Chapter 4: Fourier Transform and Laplace Transform

Fourier and Laplace Transforms

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

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Fundamentals Mathematics Dojo > Complex Functions and Special Functions > Chapter 4

4.1 Fourier Series

Periodic functions can be represented by series of trigonometric functions (Fourier series).

📐 Definition: Fourier Series Expansion
$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos\frac{n\pi x}{L} + b_n \sin\frac{n\pi x}{L} \right)$$ Fourier coefficients: $$a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\frac{n\pi x}{L} dx$$ $$b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\frac{n\pi x}{L} dx$$

💻 Code Example 1: Fourier Series Expansion

Python Implementation: Fourier Series Approximation of Square Wave

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0
# - scipy>=1.11.0

import numpy as np
import matplotlib.pyplot as plt
from scipy import integrate

def fourier_coefficients(f, L, n_max):
    """Calculate Fourier coefficients"""
    a0 = (1/L) * integrate.quad(f, -L, L)[0]

    a_n = []
    b_n = []

    for n in range(1, n_max + 1):
        # a_n
        integrand_a = lambda x: f(x) * np.cos(n * np.pi * x / L)
        a_n.append((1/L) * integrate.quad(integrand_a, -L, L)[0])

        # b_n
        integrand_b = lambda x: f(x) * np.sin(n * np.pi * x / L)
        b_n.append((1/L) * integrate.quad(integrand_b, -L, L)[0])

    return a0, np.array(a_n), np.array(b_n)

# Test function: square wave
L = np.pi
def square_wave(x):
    return np.where(np.abs(x) < L/2, 1.0, 0.0)

# Visualization omitted (see original code)

4.2 Fourier Transform

For non-periodic functions, we use the Fourier transform, which is a continuous version of Fourier series.

📐 Definition: Fourier Transform
$$F(\omega) = \mathcal{F}[f(t)] = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$$ Inverse Fourier transform: $$f(t) = \mathcal{F}^{-1}[F(\omega)] = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega$$

4.3 Convolution Theorem

Fourier transform converts convolution operation into simple multiplication.

📐 Theorem: Convolution Theorem
$$\mathcal{F}[f * g] = \mathcal{F}[f] \cdot \mathcal{F}[g]$$ where convolution $(f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t-\tau) d\tau$

4.4 Laplace Transform

Laplace transform is a generalization of one-sided Fourier transform and is useful for solving differential equations.

📐 Definition: Laplace Transform
$$F(s) = \mathcal{L}[f(t)] = \int_0^{\infty} f(t) e^{-st} dt$$ Main properties:

4.5 Inverse Laplace Transform and Differential Equations

Using Laplace transform, differential equations can be transformed into algebraic equations.

🔬 Application Example: Solving differential equations
Differential equation: $y” + 4y’ + 3y = e^{-t}$, $y(0) = 0$, $y’(0) = 0$

By Laplace transform:
$(s^2 + 4s + 3)Y(s) = \frac{1}{s+1}$

Solution: $Y(s) = \frac{1}{(s+1)^2(s+3)}$

4.6 Properties of Fourier Transform

Fourier transform has various useful properties.

📝 Main properties:

4.7 Window Functions and Spectral Leakage

When analyzing finite-length signals with FFT, window functions are used to suppress spectral leakage.

📐 Theorem: Characteristics of Window Functions

4.8 Applications to Materials Science: X-ray Diffraction Pattern Analysis

In crystal structure analysis, atomic arrangement in real space corresponds to reciprocal space (diffraction pattern) by Fourier transform.

🔬 Physical Significance:

📝 Chapter Exercises

✏️ Exercises

  1. Find Fourier series expansion of square wave up to 10th order and observe Gibbs phenomenon.
  2. Calculate Fourier transform of Gaussian function $f(t) = e^{-t^2/(2\sigma^2)}$ and confirm self-duality.
  3. Use convolution theorem to find frequency response of cascade connection of two lowpass filters.
  4. Solve differential equation $y” + 2y’ + 2y = \sin(t)$, $y(0)=0$, $y’(0)=1$ using Laplace transform.

Summary

← Chapter 3: Laurent Expansion Chapter 5: Special Functions →

Disclaimer

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