Complex Analysis

Complex Analysis

Complex analysis is the study of functions of a complex variable, and especially of those functions which are differentiable. Initially, complex analysis is concerned with generalizing the basic notions of calculus, such as limits, differentiation and integration, to complex valued functions defined on an open subset in the complex plane. Many of the first results proved in complex analysis are simple extensions of familiar facts from calculus. However, a function which is differentiable at each point of an open subset in the complex plane has many additional special properties that go far beyond merely being differentiable. If D is an open subset of the complex plane C, if ƒ:D C is a function, we say that ƒ is differentiable at the point z in D if the limit

exists. If the limit exists then the value of the limit defines the derivative ƒ(z) at the point z. If ƒ has a derivative at each point of the open set D we say that ƒ is {\it analytic} or {\it holomorphic} on D. It is important to understand that in the limit defining the derivative of ƒ the parameter h is complex. Complex analysis is the study of analytic functions.

It turns out that an analytic function ƒ:D C is also continuous, and has infinitely many continuous derivatives at each point of its domain D. Also, a function ƒ:D C is analytic on D if and only if it has a power series expansion about each point in its domain. More precisely, if ƒ is analytic on D, if w is a point in D, if r > 0 is such that the disk (w; r) = {z∈ C: |z-w| < r\} is contained in D, then there exist complex numbers c₀, c₁, c₂, ... such that ƒ(z) is given by the absolutely convergent infinite series

at each point z in (w; r). Moreover, the coefficients c₀, c₁, c₂, ... that occur in this series are given by the formula

for each nonnegative integer n.

As an example, a polynomial of degree at most N can be written as

for some complex numbers c₀, c₁, c₂, ... c_N. Thus a polynomial is given by a finite power series expanded about the point w = 0, and so defines an analytic function at each point of the complex plane C. In this case the coefficients c_n are equal to 0 when N < n.

An example of an important analytic function that is not a polynomial is the exponential function. This is defined by the power series expansion

which converges for all complex numbers z. Another common notation used to designate the exponential function is e^z, where e = 2.71828182846 ... is the base for natural logarithms. The trigonometric functions sin z and cos z can also be defined by power series expansions. These are given by

and

and again they converge for all complex numbers z. From these power series expansions it is easy to see that the functions satisfy the identity

exp iz = cos z + i sin z

for all complex numbers z. This connection between the three functions exp z, cos z and sin z is not so apparent when they are viewed as functions of a real variable.

A further example of an analytic function is provided by the power series

In this case the power series converges to (1 - z)^-1 at each point of the open disk (0; 1). But if 1 < |z| then the series diverges and so does not define a function. However, it can be shown directly from the definition of the derivative that (1 - z)^-1 is analytic on the open set D = {z ∈ C: z 1}. Thus D is the natural domain of the analytic function (1 - z)^-1, but the power series expansion about the point 0 only defines the function in a proper subset of D. If we select a different complex number, say 3 + 4i, we get the power series expansion

which converges in the disk (3 + 4i; 20) = {z ∈ C: |z - 3 - 4i| < 20}. Notice that 20 is the distance from the point 3 + 4i to the point 1 where the function (1 - z)^-1 fails to be analytic.

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