Complex Analysis: Introduction
Complex Analysis: Introduction
Complex Analysis: Introduction
Author
Richard Fearn
Title
Complex Analysis: Introduction
Description
This notebook is an introduction to complex analysis at the undergraduate level for learners with some background in mathematics, physics or engineering.
Category
Educational Materials
Keywords
complex analysis, potential flow, complex algebra, complex mapping, complex integration, Polya vector field, modular surface
URL
http://www.notebookarchive.org/2019-05-1fu5j9s/
DOI
https://notebookarchive.org/2019-05-1fu5j9s
Date Added
2019-05-03
Date Last Modified
2019-05-03
File Size
1.52 megabytes
Supplements
Rights
Redistribution rights reserved
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Richard L. Fearn (rlf@ufl.edu)
Department of Mechanical and Aerospace Engineering
University of Florida
January 15, 2017, revised May 2, 2019
Department of Mechanical and Aerospace Engineering
University of Florida
January 15, 2017, revised May 2, 2019
1 Introduction
1 Introduction
Complex analysis played an important role in the development of mathematics, and provides powerful computational tools for solving problems in physics and engineering. The approach for this introduction to the subject is to state pertinent results and to illustrate them, whenever appropriate, with interactive graphs. Proofs or partial proofs may be included if they are relatively simple and concise, and links to more detailed discussions are provided. The primary reference is Visual Complex Analysis by Tristan Needham and I recommend this book for those wanting a more extensive development of the subject from a mathematical perspective.
I assume that you have a working knowledge of calculus and some background in physics or engineering including elementary vector field theory and basic potential theory using real variables. Mathematica is used to construct interactive graphs and for occasional calculations. Cells containing the code for interactive graphs are hidden because I think that including the code would interrupt the narrative.
The topics selected hopefully lead to an understanding of the residue theorem for evaluating the integral of an analytic function of a complex variable on a closed contour. Also emphasized is the interpretation of contour integrals in terms of work and flux integrals of a vector field in preparation for solving potential models used used in physics and engineering. Mapping plays an important role in visualizing complex functions and is featured in several of the interactive graphs.
Section 2, Review of Vector Fields, provides some basic information on visualizing vector fields, integrals describing work and flux, and the divergence and curl theorems. This review can be skipped if it is not needed.
Conventions for notation in Mathematica sometimes differ from those used in mathematics and/or physics, but these differences are usually minor and should not degrade readability. I will use Mathematica conventions in input statements for computations using Mathematica, and conventions appropriate to the topic being discussed in the text. If you are not a user of Mathematica, you may find the tutorial, Some General Notations and Conventions helpful. If you are still puzzled about Mathematica usage that I do not adequately explain, try the online documentation for additional help.
Here is a list of interactive figures in this notebook and the number of the subsection in which each appears.
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Figure 1 Modular Surface is in subsection 5.1 with two examples.
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Figure 2 Pólya Vector Field is in subsection 5.2 with four examples.
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Figure 3 Combined Source and Vortex is in subsection 5.2 with one example.
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Figure 4 Complex Mapping is in subsection 5.3 with three examples.
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Figure 5 Mapping at Critical Points is in subsection 5.5 with three examples.
These examples illustrate selected properties of interest and how parameters are controlled. When describing these interactive graphs, I may refer to typical computer mouse actions, and assume that you know how to interpret these actions for your pointing device. The examples included are not intended to exhaust the information contained in the interactive graphs; this is left to your curiosity.
Please tell me of any errors that you find in this notebook. I would also appreciate any suggestions that you care to make for improving clarity.
2 Review of Vector Fields
2 Review of Vector Fields
Preview
Preview
Many quantities of interest in physics and engineering can be represented as vector fields. For example, a force field or a velocity field. Consider a vector field represented by () in a right-handed Cartesian coordinate system, where vectors are represented by their components; =(,,) and =(x,y,z). Vector fields can be visualized by plotting a collection of arrows, each arrow having components with the tail of the vector at the location . For physical problems of interest, vector fields are usually three dimensional and complicated, making visualization a challenge. The purpose of this review is to remind you of some properties of vector fields used in this notebook, and to introduce some basic graphical tools of Mathematica. If you don’t need such a review, skip it.
V
r
V
v
x
v
y
v
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r
(,,)
v
x
v
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(x,y,z)
Examples of a line integral of a vector field are the work done by a force field on an open or closed contour, and the circulation of a velocity field around a closed contour. The example chosen for a surface integral is the flux of a velocity field across the surface. The divergence and curl of vector fields are discussed including the divergence and curl theorems relating line integrals to surface integrals and surface integrals to volume integrals.
2.1 Visualizing Vector Fields with Mathematica
2.1 Visualizing Vector Fields with Mathematica
2.2 Work and Flux Integrals
2.2 Work and Flux Integrals
2.3 Divergence and Curl
2.3 Divergence and Curl
2.4 Divergence and Curl Theorems
2.4 Divergence and Curl Theorems
3 Complex Arithmetic
3 Complex Arithmetic
Preview
Preview
Except for the previous section on vector fields, the symbol is reserved for a complex variable representing a complex number . The association of a complex number with a point or vector in a plane provides a geometric interpretation of a complex number. Basic properties of addition and multiplication of complex numbers are stated, and the concept of extending the complex plane to include the point at infinity is mentioned. Complex multiplication provides a simple way to prove trigonometric identities and to derive the divergence and curl of two-dimensional vectors.
z
z=x+y
3.1 Geometric Interpretation of Complex Numbers
3.1 Geometric Interpretation of Complex Numbers
3.2 Extended Complex Plane
3.2 Extended Complex Plane
3.3 Trigonometric Identities
3.3 Trigonometric Identities
3.4 Dot and Cross Products
3.4 Dot and Cross Products
4 Complex Functions
4 Complex Functions
Preview
Preview
A complex function can be thought of as a rule that assigns to a point another complex number . That is, a complex function can be represented as an ordered pair of real functions, each a function of two real variables. Differentiation and integration of complex functions are the primary topics of this section. Looking at complex functions from different perspectives can increase our understanding of them.
z=x+y
f(x,y)=ϕ(x,y)+ψ(x,y)
A conceptual definition of a derivative in the complex plane is stated first. In general, the derivative of a complex function at a point depends on the direction of the path taken through the point of interest. The requirement that the derivative be path independent is a severe restriction. Such functions are referred to as complex differentiable and are called analytic functions of a complex variable. We will use a common convention that calls a complex function analytic even though the requirements for being analytic break down at isolated singular points. Most applications of complex analysis use functions of this type.
The complex variable and its conjugate can also be used as independent variables instead of and . Tests for determining if a complex function is analytic are stated in terms of and . The integral of a complex function along a contour in the complex plane is interpreted in terms of work and flux integrals of a vector field associated with the function. Taylor series expansions about regular points and Laurent series expansions about singular points of complex functions are demonstrated using Mathematica
(z,z)
x
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(x,y)
(z,z)
4.1 Complex Differentiable Functions
4.1 Complex Differentiable Functions
4.2 Alternative Representation of Derivatives
4.2 Alternative Representation of Derivatives
4.3 Some Properties of Analytic Functions
4.3 Some Properties of Analytic Functions
4.4 The Pólya Vector Field
4.4 The Pólya Vector Field
4.5 Power Series
4.5 Power Series
5 Visualizing Complex Functions
5 Visualizing Complex Functions
Preview
Preview
There are several ways to visualize a function of a complex variable. The modular surface is useful for locating singular points and identifying types singularities. Plotting the Pólya vector field associated with a complex function can illustrate physical quantities such as force or velocity fields in potential models. Complex mapping shows how points, curves and regions map from one complex plane to another using a complex function as a transformation. The Riemann surface is historically important as a way to make sense of multi-valued functions such as square root and logarithm, but will not be discussed in this notebook.
5.1 Modular Surface
5.1 Modular Surface
5.2 Pólya Vector Field
5.2 Pólya Vector Field
5.3 Complex Mapping
5.3 Complex Mapping
5.4 Mapping by an Analytic Function
5.4 Mapping by an Analytic Function
5.5 Critical Points of a Transformation
5.5 Critical Points of a Transformation
6 Integrating Analytic Functions
6 Integrating Analytic Functions
Preview
Preview
For applications of potential theory, the only useful complex functions are those that are analytic except at isolated pole-type singularities. Closed line integrals of analytic functions are used to describe conservation laws for physical vector fields. Cauchy’s Integral theorem states that the integral of a function on a closed contour is zero if the function is analytic everywhere within the contour. Cauchy’s integral formula evaluates the integral of a function on a closed contour if there is a simple pole-like singularity within the contour. The residue theorem summarizes the information needed to provide a concise expression for evaluating the integral of an analytic function with possible pole-type singularities within the contour.
6.1 Cauchy’s Integral Theorem
6.1 Cauchy’s Integral Theorem
6.2 Cauchy’s Integral Formula
6.2 Cauchy’s Integral Formula
6.3 Residue Theorem
6.3 Residue Theorem
Concluding Remarks
Concluding Remarks
With these basic tools of complex analysis, we are ready to look at specific applications, other powerful concepts and more advanced computational tools associated with complex analysis.
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basic two-dimensional potential flow
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Joukowski airfoil
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natural representation of infinity in geometry (Riemann sphere) and algebra (homogeneous coordinates)
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Möbius transformation
References
References
Cite this as: Richard Fearn, "Complex Analysis: Introduction" from the Notebook Archive (2019), https://notebookarchive.org/2019-05-1fu5j9s
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