Q: Which Fourier representation is suitable for $f(x) = \tan(x)$: Fourier trigonometric series, Fourier half-range expansion, or Fourier integral and why?
Well I searched and found that: $\tan(x)$ cannot be expanded as a Fourier series since $\tan(x)$ not satisfies Dirichlet’s.
Isn't there any Fourier representation for $\tan(x)$?
Thanks ^_^
$\tan(x)$ is odd with period $\pi$, so the half-range fourier coefficients would be given by $\frac{4}{\pi}\int_0^{\frac{\pi}{2}} \tan(x)\sin(\frac{nx}{2}) \mathrm{d}x$. But the integral diverges. However, if $f(x)=\tan(x)$ for $a<x<b$, where $a>-\frac{\pi}{2}$ and $b<\frac{\pi}{2}$, the periodic etension of $f(x)$ has a fourier transform.