Chapter 7.6, Problem 12E

Interpretation Introduction

Interpretation:

The orthogonality relations are to be derived, and $\langle \text{h(θ)cos}\text{kθ}\rangle =\frac{\text{1}}{2}{\text{a}}_{\text{k}}$ is to be deduced. Also, the formulas for $b_{\text{k}}$ and ${\text{a}}_0$ are to be derived.

Concept Introduction:

The expression of the Fourier series for $\text{h}\left(\text{θ}\right)$ is

$\text{h}\left(\text{θ}\right)\text{ = }{\displaystyle \sum _{\text{k}\text{=}\text{0}}^{\infty }{\text{a}}_{\text{k}}}\text{cos}\text{kθ}\text{+}{\displaystyle \sum _{\text{k}\text{=}\text{0}}^{\infty }{\text{b}}_{\text{k}}}\text{sin}\text{kθ}$

Where $\text{h}\left(\text{θ}\right)$ is the defined function, and ${\text{a}}_{\text{0}}$, ${\text{a}}_{\text{k}},{\text{b}}_{\text{k}}$ are Fourier series coefficients which can be defined as:

$\begin{array}{l}{\text{a}}_{\text{0}}\text{=}\frac{\text{1}}{\text{2π}}{\displaystyle \underset{\text{0}}{\overset{\text{2π}}{\int }}\text{h}\left(\text{θ}\right)}\text{dθ}\\ {\text{a}}_{\text{k}}\text{=}\frac{\text{1}}{\text{π}}{\displaystyle \underset{\text{0}}{\overset{\text{2π}}{\int }}\text{h}\left(\text{θ}\right)}\text{cos}\text{kθ}\text{dθ, k}\ge \text{1}\\ {\text{a}}_{\text{k}}\text{=}\frac{\text{1}}{\text{2π}}{\displaystyle \underset{\text{0}}{\overset{\text{2π}}{\int }}\text{h}\left(\text{θ}\right)}\text{sin}\text{kθ}\text{dθ, k}\ge \text{1}\end{array}$

$\text{f(x)=}\{\begin{array}{c}\text{2}{\displaystyle \underset{\text{0}}{\overset{\text{π}}{\int }}\text{f(x)dx}\text{if}\text{f(x)}\text{is}\text{even}}\\ \text{0}\text{if}\text{f(x)}\text{is}\text{odd}\end{array}$