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Fourier series

  1. Dec 13, 2016 #1
    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)
     
  2. jcsd
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