2014-53.md

course	course_year	question_number	tags	title	year
Methods	IB	53	IB 2014 Methods	Paper 4, Section II, D	2014

Let $f(x)$ be a complex-valued function defined on the interval $[-L, L]$ and periodically extended to $x \in \mathbb{R}$.

(i) Express $f(x)$ as a complex Fourier series with coefficients $c_{n}, n \in \mathbb{Z}$. How are the coefficients $c_{n}$ obtained from $f(x)$ ?

(ii) State Parseval's theorem for complex Fourier series.

(iii) Consider the function $f(x)=\cos (\alpha x)$ on the interval $[-\pi, \pi]$ and periodically extended to $x \in \mathbb{R}$ for a complex but non-integer constant $\alpha$. Calculate the complex Fourier series of $f(x)$.

(iv) Prove the formula

$$\sum_{n=1}^{\infty} \frac{1}{n^{2}-\alpha^{2}}=\frac{1}{2 \alpha^{2}}-\frac{\pi}{2 \alpha \tan (\alpha \pi)}$$

(v) Now consider the case where $\alpha$ is a real, non-integer constant. Use Parseval's theorem to obtain a formula for

$$\sum_{n=-\infty}^{\infty} \frac{1}{\left(n^{2}-\alpha^{2}\right)^{2}}$$

What value do you obtain for this series for $\alpha=5 / 2 ?$