Unit 5 — Complex Variable: Integration

This unit focuses on complex integration, contour integration, Cauchy's theorems, Taylor and Laurent series, singularities, zeros of analytic functions, residues, and residue theory.

1. Complex Integration

Let:

z = x + iy

A complex function can be written as:

f(z) = u(x,y) + iv(x,y)

A contour $C$ in the complex plane can be parameterized as:

z = z(t), \qquad a \leq t \leq b

Then:

dz = z'(t)\, dt

The complex line integral along $C$ is:

\int_C f(z)\, dz = \int_a^b f\bigl(z(t)\bigr) z'(t)\, dt

If $f(z) = u + iv$ and $dz = dx + i\, dy$ , then:

\int_C f(z)\, dz = \int_C (u + iv)(dx + i\, dy)

Expanding:

\int_C f(z)\, dz = \int_C (u\, dx - v\, dy) + i\int_C (v\, dx + u\, dy)

2. Fundamental Theorem of Complex Integration

If $f(z)$ has a primitive $F(z)$ in a domain $D$ such that:

F'(z) = f(z)

then:

\int_C f(z)\, dz = F(z_2) - F(z_1)

where $z_1$ and $z_2$ are the initial and final points of the contour.

For a closed contour ( $z_1 = z_2$ ):

\oint_C f(z)\, dz = 0

3. Cauchy's Integral Theorem

If $f(z)$ is analytic throughout a simply connected domain $D$ , then the integral around every closed contour $C$ lying inside $D$ is zero:

\oint_C f(z)\, dz = 0

If $C$ is a closed contour and $f(z)$ is analytic on and inside $C$ , then:

\oint_C f(z)\, dz = 0

This theorem is one of the fundamental results of complex integration.

4. Cauchy's Integral Formula

If $f(z)$ is analytic inside and on a simple closed contour $C$ , and $a$ lies inside $C$ , then:

f(a) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - a}\, dz

Therefore:

\oint_C \frac{f(z)}{z - a}\, dz = 2\pi i\, f(a)

provided $a$ lies inside the contour.

5. Cauchy's Integral Formula for Derivatives

The $n$ -th derivative of an analytic function can be obtained using:

f^{(n)}(a) = \frac{n!}{2\pi i} \oint_C \frac{f(z)}{(z - a)^{n+1}}\, dz

Therefore:

\oint_C \frac{f(z)}{(z - a)^{n+1}}\, dz = \frac{2\pi i}{n!}\, f^{(n)}(a)

First Derivative ( $n = 1$ )

f'(a) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{(z - a)^2}\, dz

Second Derivative ( $n = 2$ )

f''(a) = \frac{2!}{2\pi i} \oint_C \frac{f(z)}{(z - a)^3}\, dz

6. Taylor Series

If $f(z)$ is analytic in a neighborhood of $z_0$ , then it can be represented by a Taylor series:

f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n

The coefficients are:

a_n = \frac{f^{(n)}(z_0)}{n!}

Therefore:

f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z - z_0)^n

Expanded form:

f(z) = f(z_0) + f'(z_0)(z - z_0) + \frac{f''(z_0)}{2!}(z - z_0)^2 + \cdots

7. Maclaurin Series

When $z_0 = 0$ , the Taylor series becomes the Maclaurin series:

f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} z^n

f(z) = f(0) + f'(0)z + \frac{f''(0)}{2!} z^2 + \cdots

8. Common Taylor Series

Exponential Function

e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}

e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots

Sine Function

\sin z = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n+1}}{(2n+1)!}

\sin z = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \cdots

Cosine Function

\cos z = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n}}{(2n)!}

\cos z = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \cdots

Geometric Series

For $|z| < 1$ :

\frac{1}{1 - z} = \sum_{n=0}^{\infty} z^n = 1 + z + z^2 + z^3 + \cdots

9. Laurent Series

If a function is analytic in an annular region around $z_0$ , it can be represented using a Laurent series:

f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n

This can be separated into the principal part and the analytic part:

f(z) = \sum_{n=1}^{\infty} \frac{a_{-n}}{(z - z_0)^n} + \sum_{n=0}^{\infty} a_n (z - z_0)^n

The coefficients are given by:

a_n = \frac{1}{2\pi i} \oint_C \frac{f(z)}{(z - z_0)^{n+1}}\, dz

where $C$ is a positively oriented closed contour lying in the annular region of analyticity.

10. Principal Part

The negative-power terms of the Laurent series are called the principal part:

\frac{a_{-1}}{z - z_0} + \frac{a_{-2}}{(z - z_0)^2} + \frac{a_{-3}}{(z - z_0)^3} + \cdots

The coefficient $a_{-1}$ is particularly important because it is the residue.

11. Singularities

A point $z = z_0$ is a singularity of $f(z)$ if $f(z)$ is not analytic at .

The main types of isolated singularities are:

Removable singularity
Pole
Essential singularity

12. Removable Singularity

\lim_{z \to z_0} f(z)

exists and is finite, then the singularity can be removed by defining:

f(z_0) = \lim_{z \to z_0} f(z)

In the Laurent expansion, a removable singularity has no negative-power terms:

f(z) = a_0 + a_1(z - z_0) + a_2(z - z_0)^2 + \cdots

13. Pole

A function has a pole of order $m$ at $z = z_0$ if:

f(z) = \frac{\phi(z)}{(z - z_0)^m}

where $\phi(z)$ is analytic at $z_0$ and

\phi(z_0) \neq 0

Equivalently:

\lim_{z \to z_0} (z - z_0)^m f(z) = \phi(z_0) \neq 0

For a simple pole ( $m = 1$ ):

\lim_{z \to z_0} (z - z_0) f(z) \neq 0

14. Essential Singularity

A singularity is essential if the Laurent series contains infinitely many negative-power terms:

f(z) = \frac{a_{-1}}{z - z_0} + \frac{a_{-2}}{(z - z_0)^2} + \frac{a_{-3}}{(z - z_0)^3} + \cdots

with infinitely many non-zero terms in the principal part.

Example:

e^{1/z}

has an essential singularity at $z = 0$ because

e^{1/z} = 1 + \frac{1}{z} + \frac{1}{2! z^2} + \frac{1}{3! z^3} + \cdots

15. Zeros of Analytic Functions

A point $z = z_0$ is called a zero of $f(z)$ if:

f(z_0) = 0

f(z_0) = 0,\quad f'(z_0) = 0,\quad \ldots,\quad f^{(m-1)}(z_0) = 0

but

f^{(m)}(z_0) \neq 0

then $z_0$ is a zero of order $m$ .

The Taylor expansion near $z_0$ is:

f(z) = (z - z_0)^m \phi(z)

where $\phi(z_0) \neq 0$ .

A simple zero satisfies:

f(z_0) = 0, \qquad f'(z_0) \neq 0

16. Residue

The residue of $f(z)$ at $z = z_0$ is the coefficient of $\dfrac{1}{z - z_0}$ in the Laurent series.

f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n

then:

\operatorname{Res}(f, z_0) = a_{-1}

The residue can also be calculated using:

\operatorname{Res}(f, z_0) = \frac{1}{2\pi i} \oint_C f(z)\, dz

where $C$ encloses $z_0$ and no other singularity.

17. Residue at a Simple Pole

If $z_0$ is a simple pole of $f(z)$ , then:

\operatorname{Res}(f, z_0) = \lim_{z \to z_0} (z - z_0) f(z)

f(z) = \frac{\phi(z)}{\psi(z)}

where $\psi(z_0) = 0$ and $\psi'(z_0) \neq 0$ , then:

\operatorname{Res}(f, z_0) = \frac{\phi(z_0)}{\psi'(z_0)}

18. Residue at a Pole of Order $m$

If $z = z_0$ is a pole of order $m$ , then:

\operatorname{Res}(f, z_0) = \frac{1}{(m-1)!} \lim_{z \to z_0} \frac{d^{m-1}}{dz^{m-1}} \Bigl[ (z - z_0)^m f(z) \Bigr]

This formula is particularly useful for higher-order poles.

19. Cauchy's Residue Theorem

If $f(z)$ is analytic inside and on a closed contour $C$ , except for isolated singularities $z_1, z_2, \ldots, z_n$ inside , then:

\oint_C f(z)\, dz = 2\pi i \sum_{k=1}^{n} \operatorname{Res}(f, z_k)

Therefore:

\oint_C f(z)\, dz = 2\pi i \Bigl[ \operatorname{Res}(f, z_1) + \operatorname{Res}(f, z_2) + \cdots + \operatorname{Res}(f, z_n) \Bigr]

This theorem provides an efficient way to evaluate many contour integrals.

20. Residue at Infinity

The residue at infinity is defined by:

\operatorname{Res}(f, \infty) = -\operatorname{Res}\left( \frac{1}{z^2} f\left(\frac{1}{z}\right),\, 0 \right)

The sum of all residues on the extended complex plane is:

\sum_{\text{finite } z_k} \operatorname{Res}(f, z_k) + \operatorname{Res}(f, \infty) = 0

21. Application of Residue Theory to Contour Integrals

Suppose

f(z) = \frac{\phi(z)}{\psi(z)}

has isolated poles $z_1, z_2, \ldots, z_n$ inside . Then:

\oint_C \frac{\phi(z)}{\psi(z)}\, dz = 2\pi i \sum_{k=1}^{n} \operatorname{Res}\left( \frac{\phi(z)}{\psi(z)},\, z_k \right)

For simple poles:

\oint_C \frac{\phi(z)}{\psi(z)}\, dz = 2\pi i \sum_{k=1}^{n} \frac{\phi(z_k)}{\psi'(z_k)}

provided $\psi(z_k) = 0$ and $\psi'(z_k) \neq 0$ .

22. Important Series and Integration Formulas

Taylor Series

f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z - z_0)^n

Laurent Series

f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n

Laurent Coefficient

a_n = \frac{1}{2\pi i} \oint_C \frac{f(z)}{(z - z_0)^{n+1}}\, dz

Cauchy's Integral Formula

f(a) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - a}\, dz

Cauchy's Formula for Derivatives

f^{(n)}(a) = \frac{n!}{2\pi i} \oint_C \frac{f(z)}{(z - a)^{n+1}}\, dz

Residue

\operatorname{Res}(f, z_0) = a_{-1}

Simple Pole

\operatorname{Res}(f, z_0) = \lim_{z \to z_0} (z - z_0) f(z)

Pole of Order $m$

\operatorname{Res}(f, z_0) = \frac{1}{(m-1)!} \lim_{z \to z_0} \frac{d^{m-1}}{dz^{m-1}} \Bigl[ (z - z_0)^m f(z) \Bigr]

Residue Theorem

\oint_C f(z)\, dz = 2\pi i \sum_k \operatorname{Res}(f, z_k)

What You'll Learn

By completing this unit, you will be able to:

Understand complex line and contour integration.
Parameterize contours in the complex plane.
Apply Cauchy's Integral Theorem.
Apply Cauchy's Integral Formula.
Calculate higher derivatives using Cauchy's formula.
Expand analytic functions using Taylor series.
Work with Laurent series and principal parts.
Identify and classify isolated singularities.
Distinguish removable singularities, poles, and essential singularities.
Determine the order of zeros of analytic functions.
Calculate residues at simple and higher-order poles.
Apply Cauchy's Residue Theorem.
Evaluate difficult contour integrals using residues.
Understand residue behavior at infinity.

Key Takeaway

This unit develops the core techniques of complex integration and residue theory. You will learn how to evaluate contour integrals using Cauchy's theorems, represent complex functions using Taylor and Laurent series, classify singularities, and efficiently evaluate integrals using residue calculus.

Unit 5 — Complex Variable: Integration

1. Complex Integration

2. Fundamental Theorem of Complex Integration

3. Cauchy's Integral Theorem

4. Cauchy's Integral Formula

5. Cauchy's Integral Formula for Derivatives

First Derivative (n=1n = 1n=1)

Second Derivative (n=2n = 2n=2)

6. Taylor Series

7. Maclaurin Series

8. Common Taylor Series

Exponential Function

Sine Function

Cosine Function

Geometric Series

9. Laurent Series

10. Principal Part

11. Singularities

12. Removable Singularity

13. Pole

14. Essential Singularity

15. Zeros of Analytic Functions

16. Residue

17. Residue at a Simple Pole

18. Residue at a Pole of Order mmm

19. Cauchy's Residue Theorem

20. Residue at Infinity

21. Application of Residue Theory to Contour Integrals

22. Important Series and Integration Formulas

Taylor Series

Laurent Series

Laurent Coefficient

Cauchy's Integral Formula

Cauchy's Formula for Derivatives

Residue

Simple Pole

Pole of Order mmm

Residue Theorem

What You'll Learn

Key Takeaway

First Derivative ( $n = 1$ )

Second Derivative ( $n = 2$ )

18. Residue at a Pole of Order $m$

Pole of Order $m$