# Homework Help: Fourier series

Tags:
1. Dec 13, 2016

### arpon

1. The problem statement, all variables and given/known data
Can the following function be represented by a Fourier series over the range indicated:
$$f(x) = \cos^{-1}(\sin {2x}),~~~~-\infty<x<\infty$$

2. Relevant equations
The Dirichlet conditions that a function must satisfy before it can be represented
by a Fourier series are:
(i) the function must be periodic;
(ii) it must be single-valued and continuous, except possibly at a finite number
of finite discontinuities;
(iii) it must have only a finite number of maxima and minima within one
period;
(iv) the integral over one period of |f(x)| must converge.

3. The attempt at a solution
$$f(x)=\cos^{-1}(\sin{2x})=\cos^{-1}(\cos(\frac{\pi}{2}-2x))=\frac{\pi}{2}-x$$
So it does not satisfy condition (i).
But according to the book, this function fails condition (ii)

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted