Book:I.N. Sneddon/Fourier Series

Subject Matter

 * Fourier Analysis

Contents

 * Preface


 * 1. The Fourier Coefficients
 * 1. Trigonometrical series
 * 2. Fourier series
 * 3. Other types of whole-range series
 * 4. Even and odd functions
 * 5. Half-range sine series
 * 6. Half-range cosine series
 * 7. Fourier series over a general range
 * 8. Orthonormal sets of functions
 * Exercises


 * 2. A Proof of Fourier's theorem
 * 1. Piecewise-continuous functions
 * 2. Some important limits
 * 3. A Fourier theorem
 * Exercises


 * 3. Properties of Fourier series
 * 1. Integration of Fourier series
 * 2. Parseval's theorem
 * 3. The root-mean-square value of a periodic function
 * 4. Differentiation of Fourier series
 * 5. Trigonometrical polynomials and Fourier polynomials
 * 6. Gibbs's phenomenon
 * 7. Hurwitz's solution of the isopoimetric problem
 * Exercises


 * 4. Applications in the solution of partial differential equations
 * 1. The transverse vibrations of a stretched string
 * 2. Impulsive functions
 * 3. Laplace's equation
 * 4. The linear diffusion equation
 * 5. Vibrations of beams
 * Exercises


 * Answers to exercises


 * Index



Integral over $2 \pi$ of $\cos n x$
Chapter One: $\S 2$. Fourier Series

Fourier Series: $\paren {x - \pi}^2$, $\pi^2$
Chapter One: $\S 2$. Fourier Series: Example $1$

Fourier Series: $1$ over $\openint {-1} 0$, $\cos \pi x$ over $\openint 0 1$
Chapter One: $\S 3$. Other Types of Whole-Range Series: Example $3$

Fourier Series: $x$ over $\openint 0 2$, $x - 2$ over $\openint 2 4$
Chapter One: $\S 6$. Half-Range Cosine Series: Example $5$

Series Expansion for $\dfrac \pi {\sqrt 2}$
Exercises on Chapter $\text I$: $2$.

Piecewise Continuous Function with One-Sided Limits
Chapter Two: $\S 1$. Piecewise-Continuous Functions

Fourier's Theorem: Lemma 1: Mistake 1
Chapter Two: $\S 2$. Some Important Limits

Fourier's Theorem: Lemma 1: Mistake 2
Chapter Two: $\S 2$. Some Important Limits

Fourier's Theorem: Lemma 2
Chapter Two: $\S 2$. Some Important Limits

Source work progress
* : Chapter Two: $\S 3$. A Fourier Theorem: Theorem $1$


 * Revisit from $\S 2.2$: confusion over definition of piecewise differentiable. Recheck entire work, for certainty.