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

A complex function \ (f = u + \ mathrm {i} \, v \) is complex differentiable at a point \ (z = x + \ mathrm {i} y \ in D (f) \) if and only if it is real there (as a function \ (\ mathbb {R} ^ {2} \ to \ mathbb {R} ^ {2} \)) is differentiable and additionally the Cauchy-Riemann equations
$$ \ displaystyle \ frac {\ partial u} {\ partial x} = \ frac {\ partial v} {\ partial y} \ ,, \ quad \ frac {\ partial u} {\ partial y} = - \ frac {\ partial v} {\ partial x} $$
Fulfills.

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

For a positively oriented circle \ (C \) with center \ (z_ {0} \) and radius \ (R> 0 \) we get
$$ \ displaystyle \ int_ {C} (z-z_ {0}) ^ {n} \, \ mathrm {d} z = \ begin {cases} 0 & \ text {f {\ "u} r} n \ neq -1 \ 2 \ pi \ mathrm {i} & \ text {f {\ "u} r} n = -1 \ end {cases} $$

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

Cauchy's integral theorem

For a simply connected domain \ (G \) and a holomorphic function \ (f \) in it, if all considered paths lie entirely in \ (G \):
  • 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

For a holomorphic function \ (f \) in a simply connected domain \ (G \) and a closed path \ (C \) with image \ (C ^ {*} \) running entirely in \ (G \) for any \ (z_ {0} \ in G \ setminus C ^ {*} \):
$$ \ displaystyle f (z_ {0}) \, \ mathrm {Ind} _ {C} (z_ {0}) = \ frac {1} {2 \ pi \ mathrm {i}} \, \ oint_ {C } \ frac {f (z)} {z-z_ {0}} \, \ mathrm {d} z $$

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

If in a simply connected area \ (G \) \ (z_ {1}, z_ {2}, \ ldots z_ {N} \ in G \) there are finitely many points (different in pairs) and the function \ (f \) on \ (G \ setminus \ {z_ {1}, \ ldots, z_ {N} \} \) holomorphic, then for every closed path \ (C \) that is completely in \ (G \ setminus \ {z_ { 1}, \ ldots, z_ {N} \} \) runs:
$$ \ displaystyle \ oint_ {C} f (z) \ mathrm {d} z = 2 \ pi \ mathrm {i} \ cdot \ sum_ {j = 1} ^ {N} \ left (\ mathop {\ mathrm { Res}} (f, z_ {j}) \ cdot \ mathrm {Ind} _ {C} (z_ {j}) \ right) $$
This theorem is particularly useful because there is a simple formula for finding residuals at poles. One obtains for the residual of a function \ (f \) at a pole \ (k \) th order
$$ \ displaystyle \ mathop {\ mathrm {Res}} (f; z_ {0}) = \ frac {1} {(k-1)!} \, \ lim_ {z \ to z_ {0}} \ left [\ frac {\ mathrm {d} ^ {k-1}} {\ mathrm {d} z ^ {k-1}} \ left ((z-z_ {0}) ^ {k} \, f (z ) \ right) \ right]. $$

Real integrals can also be determined using the residual theorem

This works with integrals of form, for example
$$ \ displaystyle \ int_ {0} ^ {2 \ pi} R (\ cos t, \ sin t) \, \ mathrm {d} t \ ,, \ quad \ int _ {- \ infty} ^ {+ \ infty } \ frac {P (t)} {Q (t)} \, \ mathrm {d} t, $$
and
$$ \ displaystyle \ int _ {- \ infty} ^ {+ \ infty} \ frac {P (t)} {Q (t)} \, \ cos (\ alpha t) \, \ mathrm {d} t \, , \ quad \ int _ {- \ infty} ^ {+ \ infty} \ frac {P (t)} {Q (t)} \, \ sin (\ alpha t) \, \ mathrm {d} t \ ,, $$
the latter in particular play an important role in signal theory.

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

• Show identity
$$ \ displaystyle \ cos (4 \ varphi) = 8 \ cos ^ {4} \ varphi-8 \ cos ^ {2} \ varphi + 1 $$
and derive an analogous identity for \ (\ sin (4 \ varphi) \).

32.3

•• Give two areas \ (G_ {1} \) and \ (G_ {2} \) so that
  1. 1.

    Union and intersection are again areas,

  2. 2.

    the union is an area but not the average,

  3. 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

•• Is there a function \ (f (z) \) with the property
$$ \ displaystyle f \ left (\ frac {1} {n} \ right) = \ frac {1} {1- \ frac {1} {n}} \ quad \ text {f {\ "u} r} n = 2,3,4, \ ldots, $$
which (a) is holomorphic on \ (| z | <1 \), (b) on all \ (\ mathbb {C} \)

32.6

•• Find the radius of convergence without calculating the expansion of the given function around the point \ (z_ {0} \) into a power series:
  1. (a)

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

  2. (b)

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

  3. (c)

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

32.7

•• Where do the following functions \ (f \), \ (D (f) \ to \ mathbb {R} \) have singularities and what type are they (as far as they can be classified in our scheme)?
  1. (a)

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

  2. (b)

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

  3. (c)

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

32.8

• In the following we always set \ (z = x + \ mathrm {i} y \).
  • 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

• Show with the help of the Cauchy-Riemann equations that the functions \ (f \), \ (g \) and \ (h \), \ (\ mathbb {C} \ to \ mathbb {C} \) with
  • \ (f (z) = \ cos z \)

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

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

are holomorphic to the whole \ (\ mathbb {C} \).

32.11

••• Are for the function \ (f \), \ (\ mathbb {C} \ to \ mathbb {C} \) with
$$ \ displaystyle f (z) = \ begin {cases} \ frac {z ^ {5}} {| z | ^ {4}} & \ text {f {\ "u} r} z \ neq 0 \ 0 & \ text {f {\ "u} r} z = 0 \ end {cases} $$
at the point \ (z_ {0} = 0 \) the Cauchy-Riemann equations are fulfilled? Is \ (f \) in \ (z_ {0} = 0 \) complex differentiable?

32.12

•• Show that the function \ (u \), \ (\ mathbb {R} ^ {2} \ to \ mathbb {R} ^ {2} \)
$$ \ displaystyle u (x, \, y) = 2x \, (1-y) $$
is harmonic and calculate the conjugate harmonic function \ (v \) and \ (f = u + \ mathrm {i} v \) as a function of \ (z = x + \ mathrm {i} y \). (The constant of integration may be set to zero.)

32.13

•• Calculate the integrals
  • \ (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

•• Calculate the integrals
$$ \ displaystyle \ begin {aligned} \ displaystyle I_ {a, k} & \ displaystyle = \ int_ {C_ {k}} \ bar {z} \, \ mathrm {d} z & \ displaystyle I_ {b, k} & \ displaystyle = \ int_ {C_ {k}} \ mathop {\ mathrm {Re}} z \, \ mathrm {d} z \ \ displaystyle I_ {c, k} & \ displaystyle = \ int_ {C_ {k }} \ mathrm {e} ^ {\ pi z} \, \ mathrm {d} z & \ displaystyle I_ {d, k} & \ displaystyle = \ int_ {C_ {k}} z ^ {5} \, \ mathrm {d} z \ end {aligned} $$
for \ (k = 1, \, 2, \, 3, \, 4, \, 5 \) along the curves shown in Fig. 32.28:

32.16

• Calculate the integrals:
  1. 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 \).

  2. b)

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

    along the positively oriented circle

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

  3. 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

••• Show
$$ \ displaystyle \ int _ {- \ infty} ^ {+ \ infty} \ mathrm {e} ^ {- ax ^ {2} -bx} \, \ mathrm {d} x = \ sqrt {\ frac {\ pi } {4 \, | a |}} \, \ mathrm {e} ^ {\ frac {b ^ {2}} {4a}} \, \ mathrm {e} ^ {- \ frac {\ mathrm {i} } {2} \ mathop {\ mathrm {Arg}} a} $$
for \ (a, b \ in \ mathbb {C} \) and \ (\ mathop {\ mathrm {Re}} a> 0 \) by applying Cauchy's integral theorem to the integration path shown in Fig. 32.29.

32.18

••• Calculate the real integral
$$ \ displaystyle I = \ int_ {0} ^ {2 \ pi} (\ cos x) ^ {2p} \, \ mathrm {d} x \ quad \ mathrm {m \ mathrm {i} t} \; p \ in \ mathbb {N} \,. $$
for \ (p \ in \ mathbb {N} \).

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

•• Calculate the Laurent series expansion of the function
$$ \ displaystyle f (z) = \ frac {1} {(z-1) (z-2)} $$
(a) for \ (| z | <1 \), (b) for \ (1 <| z | <2 \) and (c) for \ (| z |> 2 \).

32.22

•• One decomposes the function \ (f \),
$$ \ displaystyle f (z) = \ frac {4z ^ {2} -2z + 8} {z ^ {3} -z ^ {2} + 4z-4} $$
into partial fractions and find the residuals at the poles. (Note: A zero of the denominator is \ (z = + 1 \)).

32.23

• Determine for the following functions \ (f \), \ (D (f) \ to \ mathbb {C} \) the maximum definition set and the residuals of the functions on all singularities:
  1. (a)

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

  2. (b)

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

  3. (c)

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

32.24

• Use the residual theorem to calculate the integrals over the functions \ (f \) and \ (g \)
$$ \ begin {aligned} \ displaystyle f (z) & \ displaystyle = \ frac {\ mathrm {e} ^ {\ pi z}} {z ^ {2} - (1+ \ mathrm {i}) z + \ mathrm {i}} \ \ displaystyle g (z) & \ displaystyle = \ frac {z ^ {2}} {z ^ {2} + (\ mathrm {i} -2) z-2 \ mathrm {i} } \ end {aligned} $$
along the curves \ (C_ {1} \) to \ (C_ {3} \) shown in Fig. 32.30.

32.25

•• Using the residual theorem, calculate the real integrals:
  • \ (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

•• Using the residual theorem, calculate the integrals:
$$ \ displaystyle I_ {a} = \ int _ {- \ infty} ^ {+ \ infty} \ frac {1} {t ^ {4} +1} \, \ mathrm {d} t, \ quad I_ {b } = \ int _ {- \ infty} ^ {+ \ infty} \ frac {t ^ {2}} {t ^ {6} +1} \, \ mathrm {d} t, \ quad I_ {c} = \ int _ {- \ infty} ^ {\ infty} \ frac {t \ sin t} {t ^ {2} +4} \, \ mathrm {d} t $$

32.27

• Determine the integral using the residual theorem
$$ \ displaystyle I = \ int_ {0} ^ {2 \ pi} \ frac {\ mathrm {d} t} {a + b \, \ cos t} $$
for \ (a> b \).

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 0 \). A plate can be grounded by attaching a mirror charge \ (- q \) to a suitable place.)

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

For example, if we choose \ (z_ {1} = \ pi + \ mathrm {i} \, \ ln 2 \), we get
$$ \ displaystyle \ cos (z_ {1}) = \ underbrace {\ cos (\ pi)} _ {= - 1} \, \ underbrace {\ cosh (\ ln 2)} _ {= \ frac {5} {4}} {} - {} \ underbrace {\ sin (\ pi) \, \ sinh (\ ln 2)} _ {= 0} = - \ frac {5} {4} $$
For \ (z_ {2} = 1 + \ mathrm {i} \) we get
$$ \ displaystyle \ mathrm {e} ^ {z_ {2}} = \ mathrm {e} ^ {1+ \ mathrm {i} \, \ pi} = \ mathrm {e} ^ {1} \, \ mathrm {e} ^ {\ mathrm {i} \, \ pi} = \ mathrm {e} \, (\ underbrace {\ cos \ pi} _ {= - 1} {} + {} \ mathrm {i} \, \ underbrace {\ sin \ pi} _ {= 0}) = - \ mathrm {e} <0 \,. $$

Answer 3

Analogous to the previous example we get for \ (C_ {2} \)
$$ \ displaystyle \ begin {aligned} \ displaystyle C_ {21} & \ displaystyle: & \ displaystyle z (t) & \ displaystyle = 3 + 3 \ mathrm {i} + (- 1-3 \ mathrm {i}) t \ ,, & \ displaystyle t & \ displaystyle \ in [0, \, 1] \ ,, \ \ displaystyle C_ {22} & \ displaystyle: & \ displaystyle z (t) & \ displaystyle = 2- \ mathrm { i} t \ ,, & \ displaystyle t & \ displaystyle \ in [0, \, 1] \ ,, \ \ displaystyle C_ {23} & \ displaystyle: & \ displaystyle z (t) & \ displaystyle = 1- \ mathrm {i} + \ mathrm {e} ^ {- \ mathrm {i} t} \ ,, & \ displaystyle t & \ displaystyle \ in \ left [0, \, \ frac {\ pi} {2} \ right] \ ,, \ \ displaystyle C_ {24} & \ displaystyle: & \ displaystyle z (t) & \ displaystyle = 1-2 \ mathrm {i} -t \ ,, & \ displaystyle t & \ displaystyle \ in [0, \, 1] \ ,, \ \ displaystyle C_ {25} & \ displaystyle: & \ displaystyle z (t) & \ displaystyle = - \ mathrm {i} + \ mathrm {e} ^ {- \ mathrm {i} t} \ ,, & \ displaystyle t & \ displaystyle \ in \ left [\ frac {\ pi} {2}, \, \ pi \ right] \,. \ end {aligned} $$

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.