Find a Fourier series to represent the function exp(x) for x belongs to (-pi,pi) and hence derive pi/sinh(pi).

Question

Find a Fourier series to represent the function exp(x) for x belongs to $(-\pi, \pi)$ and hence derive $frac{\pi}{\sinh(\pi)}$.

Unable to derive the pi over sinh(pi) part...how do I do it?

Answer 1

Hint: Find $$a_0=\dfrac{1}{\pi}\int_{-\pi}^{\pi} e^x dx$$ $$a_n=\dfrac{1}{\pi}\int_{-\pi}^{\pi} e^x\cos(nx) dx$$ $$b_n=\dfrac{1}{\pi}\int_{-\pi}^{\pi} e^x\sin(nx) dx$$ then $$S(f)(x)=\frac{1}{2}a_0+\sum_{n=1}^{\infty}a_n\cos(nx)+b_n\sin(nx)$$