Question

Find the half range sine series of the function f(x) = x, when 0<x<\(\frac{\pi}{2} \) and (π-x) when \(\frac{\pi}{2} \)<x< π.

(a) \(\frac{8}{\pi}[\frac{sinx}{1^{2}} – sin\frac{(3x)}{3^{2}} + sin \frac{(5x)}{5^2} – sin \frac{(7x)}{7^{2}} +……] \)

(b) \(\frac{4}{\pi}[\frac{sinx}{1^{2}} + sin\frac{(3x)}{3^{2}} + sin \frac{(5x)}{5^2} + sin \frac{(7x)}{7^{2}} +……] \)

(c) \(\frac{8}{\pi}[\frac{sinx}{1^{2}} + sin\frac{(3x)}{3^{2}} + sin \frac{(5x)}{5^2} + sin \frac{(7x)}{7^{2}} +……] \)

(d) \(\frac{4}{\pi}[\frac{sinx}{1^{2}} – sin\frac{(3x)}{3^{2}} + sin \frac{(5x)}{5^2} – sin \frac{(7x)}{7^{2}} +……] \)

I had been asked this question in an internship interview.

My doubt is from Fourier Half Range Series in section Fourier Series of Engineering Mathematics

Answer 1

The correct option is (d) \(\frac{4}{\pi}[\frac{sinx}{1^{2}} – sin\frac{(3x)}{3^{2}} + sin \frac{(5x)}{5^2} – sin \frac{(7x)}{7^{2}} +……] \)

To elaborate: bn = \(\frac{2}{\pi}[\int_{0}^{\pi⁄2}xsinnxdx+\int_{\pi⁄2}^{\pi}(\pi-x)sinnxdx\)]

= \(\frac{4}{\pi}\frac{sin\left(n\frac{\pi}{2}\right)}{n^2} \) [this term is zero whenever n is even and when odd, it gives 1 or -1]  

= \(\frac{4}{\pi}[\frac{sinx}{1^{2}} – sin\frac{(3x)}{3^{2}} + sin \frac{(5x)}{5^2} – sin \frac{(7x)}{7^{2}} +……]. \)