An Introduction to Complex Analysis

We begin this lecture with the definition of complex numbers and then introduce basic operations-addition, subtraction, multiplication, and division of complex numbers. Next, we shall show how the complex numbers can be represented on the

xy

-plane. Finally, we shall define the modulus and conjugate of a complex number.

In this lecture, we shall first show that complex numbers can be viewed as two-dimensional vectors, which leads to the triangle inequality. Next, we shall express complex numbers in polar form, which helps in reducing the computation in tedious expressions.

In this lecture, we shall first show that every complex number can be written in exponential form, and then use this form to raise a rational power to a given complex number. We shall also extract roots of a complex number. Finally, we shall prove that complex numbers cannot be ordered.

In this lecture, we collect some essential definitions about sets in the complex plane. These definitions will be used throughout without further mention.

In this lecture, first we shall introduce a complex-valued function of a complex variable, and then for such a function define the concept of limit and continuity at a point.

In this lecture, using the fundamental notion of limit, we shall define the differentiation of complex functions. This leads to a special class of functions known as analytic functions. These functions are of great importance in theory as well as applications, and constitute a major part of complex analysis. We shall also develop the Cauchy-Riemann equations which provide an easier test to verify the analyticity of a function.

In this lecture, we shall first prove Theorem 6.5 and then through simple examples demonstrate how easily this result can be used to check the analyticity of functions. We shall also show that the real and imaginary parts of an analytic function are solutions of the Laplace equation.

We have already seen that the complex exponential function e

z

= e

x

(cos y + i sin y) is entire, and d(e

z

)/dz = e

z

. In this lecture, we shall first provide some further properties of the exponential function, and then define complex trigonometric and hyperbolic functions in terms of e

z

.

In this lecture, we shall introduce the complex logarithmic function, study some of its properties, and then use it to define complex powers and inverse trigonometric functions.

In this lecture, we shall present a graphical representation of some elementary functions. For this, we will need two complex planes representing, respectively, the domain and the image of the function.

In this lecture, we shall study graphical representations of the Möbius transformation, the trigonometric mapping sin z, and the function z

1/2

.

In this lecture, we define a few terms that will be used repeatedly in complex integration. We shall also state Jordan’s Curve Theorem, which seems to be quite obvious; however, its proof is rather complicated.

In this lecture, we shall introduce integration of complex-valued functions along a directed contour. For this, we shall begin with the integration of complex-valued functions of a real variable. Our approach is based on Riemann integration from calculus. We shall also prove an inequality that plays a fundamental role in our later lectures.

The main result of this lecture is to provide conditions on the function

f

so that its contour integral is independent of the path joining the initial and terminal points. This result, in particular, helps in computing the contour integrals rather easily.

In this lecture, we shall prove that the integral of an analytic function over a simple closed contour is zero. This result is one of the most fundamental theorems in complex analysis.

In this lecture, we shall show that the integral of a given function along some given path can be replaced by the integral of the same function along a more amenable path.

In this lecture, we shall present Cauchy’s integral formula that expresses the value of an analytic function at any point of a domain in terms of the values on the boundary of this domain, and has numerous important applications. We shall also prove a result that paves the way for the Cauchy’s integral formula for derivatives given in the next lecture.

In this lecture, we shall show that, for an analytic function in a given domain, all the derivatives exist and are analytic. This result leads to Cauchy’s integral formula for derivatives. Next, we shall prove Morera’s Theorem, which is a converse of the Cauchy–Goursat Theorem. We shall also establish Cauchy’s inequality for the derivatives, which plays an important role in proving Liouville’s Theorem.

In this lecture, we shall prove the Fundamental Theorem of Algebra, which states that every nonconstant polynomial with complex coefficients has at least one zero. Then, as a consequence of this theorem, we shall establish that every polynomial of degree

n

has exactly

n

zeros, counting multiplicities. For a given polynomial, we shall also provide some bounds on its zeros in terms of the coefficients.

In this lecture, we shall prove that a function analytic in a bounded domain and continuous up to and including its boundary attains its maximum modulus on the boundary. This result has direct applications to harmonic functions.

In this lecture, we shall collect several results for complex sequences and series of numbers. Their proofs require essentially the same arguments as in calculus.

In this lecture, we shall prove some results for complex sequences and series of functions. These results will be needed repeatedly in later lectures.

Power series are a special type of series of functions that are of fundamental importance. For a given power series we shall introduce and show how to compute its radius of convergence. We shall also show that within its radius of convergence a power series can be integrated and differentiated term-by-term.

In this lecture, we shall prove Taylor’s Theorem, which expands a given analytic function in an infinite power series at each of its points of analyticity. The novelty of the proof comes from the fact that it requires only Cauchy’s integral formula for derivatives.

In this lecture, we shall expand a function that is analytic in an annulus domain. The resulting expansion, although it resembles a power series, involves positive as well as negative integral powers of (z – z

0

). From an applications point of view, such an expansion is very useful.

In this lecture, we shall use Taylor’s series to study zeros of analytic functions. We shall show that unless a function is identically zero, about each point where the function is analytic there is a neighborhood throughout which the function has no zero except possibly at the point itself; i.e., the zeros of an analytic function are isolated. We begin by proving the following theorem.

The uniqueness result established in Corollary 26.5 will be used here to discuss an important technique in complex function theory known as analytic continuation. The principal task of this technique is to extend the domain of a given analytic function.

The cross ratio defined in Problem 11.9 will be used here to introduce the concept of symmetry of two points with respect to a line or a circle. We shall also prove

Schwarz’s Reflection Principle

, which is of great practical importance for analytic continuation.

In this lecture, we shall define, classify, and characterize singular points of complex functions. We shall also study the behavior of complex functions in the neighborhoods of singularities.

Expanding a function in a Laurent series is often difficult. Therefore, in this lecture we shall find the behavior of an analytic function in the neighborhood of an essential singularity. We shall also discuss zeros and singularities of analytic functions at infinity.

In this lecture, we shall use Laurent’s expansion to establish Cauchy’s Residue Theorem, which has far-reaching applications. In particular, it generalizes Cauchy’s integral formula for derivatives (18.5), so that integrals that have a finite number of isolated singularities inside a contour can be integrated rather easily.

In this lecture and the next, we shall show that the theory of residues can be applied to compute certain types of definite as well as improper real integrals. Some of these integrals occur in physical and engineering applications, and often cannot be evaluated by using the methods of calculus.

We begin this lecture with two examples where we need to use appropriate contours to evaluate certain proper and improper real integrals. We shall also prove Jordan’s Lemma, which plays a fundamental role in the computation of integrals involving rational functions multiplied by trigonometric functions.

In previous lectures, when evaluating the real improper integrals we assumed that the integrand has no singularity over the whole interval of integration. In this lecture, we shall show that by using indented contours some functions which have simple poles at certain points on the interval of integration can be computed.

In previous lectures, we successfully applied contour integration theory to evaluate integrals of real-valued functions. However, often it turns out that the extension of a real function to the complex plane is a multi-valued function. In this lecture, we shall show that by using contours cleverly some such functions can also be integrated.

One of the remarkable applications of the Residue Theorem is that we can sum ∑

n∈Z

f

(

n

) for certain types of functions

f

(

z

)

.

For this, we shall prove the following result.

We begin this lecture with an extension of Theorem 26.3 known as the Argument Principle. This result is then used to establish Rouché’s Theorem, which provides locations of the zeros and poles of meromorphic functions. We shall also prove an interesting result due to Hurwitz.

In this lecture, we shall use the Rouché Theorem to investigate the behavior of the mapping

f

generated by an analytic function

w

=

f

(

z

)

.

Then, we shall study some properties of the inverse mapping f

-1

. We shall also discuss functions that map the boundaries of their domains to the boundaries of their ranges. Such results are of immense value for constructing solutions of the Laplace equation with boundary conditions.

In Lecture 10, we saw that the nonconstant linear mapping (10.1) is an expansion or contraction and a rotation, followed by a translation. Thus, under a linear mapping, the angle between any two intersecting arcs in the

z

-plane is equal to the angle between the images in the

w

-plane. Mappings that have this

angle-preserving

property are called

conformal mappings

. These mappings are of immense importance in solving boundary value problems involving Laplace’s equation. We begin with the following definitions.

In this lecture, we shall employ earlier results to establish some fundamental properties of harmonic functions. The results obtained strengthen our understanding of harmonic functions and are of immense help in solving boundary value problems for the Laplace equation. We begin by proving the following result.

In this lecture, we shall provide an explicit formula for the derivative of a conformal mapping that maps the upper half-plane onto a given bounded or unbounded polygonal region (boundary contains a finite number of line segments). The integration of this formula (often a formidable task unless done numerically) and then its inversion (another nontrivial task) yields a conformal mapping that maps a polygonal region onto the upper halfplane. Such mappings are often applied in physical problems such as in heat conduction, fluid mechanics, and electrostatics.

In this lecture, we shall introduce infinite products of complex numbers and functions and provide necessary and sufficient conditions for their convergence.

In this lecture, we shall provide representations of entire functions as finite/infinite products involving their finite/infinite zeros. We begin with the following simple cases.

In this lecture, we shall construct a meromorphic function in the entire complex plane with preassigned poles and the corresponding principal parts.

Recall from Lecture 8 that a complex number ω ≠ 0 is a

period

of a function

f

(

z

) if

f

(

z

+ ω) =

f

(

z

) for all

z

. For example,

e

z

has the period 2π

i

, and sin

z

and cos

z

have the period 2π. If ω

1

and ω

2

are periods of

f

(

z

), then

$$f(z + \omega _1 + \omega _2 )\, = \,f(z + \omega _1 ){\rm } = {\rm }f(z);$$

i.e.,

ω

1

+

ω

2

is also a period. In particular, if

ω

is a period, then

nω

is also a period, where

n

is any integer.

The Riemann zeta function is one of the most important functions of classical mathematics, with a variety of applications in analytic number theory. In this lecture, we shall study some of its elementary properties.

Let

S

be the class of functions that are analytic and one-to-one in the unit disk

B

(0, 1) and are normalized by the conditions

f

(0) = 0 and

f

’(0) = 1. The class

S

has many interesting properties.

A Riemann surface is an ingenious construct for visualizing a multivalued function. We treat all branches of a multi-valued function as a single-valued function on a domain that consists of many sheets of the

z

plane. These sheets are then glued together so that in moving from one sheet to another we pass continuously from one branch of the multi-valued function to another. This glued structure of sheets is called a Riemann surface for the multi-valued function. For example, in a multi-story car park, floors can be thought of as sheets of the

z

-plane, that are glued by the ramps on which cars can go from one level to another. Riemann surfaces have proved to be of inestimable value, especially in the study of algebraic functions. Although there is much literature on the subject, in this lecture we shall construct Riemann surfaces for some simple functions.

In this lecture, we shall discuss the geometric and topological features of the complex plane associated with dynamical systems whose evolution is governed by the iterative scheme

$$z_n {\rm + 1}\,{\rm = }\,f{\rm (}zn{\rm ), }\,z_{0\,} {\rm = }\,p{\rm }\,{\rm where}\,{\rm }f{\rm (}z{\rm )}$$

is a complex valued function and

$$p\, \in \,C.$$

Such systems occur in physical, engineering, medical, and aesthetic problems, especially those exhibiting chaotic behavior.

The problem of complex numbers dates back to the 1st century, when Heron of Alexandria (about 75 AD) attempted to find the volume of a frustum of a pyramid, which required computing the square root of 81

-

144 (though negative numbers were not conceived in the Hellenistic world). We also have the following quotation from Bhaskara Acharya (working in 486 AD), a Hindu mathematician: “The square of a positive number, also that of a negative number, is positive: and the square root of a positive number is two-fold, positive and negative; there is no square root of a negative number, for a negative number is not square.” Later, around 850 AD, another Hindu mathematician, Mahavira Acharya, wrote: “As in the nature of things, a negative (quantity) is not a square (quantity), it has therefore no square root.” In 1545, the Italian mathematician, physician, gambler, and philosopher Girolamo Cardano (1501-76) published his

Ars Magna

(The Great Art), in which he described algebraicmethods for solving cubic and quartic equations. This book was a great event in mathematics. In fact, it was the first major achievement in algebra in 3000 years, after the Babylonians showed how to solve quadratic equations. Cardano also dealt with quadratics in his book.