3.1 Taylor Series and Maclaurin Expansion
Analytic functions can be expanded into Taylor series within the circle of convergence.
$$f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z - z_0)^n$$ 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
# 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.
$$f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n$$ 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.
- Removable singularity: Principal part is 0 → $\lim_{z \to z_0} f(z)$ is finite
- Pole of order $m$: Principal part has finite terms up to $(z-z_0)^{-m}$
- Essential singularity: Principal part has infinite terms
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.
$$\text{Res}(f, z_0) = a_{-1}$$ 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.
$$\oint_C f(z) dz = 2\pi i \sum_{k} \text{Res}(f, z_k)$$ 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.
$$\int_{-\infty}^{\infty} \frac{P(x)}{Q(x)} dx = 2\pi i \sum_{\text{upper half-plane}} \text{Res}(f, z_k)$$ 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.
$$z = e^{i\theta}, \quad \cos\theta = \frac{z + z^{-1}}{2}, \quad \sin\theta = \frac{z - z^{-1}}{2i}$$ $$\int_0^{2\pi} R(\cos\theta, \sin\theta) d\theta = \oint_{|z|=1} R\left(\frac{z+z^{-1}}{2}, \frac{z-z^{-1}}{2i}\right) \frac{dz}{iz}$$
3.8 Applications to Real Integrals (3): Fourier-type Integrals
Residue theorem is also effective for integrals containing $e^{iax}$.
$$\int_{-\infty}^{\infty} f(x) e^{iax} dx = 2\pi i \sum_{\text{Im}(z_k)>0} \text{Res}(f(z)e^{iaz}, z_k) \quad (a > 0)$$
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.
- Poles of Green's function → Lattice vibration modes (phonons)
- Spectral function → Density of states
- Complex frequency → Damping of vibrations
📝 Chapter Exercises
- Find the Laurent expansion of $f(z) = \frac{e^z}{z^3}$ around $z=0$.
- Calculate residues of $f(z) = \frac{1}{z(z-1)(z-2)}$ at all singularities.
- Calculate $\int_{-\infty}^{\infty} \frac{dx}{1+x^4}$ using residue theorem.
- Calculate $\int_0^{2\pi} \frac{d\theta}{3 + 2\cos\theta}$ using residue theorem.
Summary
- Laurent series provides function representation near singularities
- Complex real integrals can be calculated using residue theorem
- Wide applications in physics (quantum mechanics, statistical mechanics)
- Understanding residues is important even in numerical computation