Chapter 3: Laurent Series and Residue Theorem

Laurent Series and Residue Theorem

📖 Reading Time: 20-25 minutes 📊 Difficulty: Beginner 💻 Code Examples: 0 📝 Exercises: 0

🌐 EN | 🇯🇵 JP | Last sync: 2025-11-16

Fundamentals Mathematics Dojo > Complex Functions and Special Functions > Chapter 3

3.1 Taylor Series and Maclaurin Expansion

Analytic functions can be expanded into Taylor series within the circle of convergence.

📐 Definition: Taylor Series Expansion
$$f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z - z_0)^n$$ Maclaurin expansion ($z_0 = 0$): $$f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} z^n$$

💻 Code Example 1: Calculation of Taylor Series Expansion

Python Implementation: Function Approximation by Taylor Series

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

"""
Example: 💻 Code Example 1: Calculation of Taylor Series Expansion

Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import factorial
import sympy as sp

# Symbolic computation with SymPy
z = sp.Symbol('z')
z0 = sp.Symbol('z0')

# Function definitions
functions_sym = {
    'e^z': sp.exp(z),
    'sin(z)': sp.sin(z),
    'cos(z)': sp.cos(z),
    '1/(1-z)': 1/(1-z),
}

print("=== Taylor Series Expansion (Maclaurin expansion, z0=0) ===\n")

for name, f_sym in functions_sym.items():
    print(f"f(z) = {name}")
    # Taylor expansion (up to 10th order)
    taylor_series = sp.series(f_sym, z, 0, n=6).removeO()
    print(f"Taylor series: {taylor_series}")
    print()

# Visualization omitted (see original code)

3.2 Laurent Series Expansion

In regions containing singularities, expansion is done with Laurent series including negative powers.

📐 Definition: Laurent Series Expansion
$$f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n$$ Separated into regular part and principal part: $$f(z) = \underbrace{\sum_{n=0}^{\infty} a_n (z - z_0)^n}{\text{Regular part}} + \underbrace{\sum{n=1}^{\infty} \frac{a_{-n}}{(z - z_0)^n}}_{\text{Principal part}}$$

3.3 Classification of Singularities

Singularities are classified into three types: removable singularity, pole, and essential singularity.

📐 Theorem: Classification of Singularities

3.4 Calculation of Residues

The residue is the coefficient of $(z-z_0)^{-1}$ in Laurent expansion and is important for calculating complex integrals.

📐 Definition: Residue
$$\text{Res}(f, z_0) = a_{-1}$$ where $a_{-1}$ is coefficient of $(z-z_0)^{-1}$ in Laurent expansion $f(z) = \sum a_n (z-z_0)^n$

For pole of order $m$: $$\text{Res}(f, z_0) = \frac{1}{(m-1)!} \lim_{z \to z_0} \frac{d^{m-1}}{dz^{m-1}} [(z-z_0)^m f(z)]$$

3.5 Residue Theorem

The residue theorem allows calculating complex integrals as sum of residues.

📐 Theorem: Residue Theorem
$$\oint_C f(z) dz = 2\pi i \sum_{k} \text{Res}(f, z_k)$$ where $z_k$ are singularities inside $C$

3.6 Applications to Real Integrals (1): Rational Functions

Using residue theorem, complex real integrals can be calculated by converting to complex integrals.

🔬 Application Example: Real integrals of rational functions
$$\int_{-\infty}^{\infty} \frac{P(x)}{Q(x)} dx = 2\pi i \sum_{\text{upper half-plane}} \text{Res}(f, z_k)$$ converges when $\deg Q \geq \deg P + 2$

3.7 Applications to Real Integrals (2): Integrals with Trigonometric Functions

By substitution $z = e^{i\theta}$, integrals containing trigonometric functions can be converted to complex integrals.

📐 Definition: Transformation of Trigonometric Integrals
$$z = e^{i\theta}, \quad \cos\theta = \frac{z + z^{-1}}{2}, \quad \sin\theta = \frac{z - z^{-1}}{2i}$$ $$\int_0^{2\pi} R(\cos\theta, \sin\theta) d\theta = \oint_{|z|=1} R\left(\frac{z+z^{-1}}{2}, \frac{z-z^{-1}}{2i}\right) \frac{dz}{iz}$$

3.8 Applications to Real Integrals (3): Fourier-type Integrals

Residue theorem is also effective for integrals containing $e^{iax}$.

📐 Theorem: Fourier-type Integrals
$$\int_{-\infty}^{\infty} f(x) e^{iax} dx = 2\pi i \sum_{\text{Im}(z_k)>0} \text{Res}(f(z)e^{iaz}, z_k) \quad (a > 0)$$

3.9 Applications to Materials Science: Lattice Vibrations and Phonon Dispersion

In solid state physics, complex function theory is used when analyzing dispersion relations of lattice vibrations.

📝 Physical Significance:

📝 Chapter Exercises

✏️ Exercises

  1. Find the Laurent expansion of $f(z) = \frac{e^z}{z^3}$ around $z=0$.
  2. Calculate residues of $f(z) = \frac{1}{z(z-1)(z-2)}$ at all singularities.
  3. Calculate $\int_{-\infty}^{\infty} \frac{dx}{1+x^4}$ using residue theorem.
  4. Calculate $\int_0^{2\pi} \frac{d\theta}{3 + 2\cos\theta}$ using residue theorem.

Summary

← Chapter 2: Complex Integration Chapter 4: Fourier Transform →

Disclaimer

← Previous Chapter Back to Series Index Next Chapter →