i

Consider the integration along the curve ACDB

Along AC, x = 0, z = iy

3

100010

0

210

0

2 idyyidzz

ii

Along CDB, r = 10, z = 10ei

3

64352101004

3tan

2

1268

10

21 i

dieedzz iii

i

3

936352 i

Independent of path

Evaluate around a circle with its centre at the originC z

dz2

Let z = rei

02

0

2

0

2

0 222

i

e

r

ide

r

i

er

dire

z

dz ii

i

i

C

Although the function is not analytic at the origin, 02C z

dz

Evaluate around a circle with its centre at the originC z

dz

Let z = rei

iire

dire

z

dzi

i

C

220

2

0

Cauchy’s integral formula

C

a

A complex function f(z) is analytic upon and within the solid linecontour C. Let a be a point within the closed contour such that f(z) isnot zero and define a new function g(z):

az

zfzg

)()(

g(z) is analytic within the contour C except at the point a (simple pole).

If the pole is isolated by drawing a circle around a and joining to C,the integral around this modified contour is 0 (Cauchy’s theorem).

The straight dotted lines joining the outside contour C and the innercircle are drawn very close together and their paths are synonymous.

Since integration along them will be in opposite directions and g(z) isanalytic in the region containing them, the net value of the integralalong the straight dotted lines will be zero:

C az

dzzf

az

dzzf

0)()(

Let the value of f(z) on be ; where is a small quantity. )()( afzf

C az

dz

az

dzaf

az

dzzf

0

)()(

0, where is small

ireaz

2

0)(2)( aif

re

dreaif

i

i

C az

dzzf

iaf

)(

2

1)(

Cauchy’s integral formula: It permits the evaluationof a function at any point within a closed contour whenthe value of the function on the contour is known.

The theory of residues

• The theory of residues is an extension of Cauchy’s theorem for the case when f(z) has a singularity at some point within the contour C.

If a coordinate system with its origin at the singularity of f(z) and noother singularities of f(z). If the singularity at the origin is a pole oforder N, then:

)()( zfzzg N

will be analytic at all points within the contour C.g(z) can then be expanded in a power series in z and f(z) will thus be:

0

111 ...)(

n

nnN

NNN zC

z

B

z

B

z

Bzf Laurent expansion of the complex function

The infinite series of positive powers of z is analytic within and uponC and the integral of these terms will be zero by Cauchy’s theorem.

the residue of the function at the pole

If the pole is not at the origin but at z0 )( 0zzz

C

iBdzzf 12)(

Evaluate around a circle centred at the origin C

z

az

dze3)(

If |z| < |a|, the function is analytic within the contour

0)( 3

Cz

az

dzeCauchy’s theorem

If |z| > |a|, there is a pole of order 3 at z = a within the contour.Therefore transfer the origin to z = a by putting = z - a.

dede

az

dzeC

a

C

a

C

z

...!4!3

1

!2

111

)( 2333

non-zero term, residue = ae

2

1

aa

CieeiiBdzzf )

2

1(22)( 1

Evaluation of residues without the Laurent Expansion

)(

)()(

zg

zFzf

The complex function f(z) can be expressed in terms of a numeratorand a denominator if it has any singularities:

If a simple pole exits at z = a, then g(z) = (z-a)G(z)

...)(...)()( 101

nn azbazbb

az

Bzf Laurent expansion

multiply both sides by (z-a)

...)(...)()())(( 12101 n

n azbazbazbBazzf

az

azazzfB |))((1

)(

)(1 aG

aFB

Evaluate the residues of 122 zz

z

)3)(4(12)(

2

zz

z

zz

zzf Two poles at z = 3 and z = - 4

)(

)(1 aG

aFB

The residue at z = 3:B1= 3/(3+4) = 3/7

The residue at z = - 4:B1= - 4/(- 4 - 3) = 4/7

Evaluate the residues of 22 wz

e z

))(()(

22 iwziwz

e

wz

ezf

zz

Two poles at z = iw and z = - iw

)(

)(1 aG

aFB

The residue at z = iw:B1= eiw/2iw

The residue at z = - iw:B1= -eiw/2iw

If the denominator cannot be factorized, the residue of f(z) at z = a is

0

0

)(

)()(1

azzg

zFazB indeterminate

L’Hôpital’s rule

)('

)(

)('

)(')()(

/)(

/)()(1 zg

zF

zg

zFazzF

dzzdg

dzzFazdB azaz

Evaluate around a circle with centre at the origin

and radius |z| < /n

)(

)(

sin zg

zFdz

nz

eC

z

nnzn

e

nzdzd

eB z

z

z

z 1

cossin001

n

idznz

eC

z 12

sin

Evaluation of residues at multiple poles

If f(z) has a pole of order n at z = a and no other singularity, f(z) is:

naz

zFzf

)(

)()(

where n is a finite integer, and F(z) is analytic at z = a.

F(z) can be expanded by the Taylor series:

...)(!

)(...)(''

!2

)()(')()()(

2

aFn

azaF

azaFazaFzF n

n

Dividing throughout by (z-a)n

...)()!1(

)(...

)(

)('

)(

)()(

1

1

azn

aF

az

aF

az

aFzf

n

nnThe residue at z = a is thecoefficient of (z-a)-1

The residue at a pole of order n situated at z = a is:

azn

n

n

az

n

zfazdz

d

nn

aFB

)()()!1(

1

)!1(

)(1

11

1

Evaluate around a circle of radius |z| > |a|.dzaz

zC 3)(

2cos

3)(

2cos

az

z

has a pole of order 3 at z = a, and the residue is:

aaz

zaz

dz

d

zfazdz

d

nn

aFB

az

azn

n

n

az

n

2cos2)(

2cos)(

!2

1

)()()!1(

1

)!1(

)(

33

2

2

1

11

1

)2cos2(2)(

2cos3

aidzaz

zC