Complex Analysis : Review
Complex Algebra crops up in a ton of places in applied math. I came across an application of complex algebra in the context of Inverse Laplace transforms and could not understand some aspects of it. Looked around a bit and stumbled on this book that seemed to explain the principles of complex analysis in an elegant way. Let me summarize some of the main points in the book.
Chapter 1: What do I need to know?
The chapter provided a quick recap of all the prerequisites needed for working through the book. All the elementary results pertaining to set theory and calculus are provided, for formality sake.
Chapter 2: Complex Numbers
Why does one need complex numbers? In one sense, the set of complex numbers is the natural extension of the sets N, Z, Q(the set of natural numbers, integers and rational numbers) . Denoted by C, this set encompasses all the roots of a polynomial equation. The Fundamental Theory of Algebra says there is no need to look beyond C , for this set contains all the roots of a polynomial equation. The proof of this theorem is given in Chapter 7 of this book after enough theorems are stated and proved. This chapter is brief and lists down all the basic properties of complex numbers that one comes across in a typical high school level math curriculum.
Chapter 3: Prelude to Complex Analysis
All the important concepts developed on the real line depends on three properties of R, i.e. 1)it is a field, 2) there is a notion of distance between two points and 3) there are no gaps(Least Upper Bound Axiom).In C, the concept of order is tweaked to compare the two complex numbers, i.e. use absolute value of the complex numbers to compare one another. Like R, the Completeness property does make sense, i.e. every Cauchy sequence in C is convergent.
To create the “interval” equivalent in C, a few terms are defined such as

Neighborhood of point c in C

Open subset in C / open disc

Closed subset of C / closed disc

Boundary of a subset

Punctured disc
Functions in the complex plane need to be visualized differently from those on the real line. One needs to draw two planes, one the z plane and the second f(z) plane, so as to see the effects of transformation. The concept of limit for a function at any point on C, is defined in terms of punctured disc at that point.
Chapter 4: Differentiation
The notion of derivative in the complex plane comes with an additional requirement that the rate of change of function has to be the same in possible directions. This requirement is formalized by Cauchy – Riemann equations for certain functions. The CauchyRiemann equations arise from the requirement that the rate of change of the function at a point c must be the same in the x and y directions. So, what about the rate of change in other directions? Well, Cauchy Riemann conditions serve as a “necessary and sufficient condition” for ONLY those functions, for which the partial derivatives exist and are continuous throughout the open disc.
The chapter subsequently introduces the terms holomorphic and entire function, the former denotes the property of differentiability in a subset of C and the latter denotes the differentiability over the entire C. One of the most important theorems of real analysis is the Mean value theorem. One cannot directly translate the mean value theorem on to complex plane C. It has to tweaked a bit and this tweak goes by the name, “Goursat’s Lemma”.
A brief introduction to Power Series is given and one finds that most of the ideas in the real line space and can be easily extended to complex plane. All the ideas such Radius of convergence, procedures to compute radius of convergence in C are analogous to the ones on R. The radius of convergence plays a key role in computing higher order derivatives, for they exist only within the radius of convergence. The exponential function, sine, cos, cosh, sinh have infinity as the radius of convergence. This means that they are infinitely differentiable everywhere on C. Hence these functions fall under entire function category
Logarithmic function of complex number is vastly different from the logarithm of a real valued function. The log function in C is a multi valued function. To pin down and work with a single value, one computes the principal logarithm. Also basic identities involving logarithms fail unless one takes in to consideration the multiplevalued nature of the logarithmic function. Basically one needs to keep in mind that functions such as (1+i)i are multiple valued functions and be careful about using the identities from R. The formula for the differentiation of the real function log(x) is easily extended to the complex plane as all the values of the multifunction logarithm on C differ by a constant.
The chapter introduces branch points and cuts. A branch of a multiple valued function f is any single valued function F that is analytic in some domain and at each point z in the domain the value F(z) is one of the values of f (z). A branch cut is a line or curve that is removed from the complex plane to define a branch of a multiple valued function. A branch point is a point that is on all branch cuts for a particular function. The two most common multifunctions used in the book are Log z and z(1/n), whose branch points are 0 and infinity. The idea of defining branch points is that on any contour that does not go through branch points, the functions changes continuously and returns to its original value.
There is a section on singularities that deals with explaining the terms such as pole, order of the pole, meromorphic in a subset etc. These terms are useful for describing functions that are differentiable in a subset of C, at all points excepting the poles.
Chapter 5 : Complex Integration
The chapter starts off by proving Heine Borel theorem that says that for a closed bounded subset of C, every open covering of S contains a finite sub covering of S. This theorem is then used to prove the existence of supremum and infimum of a complex function whose domain is a closed bounded set. In real analysis, integration involves computing the integral value over an interval/ set. In complex analysis, integration is computed along a curve or a path. In order to represent a path, parametric representation is much better than the standard representation of x and y coordinates. Why? If a curve crosses itself or becomes vertical, the standard approach has problems. The complex integration is done over a curve and hence the right parameterization of the curve can make the computations quick and easy. The chapter introduces terms to verbalize the various curves around which the integration can be performed

Simple curve : One that doesn’t cross itself

Closed curve : end points are the same

Simple closed curve : One that doesn’t cross itself and whose ends points are the same

Rectifiable curve: A curve whose length can be defined in terms of supremum. For any rectifiable curve, one has a convenient expression for computing the length.

Smooth curve : The parameterized function has continuous derivatives in the domain

Contour : Piecewise smooth ,simple and closed

Interior of a contour

Exterior of a contour

Convex contour: All the line segments joining two points in the interior of the contour lie completely in the interior of the contour.
After introducing the terminology and types of curves around which a complex function might be integrated, the chapter takes a detour and talks about various bounds of a complex integral. Instead of evaluating explicitly, there are certain inequalities that are stated and proved. The chapter ends with a discussion of uniform convergence that is a key idea behind swapping differential and summation operators on an infinite series.
Chapter 6: Cauchy’s Theorem
An entire chapter on this theorem obviously means that it is one of most important theorems in complex analysis. The theorem states that if one integrates a holomorphic function on contour, the resulting value is 0. The proof is easy to follow for contours that are triangles and polygons. For a generic contour, the proof requires a greater effort from the reader. The other highlight of this chapter is the deformation theorem. It states the integral of function of a curve between two end points is independent of the smooth curve between the end points. Also if a contour A lies inside contour B, the integral around both the contours gives the same value.
Chapter 7: Some consequences of Cauchy’s Theorem
This chapter reveals a fundamental difference between complex analysis and real analysis. For an analytic real function f, we can make no deduction at all about the values of the function in (a,b) from its values at a and at b. In complex analysis it is possible to compute the value of a holomorphic function inside a contour by its values on the contour. The theorem that makes this computation possible is Cauchy’s Integral formula. The following expression enables one to compute the value of a function at any point a in the interior of the contour.
One can also derive an expression for the nth order derivative of the function at a point, i.e.
The chapter has a section that proves the fundamental theorem of algebra by using Liouville’s theorem that states any bounded entire function is essentially a constant function. The chapter ends with a discussion of Taylor series expansion of a holomorphic function. Also the Taylor series of the function is unique (once one chooses the center around which expansion is desired).
Chapter 8: Laurent Series and the Residue Theorem
If a function has a singularity at a point c, then it cannot have a Taylor series centered on c. Instead it has a Laurent series, a generalized version of a Taylor series in which there are negative as well as positive powers of (zc). The connection between Laurent series and contour integrals exists via residues. Once the function has been expanded as Laurent series, one reads off the coefficient of 1/z to compute the residue and hence the relevant integral. The concept of singularities can be understood better from a Laurent series perspective. If the coefficients of the series do not die out, then the singularity is called essential singularity. Given this background, the chapter states the residue theorem where an integral around a contour is evaluated using the residues at various poles of the function being integrated. A few tricks are shown to compute the residues of various residues of a function. The chapter will give the reader enough practice to compute complex function integrals around contours whose interior comprises the poles of the function.
Chapter 9: Application of Contour Integration
This chapter brings the ideas and principles of all the previous chapters in solving various contour integration problems. The first application is solving real integrals that are reasonably tedious to evaluate. In all such cases, one can transform the integral to an integral on a semi circular contour and use Cauchy residue theorem to evaluate it elegantly. The second application that the chapter shows is that of rewriting an integral around a unit circle. This is a familiar trick that one usually comes across in elementary calculus. The third application discussed in the chapter is the use of Jordan’s lemma to expand the range of real integrals that can be evaluated using Cauchy residue theorem. The fourth application involves choosing special contours to make integral computations easier. The chapter ends with the application of residue theorem to summation of infinite series.
The last three chapters give an introduction to some advanced topics in complex analysis. I have skipped the last three chapters of the book, but a serious math buff might want to go over those chapters as they give an exposure to the open problems, yet to be cracked in this field.
[ Takeaway :
Complex Analysis, contrary to its name, makes computations easier in a variety of applied math problems. This book is structured in a way that it highlights all the essential theorems of the subject and interlaces them with abundant examples and illustrations.