Complex Analysis

Preliminaries to Complex Analysis

Functions on the complex plane

Continuous functions

• Definition
Let be a function defined on a set of complex numbers. We say that is continuous at the point if for every there exists such that whenever and then .
The function is said to be continuous on if it is continuous at every point of .

• Theorem 2.1
A continuous function on a compact set is bounded and attains a maximum and minimum on .

Holomorphic functions

• Definition
Let be an open set in and a complex-valued function on . The function is holomorphic at the point if the quotient
converges to a limit when . Here and with , so that the quotient is well defined. The limit of the quotient, when it exists, is denoted by , and is called the derivative of at :
It should be emphasized that in the above limit, is a complex number that may approach from any direction.

• Definition
The function is said to be holomorphic on if is holomorphic at every point of . If is a closed subset of , we say that is holomorphic on if is holomorphic in some open set containing . Finally, if is holomorphic in all of we say that is entire.

• Proposition 2.2
If and are holomorphic in , then:
1. is holomorphic in and .
2. is holomorphic in and .
3. If , then is holomorphic at and
Moreover, if and are holomorphic, the chain rule holds

• Cauchy-Riemann equations

• Definition

• Proposition 2.3
If is holomorphic at , then
Also, if we write , then is differentiable in the sense of real variables, and

• Theorem 2.4
Suppose is a complex-valued function defined on an open set . If and are continuously differentiable and satisfy the Cauchy-Riemann equations on , then is holomorphic on and .

• Theorem 2.5
Given a power series , there exists such that:
1. If the series converges absolutely.
2. If the series diverges.
Moreover, if we use the convention that and , then is given by Hadamard’s formula
The number is called the radius of convergence of the power series, and the region the disc of convergence. In particular, we have in the case of the exponential function, and for the geometric series.

• Definition
Further examples of power series that converge in the whole complex plane are given by the standard trigonometric functions; these are defined by
and they agree with the usual cosine and sine of a real argument whenever . A simple calculation exhibits a connection between these two functions and the complex exponential, namely,
These are called the Euler formulas for the cosine and sine functions.

• Theorem 2.6
The power series defines a holomorphic function in its disc of convergence. The derivative of is also a power series obtained by differentiating term by term the series for , that is,
Moreover, has the same radius of convergence as .

• Corollary 2.7
A power series is infinitely complex differentiable in its disc of convergence, and the higher derivatives are also power series obtained by termwise differentiation.

• Definition
A function defined on an open set is said to be analytic (or have a power series expansion) at a point if there exists a power series centered at , with positive radius of convergence, such that
If has a power series expansion at every point in , we say that is analytic on .

Integration along curves

• Definition
Given a smooth curve in parametrized by , and a continuous function on , we define the integral of along by
By definition, the length of the smooth curve is

• Proposition 3.1
Integration of continuous functions over curves satisfies the following properties:
1. It is linear, that is, if , then
2. If is with the reverse orientation, then
3. One has the inequality

• Theorem 3.2
If a continuous function has a primitive in , and is a curve in that begins at and ends at ,then

• Corollary 3.3
If is a closed curve in an open set , and is continuous and has a primitive in , then

• Corollary 3.4
If is holomorphic in a region and , then is constant.

Cauchy’s Theorem and Its Applications

Goursat’s theorem

• Theorem 1.1
If is an open set in , and a triangle whose interior is also contained in , then
whenever is holomorphic in .

• Corollary 1.2
If is holomorphic in an open set that contains a rectangle and its interior, then

Local existence of primitives and Cauchy’s theorem in a disc

• Theorem 2.1
A holomorphic function in an open disc has a primitive in that disc.

• Theorem 2.2 (Cauchy’s theorem for a disc)
If is holomorphic in a disc, then
for any closed curve in that disc.

• Corollary 2.3
Suppose is holomorphic in an open set containing the circle and its interior. Then

• Jordan定理

Evaluation of some integrals

Cauchy’s integral formulas

• Theorem 4.1
Suppose is holomorphic in an open set that contains the closure of a disc . If denotes the boundary circle of this disc with the positive orientation, then

• Corollary 4.2
If is holomorphic in an open set , then has infinitely many complex derivatives in . Moreover, if is a circle whose interior is also contained in , then
for all in the interior of .

• Corollary 4.3 (Cauchy inequalities)
If is holomorphic in an open set that contains the closure of a disc centered at and of radius , then

• Theorem 4.4
Suppose is holomorphic in an open set . If is a disc centered at and whose closure is contained in , then has a power series expansion at
for all , and the coefficients are given by

• Corollary 4.5 (Liouville’s theorem)
If is entire and bounded, then is constant.

• Corollary 4.6
Every non-constant polynomial with complex coefficients has a root in .

• Corollary 4.7
Every polynomial of degree has precisely roots in . If these roots are denoted by , then can be factored as

• Theorem 4.8
Suppose is a holomorphic function in a region that vanishes on a sequence of distinct points with a limit point in . Then is identically . In other words, if the zeros of a holomorphic function in the connected open set accumulate in , then .

• Corollary 4.9
Suppose and are holomorphic in a region and for all in some non-empty open subset of (or more generally for in some sequence of distinct points with limit point in ). Then throughout .

Zeros and poles

• Theorem 1.1
Suppose that is holomorphic in a connected open set , has a zero at a point , and does not vanish identically in . Then there exists a neighborhood of , a non-vanishing holomorphic function on , and a unique positive integer such that

• Theorem 1.2
If has a pole at , then in a neighborhood of that point there exist a non-vanishing holomorphic function and a unique positive integer such that

• Theorem 1.3
If has a pole of order at , then
where is a holomorphic function in a neighborhood of .
The sum
is called the principal part of at the pole , and the coefficient is the residue of at that pole. We write .The importance of the residue comes from the fact that all the other terms in the principal part, that is, those of order strictly greater than , have primitives in a deleted neighborhood of . Therefore, if denotes the principal part above and is any circle centered at , we get

• Theorem 1.4
If has a pole of order at , then
The theorem is an immediate consequence of formula (1), which implies

The residue formula

• Theorem 2.1
Suppose that is holomorphic in an open set containing a circle and its interior, except for a pole at inside . Then