# How do you solve a complex integral

## Function theory - of complex relationships

### Summary

### Complex functions and differentiability

Many relationships known from the real also apply unchanged in the complex, but caution is advised, especially with general powers and logarithms. Complex differentiability is a much stronger condition than real.

### Exponential and trigonometric functions are defined by power series

The real power series definition of the elementary functions also applies unchanged to complex arguments. The logarithm can also be transferred to the complex, but there it is a function with several branches. Arc and area functions can be expressed using the complex logarithm.

### Complex differentiability is a strict condition

Although \ (\ mathbb {C} \) and \ (\ mathbb {R} ^ {2} \) are equivalent on the geometric level, algebraic properties are still important in \ (\ mathbb {C} \). Correspondingly, a differentiable function \ (\ mathbb {R} ^ {2} \ to \ mathbb {R} ^ {2} \) does not have to be complex differentiable. Complex differentiable functions satisfy the following equations in addition to real differentiability.

### Cauchy-Riemann equations

**Cauchy-Riemann equations**

From the Cauchy-Riemann equations it follows that the real and imaginary parts of holomorphic functions are harmonic, \ (\ Delta u = \ Delta v = 0 \). This has important applications in electrostatics, fluid mechanics, and other areas.

### Conforming images are given angles and orientation

In the complex, conformal mappings are precisely the holomorphic functions. This allows solutions to problems with complex geometry to be transferred to problems with simple geometry.

### Complex curve integrals

Integrals over complex-valued functions do not cause any difficulties, the linearity of the integration is carried over to complex-valued functions of a real variable. Most of the time, however, you want to integrate paths in the complex plane, which can be described using curves.

### Curve integrals can be reduced to integrals of complex-valued functions of a real variable

This is done through parameterization. An integral turns out to be particularly important.

### Central integral of function theory

### Cauchy's integral theorem makes statements about the path independence of curve integrals

### Cauchy's integral theorem

For every closed path \ (C \) \ (\ oint_ {C} f (z) dz = 0 \).

- For any two paths \ (C_ {1} \) and \ (C_ {2} \) with the same start and end point$$ \ displaystyle \ int_ {C_ {1}} f (z) \, \ mathrm {d} z = \ int_ {C_ {2}} f (z) \, \ mathrm {d} z \,. $$
- For \ (f (z) \) there is an antiderivative \ (F (z) \) with \ (\ frac {\ mathrm {d} F} {\ mathrm {d} z} = f \) and$$ \ displaystyle \ int_ {z_ {A} \ to z_ {E}} f (z) \, \ mathrm {d} z = F (z_ {E}) - F (z_ {A}) \,. $ $

### Cauchy's integral formula allows fundamental statements about holomorphic functions

Cauchy's integral formula can be derived from Cauchy's integral theorem. In this turn numbers \ (\ mathrm {Ind} _ {C} (z) \) appear, which describe how often a curve \ (C \) revolves around a certain point \ (z_ {0} \).

### Cauchy's integral formula

### Holomorphic functions have other amazing properties as well

The “inner cohesion” of holomorphic functions is very strong, for example the following sentence applies.

### Identity set

If two holomorphic functions in a domain \ (G \) coincide on a set that has an accumulation point in \ (G \), or if all derivatives of the two functions coincide at one point, then these functions in \ (G \) identical.

Every function is infinitely differentiable in its holomorphic domain. In the interior of the holomorphic area, there can be no real maximum values.

### Laurent series and residual theorem

Laurent series are a generalization of power series, in which negative powers are also included. A Laurent series converges, if at all, in a circular ring.

### Singularities can be classified using Laurent series

Depending on the appearance of the Laurent series, isolated singularities can be subdivided into liftable singularities, poles and essential singularities.

### The residual theorem allows integrals to be determined in the simplest possible way

### Residual theorem

### Real integrals can also be determined using the residual theorem

### Bonus material

In the bonus material we make some comments on complex differentiability, in particular we present the Wirtinger operators as useful aids. A particularly useful class of conformal mappings are the Möbius transformation, which we discuss as well as Schwarz-Christoffel's mapping formula.

In addition, we investigate the question of how “branched” functions can be clearly defined and how a given holomorphic function can be holomorphically extended into a larger area.

### tasks

The tasks are divided into three categories: You can use the comprehension questions to check whether you have understood the terms and central statements, with the arithmetic tasks you practice your technical skills and the application problems give you the opportunity to try out what you have learned on practical questions.

A point system differentiates between easy tasks •, moderately difficult •• and demanding ••• tasks. Solution hints at the end of the book will help you if you get stuck with a task at all. There you will also find the solutions - but do not deceive yourself and only look up when you have come to a solution yourself. You can find detailed solutions, evidence, and illustrations on the book's website.

Have fun and success with the tasks!

### Questions of understanding

### 32.1

• Show that the sum of the \ (n \) th roots of unity for \ (n \ geq 2 \) always equals zero and interpret this result for \ (n \ geq 3 \) geometrically.

### 32.2

### 32.3

- 1.
Union and intersection are again areas,

- 2.
the union is an area but not the average,

- 3.
are neither union nor intersection of regions.

### 32.4

• Show that the “accumulation point condition” in the identity theorem is actually necessary, that two holomorphic functions that match on an infinite set \ (M \) do not have to be the same if \ (M \) has no accumulation point.

### 32.5

### 32.6

- (a)
\ (f (z) = \ frac {z} {(z- \ mathrm {i}) (z + 2)} \) around \ (z_ {0} = 0 \)

- (b)
\ (f (z) = \ mathop {\ mathrm {Log}} (z) \) around \ (z_ {0} = 2 + \ mathrm {i} \)

- (c)
\ (f (z) = 1 / \ sin (\ frac {1} {z}) \) around \ (z_ {0} = \ frac {1} {\ pi} + \ mathrm {i} \)

### 32.7

- (a)
\ (f (z) = \ frac {1} {z ^ {8} + z ^ {2}} \).

- (b)
\ (f (z) = \ frac {1} {\ cos \ frac {1} {z}} \),

- (c)
\ (f (z) = \ frac {\ sin \ frac {1} {z}} {z ^ {2} +1} \).

### 32.8

Write the expression \ (x ^ {3} + xy ^ {2} \) on \ (z \) and \ (\ bar {z} \) as well as \ (z ^ {2} \ bar {z} \) to \ (x \) and \ (y \).

Verify the relation \ (\ mathrm {e} ^ {\ mathrm {i} z} = \ cos z + \ mathrm {i} \ sin z \) for the complex number \ (z = \ pi + \ mathrm {i} \ ).

- Calculate$$ \ displaystyle \ mathop {\ mathrm {Re}} (\ mathrm {e} ^ {(z ^ {3})}) \ quad \ mathrm {and} \ quad \ mathop {\ mathrm {Im}} (\ mathrm {e} ^ {(z ^ {3})}) $$for \ (z = x + \ mathrm {i} y \) and especially for \ (z_ {1} = \ sqrt [3] {\ pi} + \ mathrm {i} \ sqrt [3] {\ pi} \) .
Calculate \ (\ mathop {\ mathrm {Log}} z_ {k} \) for the complex numbers \ (z_ {1} = \ mathrm {i} \), \ (z_ {2} = \ sqrt {2} + \ sqrt {2} \, \ mathrm {i} \) and \ (z_ {3} = z_ {1} \ cdot z_ {2} \).

- Check that the two limits$$ \ displaystyle G_ {1} = \ lim_ {z \ to 1} \ frac {z ^ {2} -1} {z + 2} \ qquad G_ {2} = \ lim_ {z \ to 0} \ frac {\ bar {z}} {z} $$exist and compute them if necessary.

### 32.9

•• Show that \ (f \), \ (\ mathbb {C} \ to \ mathbb {C} \), \ (f (z) = \ mathop {\ mathrm {Im}} z \) for none \ (z \ in \ mathbb {C} \) is complex differentiable by (a) forming the corresponding limit values, (b) checking the Cauchy-Riemann equations.

### 32.10

\ (f (z) = \ cos z \)

\ (g (z) = z ^ {2} + (1+ \ mathrm {i}) z-1 \)

\ (h (z) = \ mathrm {e} ^ {\ sin z} \)

### 32.11

### 32.12

### 32.13

\ (I_ {1} = \ int_ {0} ^ {\ pi} \ frac {\ mathrm {e} ^ {\ mathrm {i} t} +1} {\ mathrm {e} ^ {\ mathrm {i} t} + \ mathrm {e} ^ {- \ mathrm {i} t}} \, \ mathrm {d} t \)

\ (I_ {2} = \ int_ {0} ^ {1} (t ^ {3} + (\ mathrm {i} +1) t ^ {2} + (\ mathrm {i} -1) t + 2 \ mathrm {i}) \, \ mathrm {d} t \)

\ (I_ {3} = \ int_ {0} ^ {1} \ frac {2t} {t ^ {2} + (1+ \ mathrm {i}) t + \ mathrm {i}} \, \ mathrm {d } t \)

### 32.15

### 32.16

- a)
\ (I_ {1, k} = \ oint_ {C_ {k}} \ frac {\ mathrm {e} ^ {z}} {z-2} \, \ mathrm {d} z \)

along the positively oriented circles

\ (C_ {1} \): \ (| z | = 3 \) and \ (C_ {2} \): \ (| z | = 1 \).

- b)
\ (I_ {2} = \ oint_ {C} \ frac {\ sin 3z} {z + \ frac {\ pi} {2}} \, \ mathrm {d} z \)

along the positively oriented circle

\ (C \): \ (| z | = 5 \).

- c)
\ (I_ {3} = \ oint_ {C} \ frac {\ mathrm {e} ^ {3z}} {z- \ pi i} \, \ mathrm {d} z \)

along the positively oriented curve

\ (C \): \ (| z-2 | + | z + 2 | = 6 \).

### 32.17

### 32.18

### 32.19

•• Determine the Laurent series expansion of the function \ (f \), \ (\ dot {\ mathbb {C}} \ to \ mathbb {C} \), \ (f (z) = \ sin (\ frac { 1} {z ^ {2}}) \) by \ (z = 0 \).

### 32.20

•• Develop the function \ (f \), \ (f (z) = \ frac {1} {z ^ {2} -2 \ mathrm {i} z} \) in Laurent series around the points \ ( z_ {1} = 0 \) and \ (z_ {2} = 2 \ mathrm {i} \) (two areas each).

### 32.21

### 32.22

### 32.23

- (a)
\ (f (z) = \ frac {\ mathrm {e} ^ {\ pi z}} {z ^ {2} +1} \)

- (b)
\ (f (z) = \ frac {1} {z ^ {4} + 2z ^ {2} -3} \)

- (c)
\ (f (z) = \ frac {4z ^ {2} -5z + 3} {z ^ {3} -2z ^ {2} + z} \)

### 32.24

### 32.25

\ (I_ {1} = \ int_ {0} ^ {\ pi} \ sin ^ {2} t \, \ mathrm {d} t \)

\ (I_ {2} = \ int_ {0} ^ {2 \ pi} \ frac {\ cos t} {5-4 \ cos t} \, \ mathrm {d} t \)

\ (I_ {3} = \ int_ {0} ^ {\ pi} \ frac {\ cos 3t} {5-4 \ cos t} \, \ mathrm {d} t \)

### 32.26

### 32.27

### Application problems

### 32.28

••• Two infinitely extended earthed plates are attached parallel at a distance \ (a \), between them charge \ (q \) is fixed. Determine the potential between the plates as a function of the distance between the charge and one of the plates.

(Notes: The figure \ (w = \ mathrm {e} ^ {\ frac {\ pi z} {a}} \) forms the stripe \ (0

### Answers to the self-questions

### Answer 1

If real and imaginary parts are simply represented as surfaces, this is of course possible. In the second type of representation, however, the amount corresponds to the amount, and this can never be negative. In this case, the area must never be below the \ (x \) - \ (y \) plane.

### Answer 2

### Answer 3

### Answer 4

Generally not. Only if the inner radius is \ (R_ {1} = 0 \), i.e. one considers the innermost possible ring and additionally all coefficients \ (a _ {- n} \) with \ (n \ in \ mathbb {N} \) disappear, there is convergence.

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