A Catalogue of Analytic Functions

7/29/2019 A Catalogue of Analytic Functions

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A catalogue of analytic functions.

Exponential in basis

Exponential in basis .

Definition 4.1.1 If , with real , we define:

.

For example, and .

Note that the main requirement is fulfilled:

Proposition 4.1.2

(i) The exponential is defined on and .(ii) The exponential is an entire function.

(iii) .

(iv) .

(v) .

Proof.

(i) Let . Then . As for any real number

, , we have .

(ii) As above, let . Then we have:


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It can be easily shown that the functions and verify the Cauchy-Riemann equationsover the whole plane. As the plane is open, the exponential function is differentiable at

every point of an open set, whence analytic at every point. This means that the function is

an entire function.

(iii) Denote and , with . Then we have:

(iv) Proceed as for (iii).

(v) For any , we have:

Example 4.1.3 Let and . Then:

Example 4.1.4 Solve the equation in .

Let , where are real numbers. We have:


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As , for any , we have , i.e. . We consider now

two cases: (i) If , with , we have . The first equation has one

solution, given by .

(ii) If , with , we have . The first equation implies now

that , and has no solution.

We conclude: the solution set of the given equation in is .

Example 4.1.5 Solve the equation in . Let , where are realnumbers. We have:

As , for any , we have , i.e. . We consider now

two cases: (i) If , for , we have . The second equationimplies , i.e. .

(ii) If , for , we have .The second equation

implies ,which has no real solution.

We conclude: the solution set of the given equation in is .

Trigonometric functions


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Trigonometric functions.

Definition 4.2.1

1. .

2. .

3. .

Example 4.2.2

Proposition 4.2.3

1. .

2.

3.

4.

5.

Example 4.2.4

Proposition 4.2.5

1. .

2. .

3. .

Example 4.2.6


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Hyperbolic functions

Hyperbolic functions.

Definition 4.3.1

1. .

2. .

3. .

Example 4.3.2

Proposition 4.3.3

1. .

2.

3.

4.

5. .

Example 4.3.4 (A nice equation) We solve the equation .

Let , where . Then:

1.
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2.

We have now:

3.

4. We take care of the second equation:

5.

6. We substitute into the first equation:

1. . This equation has no real solution.

2. . This equation is equivalent to

3.

Its (real) solutions are and .

4. . This equation has no solution.

In conclusion, the complex solutions of the equation are:

and

Example 4.3.5 (A nice equation - second way) We solve the equation .


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We substitute and solve for the equation . The solutions are:

and . In order to find the corresponding values of , we

need logarithms. They will be defined in the next paragraph (v.i. 4). This exercise will be

finished in example 4.6.

The logarithm of a complex number


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The logarithm of a complex number.

Definition 4.4.1

,

where is defined up to an additive multiple of .

Example 4.4.2

We denote Log the principal value of , i.e. the value corresponding to the principal value

of (recall that ).

Example 4.4.3

Log

Log

Proposition 4.4.4

1. .

2. .

3. .

Proposition 4.4.5 The logarithmic function is analytic on its domain.

For a proof, use Cauchy-Riemann equations (v.s. 3).

Example 4.4.6


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Analyticity of the logarithmic function

Analyticity of the logarithmic function.

Definition 4.5.1 Let be a multivalued function. A branch of is a single valued

function, which is analytic on some domain . At every point of , the value of the

branch is exactly one of the multiple values of at that point.

Example 4.5.2 The principal part Log of the function is a branch of , defined

over .

Complex exponentials

Complex exponentials.

Definition 4.6.1

For any and every

Example 4.6.2

.

Proposition 4.6.3 .

For the proof, we use the definition . Then we have:
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Proposition 4.6.4 .

For the proof, we use the definition . Then we have:

Inverse trigonometric functions

Inverse trigonometric functions.

Theorem 4.7.1 For any complex number :

Example 4.7.2

Inverse hyperbolic functions


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Inverse hyperbolic functions.

Theorem 4.8.1 For any complex number :

Example 4.8.2

A Catalogue of Analytic Functions

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