Book:I.N. Sneddon/Fourier Series

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I.N. Sneddon: Fourier Series

Published $\text {1961}$, Routledge and Kegan Paul


Subject Matter


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 isoperimetric 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


Next


Cited by


Errata

Integral over $2 \pi$ of $\cos n x$

Chapter One: $\S 2$. Fourier Series


$\ds \int_0^{2 \pi} \cos n x \rd x = 0$


Fourier Series: $\paren {x - \pi}^2$, $\pi^2$

Chapter One: $\S 2$. Fourier Series: Example $1$


$\map f x \sim \ds \frac 2 3 \pi^2 + 2 \sum_{n \mathop = 1}^\infty \sqbrk {\frac {\cos n x} {n^2} + \set {\frac {\paren {-1}^n \pi} n - \frac {2 \paren {1 - \paren {-1}^n} } {\pi n^3} } \sin n x}$


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 $3$


By formulae $(3)$ the Fourier coefficients are:
$a_0 = \ds \int_{-1}^0 \rd x + \int_0^1 \rd x = 1$
$a_n = \ds \int_{-1}^0 \map \cos {n \pi x} \rd x + \int_0^1 \map \cos {n \pi x} \map \cos {\pi x} \rd x = 0$
$b_n = \ds \int_{-1}^0 \map \sin {n \pi x} \rd x + \int_0^1 \map \sin {n \pi x} \map \cos {\pi x} \rd x$
$= \dfrac {\map \cos {-n \pi} - 1} {n \pi} + \dfrac 1 {2 \pi} \set {\dfrac {1 - \map \cos {n + 1} \pi} {n + 1} + \dfrac {1 - \map \cos {n - 1} \pi} {n - 1} }$


Fourier Series: $x$ over $\openint 0 2$, $x - 2$ over $\openint 2 4$

Chapter One: $\S 6$. Half-Range Cosine Series: Example $5$


Find the half-range cosine series for
$\map f x = \begin{cases}

1 , 0 < x < 2 \\ x - 2 , 2 < x < 4 \end{cases}$

for the half-range $0 < x < 4$.

... and the required series is

$\map S x = \ds 1 + \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{r - 1} } {2 r - 1} \set {1 + \frac {4 \paren {-1}^r} {\paren {2 r - 1} \pi} } x \cos \frac {\paren {2 r - 1} \pi x} 4$.


Series Expansion for $\dfrac \pi {\sqrt 2}$

Exercises on Chapter $\text I$: $2$.


Deduce that
$\ds \sum_{n \mathop = 1}^\infty \paren {-1}^{r - 1} \frac {r - \frac 1 2} {r^2 - r + \frac 3 {16} } = \frac \pi {\sqrt 2}$


Piecewise Continuous Function with One-Sided Limits

Chapter Two: $\S 1$. Piecewise-Continuous Functions


A function $\map \psi x$ is said to be piecewise-continuous in a finite interval $\tuple {a, b}$ if:
$\text{(i)}$ the interval $\tuple{a, b}$ can be subdivided into a finite number, $m$ say, of intervals $\tuple {a, a_1}, \tuple {a_1, a_2}, \dotsc, \tuple {a_r, a_{r + 1} }, \dotsc, \tuple {a_{m - 1}, b}$, in each of which $\map f x$ is continuous;
$\text{(ii)}$ $\map f x$ is finite at the end-points of such an interval.


Fourier's Theorem: Lemma 1: Mistake 1

Chapter Two: $\S 2$. Some Important Limits


$\ds \int^b \map \psi u \sin N u \rd u = \sum_{r \mathop = 0}^{m - 1} \int_{a_r}^{a_{r + 1} } \map \psi u \sin N u \rd u$


Fourier's Theorem: Lemma 1: Mistake 2

Chapter Two: $\S 2$. Some Important Limits


... If $M$ is the greatest of the finite numbers $1 M_0 1, 1 M_1 1, \ldots, 1 M_{m - 1} 1$ we have
$\ds 1 \int_a^b \map \psi u \sin N u \rd u 1 < \dfrac {M m} N$


Fourier's Theorem: Lemma 2

Chapter Two: $\S 2$. Some Important Limits


we find that:
$\ds \int_0^a \map \psi u \frac {\sin N u} u \rd u = \map \psi {0^+} \int_0^a \frac {\sin N u} u + \int_0^a \map \phi u \sin N u \rd u$
where:
$\map \phi u = \dfrac {\map \psi u - \map \psi {0^+} } u$.


Source work progress

Revisit from $\S 2.2$: confusion over definition of piecewise differentiable.
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