Table of Contents
Table of Contents
“Exercises of Complex Analysis”
INTRODUCTION
THEORETICAL OUTLINE
EXERCISES
“Exercises of Complex Analysis”
“Exercises of Complex Analysis”
SIMONE MALACRIDA
In this book, exercises are carried out regarding the following mathematical topics:
monodromy and polychromy in complex analysis
complex integrals and series
remarkable theorems of complex analysis
Initial theoretical hints are also presented to make the performance of the exercises understood.
Simone Malacrida (1977)
Engineer and writer, has worked on research, finance, energy policy and industrial plants.
ANALYTICAL INDEX
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INTRODUCTION
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I – THEORETICAL OUTLINE
Property
Monodromy and polydromy
Complex integration
Euler functions
Complex series
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II – EXERCISES
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Exercise 17
Exercise 18
Exercise 19
Exercise 20
Exercise 21
Exercise 22
Exercise 23
Exercise 24
Exercise 25
Exercise 26
Exercise 27
Exercise 28
Exercise 29
Exercise 30
Exercise 31
Exercise 32
Exercise 33
Exercise 34
Exercise 35
INTRODUCTION
INTRODUCTION
In this workbook, some examples of calculations related to complex analysis are carried out.
Furthermore, the main theorems of this analysis and their practical use in order to solve problems are presented.
Complex analysis, extending what has already been learned in the set of real numbers, constitutes a cornerstone in the natural setting of the complex field.
The peculiarities of this analysis make it so specific as to require a reflection in its own right with respect to the other fields of mathematics.
In order to understand in more detail what is presented in the resolution of the exercises, the theoretical reference context is recalled in the first chapter.
What is exposed in this workbook is generally addressed in advanced mathematical analysis courses (analysis 3).
I
THEORETICAL OUTLINE
THEORETICAL OUTLINE
Property
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If the function considered in mathematical analysis is a complex variable and not real, we speak of complex analysis.
Given an open subset of the complex plane, a function is differentiable in the complex sense if such a limit exists:
This limit means that, for each sequence of complex numbers that converge to a given point, the limit of the incremental ratio must tend to the same value.
A function is said to be holomorphic in an open set if it is differentiable in the complex sense at every point in the set .
A function is differentiable in the complex sense if it is differentiable and if this relation holds:
Continuity in the complex sense is defined in the same way as for the case of real functions.
It is possible to relate the differentiability between complex functions and real functions simply by recalling the Cartesian form of complex numbers:
A function is holomorphic if and only if it satisfies the Cauchy-Riemann equations:
The components u(x,y) and v(x,y) of a holomorphic function are harmonic functions.
A holomorphic function is differentiable infinite times, while the Wirtinger derivative of a holomorphic function is zero:
A complex function of several variables is holomorphic if and only if it can be developed as a convergent power series (this condition is more stringent than the Cauchy-Riemann equations alone for complex functions of one variable) i.e. every holomorphic function is analytic.
By Liouville's theorem, every bounded holomorphic function defined on the entire complex plane is a constant.
Furthermore, each analytic function of real variable extends uniquely to a holomorphic function.
This procedure is called analytic extension and can be applied to functions such as the exponential and the majority of trigonometric functions.
the other hand, we speak of analytic continuation when it is possible to extend the domain of definition of a holomorphic function, maintaining the same starting holomorphic function in the original domain.
Typically, analytic continuation is not a one-time operation.
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A holomorphic function having always non-zero derivative is a conformal map .
We define a conformal map as a function which preserves the angles and their orientation, but not necessarily the dimensions.
Anti-holomorphic functions are those complex functions that are holomorphic with respect to the complex conjugate of the argument.
A function that is both holomorphic and anti-holomorphic is constant.
An anti-holomorphic function preserves the angles, but not their orientation, and is therefore not a conformal map.
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A Riemann surface is a connected topological manifold, Hausdorff, two-dimensional and has a complex structure such that the function describing the manifold is holomorphic.
For example, every open subset of the complex plane is a Riemann surface.
A Riemann surface is orientable, as it is described by a holomorphic function which is a conformal map.
A Riemann sphere is a Riemann surface that is obtained by adding a point at infinity to the complex plane.
In essence it is the generalization of the projective line in the complex plane.
A Riemann sphere is the simplest compact Riemann surface.
The Riemann uniformization theorem states that a Riemann surface admits a Riemannian metric with constant curvature which induces the same conformal structure given by the original structure of the Riemann surface .
The value of the curvature can be 1, 0 or -1 and we speak, respectively, of elliptical, flat or hyperbolic metric.
Only if the metric is flat can it be scaled with a multiplicative factor, while in all other cases the metric is unique.
A biolomorphism is a holomorphic function which is injective, surjective and whose inverse is holomorphic.
The biolomorphism relation generalizes the isomorphism relation for the case of complex analysis.
A biolomorphism between Riemann surfaces is given by a bijective holomorphic function.
If the Riemann surface is simply connected, then the Riemann uniformization theorem reduces to the Riemann map theorem which states that the surface is biolomorphic to one of the following models: the Poincaré disk, the complex plane or the Riemann, whose curvatures are -1.0 and 1.
In all these cases, biolomorphisms are also isometries. The biolomorphisms of the Riemann sphere are the Mobius transformations, those of the complex plane are the translations, those of the Poincaré disk are the so-called Fuchsian groups.
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Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 22.04.2023
ISBN: 978-3-7554-3994-3
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