Complex Analysis

The following sections are included:

• Preface

• Contents

Let ℝ be the set of all real numbers. Then a complex number is of the form a + ib, where a and b are in ℝ and i² = −1. We denote the set of all complex numbers by ℂ…

The notion of a phase or an argument of a complex number is what makes complex numbers have a flavor different from real numbers. To see what it is, let z = x + iy be a nonzero complex number. Then in terms of polar coordinates, z can be identified as (r, θ), where

We all know what e^θ is if θ is a real number. What is e^iθ? To answer this question, we again use the rule that complex numbers are treated as real numbers in calculations…

We need to study sets in the complex plane ℂ because they come up as domains of functions…

In this book we study complex-valued functions defined on subsets of the complex plane ℂ. The subsets that are of most interest to us are known as domains. Domains are open and connected subsets of ℂ. A domain with some or all of its boundary points is known as a region…

We begin with the definition of the function e^z for all z ∈ ℂ. Guided by the properties of the exponential function e^x for all x ∈ ℝ, we expect the function e^z to be an entire function such that

A very interesting function in complex analysis is the logarithmic function

As in calculus, we want to define log z as the inverse of the exponential function studied in the preceding chapter…

Let γ be a curve in ℂ. Suppose that it is parametrized by a continuous complex-valued function z = z(t) for all t in [a, b] such that

(1) z(t) has a continuous derivative on [a, b],

(2) z′(t) ≠ 0 for all t in [a, b],

(3) z(t) is one-to-one on [a, b].

Then we call γ a smooth arc…

A criterion for a continuous complex-valued function w = f(z) on a domain D to have an antiderivative on D is that the integral of f on every closed contour in D is equal to zero. How is it possible to check that the integral on every closed contour is equal to zero? It seems to be a tall order indeed. We show in this chapter that for holomorphic functions w = f(z) on simply connected domains D, ∫_Γ f(z) dz = 0 for every closed contour Γ in D. This surprising result is known as Cauchy's integral theorem. We give in the appendix of this chapter a proof of this result when the closed contour is a rectangle. The full-blown case is a consequence of a version of Cauchy's integral theorem based on continuous deformations of closed contours. We state this more general version of Cauchy's integral theorem without proof. Then we work on some examples to appreciate the power of Cauchy's integral theorem…

We give in this chapter a formula that is a landmark in complex analysis. It is known as Cauchy's integral formula. Let us emphasize the distinction between Cauchy's integral formula and Cauchy's integral theorem given in the last chapter. The latter is a result on the invariance of contour integrals, while the former, as we shall see in due course, allows us to compute values of a holomorphic function inside a contour in terms of the values of the function on the contour. It reveals the mean value property of holomorphic functions. See Exercise (3) in this chapter…

The first goal of this chapter is to prove that holomorphic functions can be locally represented by Taylor series…

Up to this point, we know that a holomorphic function on an open disk can be represented by a power series. What happens if a function is holomorphic only on an annulus?…

If w = f(z) is a holomorphic function on and inside a simple closed contour Γ, then

Residues can be used to evaluate a great variety of integrals arising in many areas of mathematical analysis. The trigonometric integrals to be evaluated in this chapter are of the form

Let f be a continuous complex-valued function on [0,∞). Then the improper integral of f on [0,∞) is defined by

Let f be a continuous complex-valued function on (−∞, ∞). Then we define the Fourier transform of f on (−∞, ∞) by

In this chapter we look at Cauchy principal values of improper integrals , where the integrand f has a finite number of local singularities on (−∞, ∞). These improper integrals are called singular integrals on (−∞, ∞)…

By way of motivation, let α be a real number, but not an integer. Let f(z) = z^α, where the branch z^α = e^{α log₀ z} is taken. To recall,

Let D₁ and D₂ be domains in ℂ. Then we say that D₁ and D₂ are biholomorphic or conformally equivalent if there exists a bijective holomorphic function f : D₁ → D₂. We call a bijective holomorphic function f : D₁ → D₂ a biholomorphism or a conformal mapping…

In order to have a deeper geometric understanding of ⅅ and ℍ, we need Schwarz's lemma. To prove Schwarz's lemma, we use the maximum modulus principle, which depends on an interesting property of the zeros of holomorphic functions…

Let D be a domain in ℂ. A biholomorphism f : D → D is called an automorphism of D. We denote by Aut(D) the set of all automorphisms of D. It can be shown that Aut(D) is a group with respect to the usual composition of mappings. We call Aut(D) the automorphism group of D. In the two cases of interest to us, Aut(ⅅ) and Aut(ℍ) can be seen easily to be groups by explicit formulas in this chapter and the following chapter respectively…

Let F : ℍ → ⅅ be the Cayley transform studied in Chapter 19. To recall,

Let us recall that a harmonic function u on a domain D of the complex plane ℂ is a real-valued function in C²(D) such that

The following sections are included:

• Bibliography

• Index