Question

Consider the $2\pi$ periodic odd function defined on $[0,\pi ]$ by $f(\Theta )=\Theta (\pi -\Theta )$ .

a) Draw the graph of f.

b) Compute the Fourier Coefficients of f, and show that $f(\Theta )=\frac{8}{\pi}$ $\sum_{}^{k odd\geq 1}$    $sink\Theta / k^3$

Answer 1

Solution: Consider the 27-periodic oddfunction defined on [0,7] by f(0) = θ(7-0) We have to draw the graph of s. 1.5 (0,0) (x,0) 0 1.5 3

We have to compute the coefficients of f and show that f(θ)--Σ sin ko Smce sine term is presentin _ 2 we have to consider the coefficient b, where b.-(e)sin(k0) cos(ke 8cos(ke πk sin (kz) + cos(kr)-1 2 sin (ke) 2ksin (kr)+2cos(k)-2 2 [-rksin(kz)-2cos(kr)+2

We know that sin (kr)-0VkEN 2 [-2 cos(kz)+2 k3 h-212-2cos(km) 41-cos(kn Again we know that cos(kr)=(-1) bk = 0vk is even And when k is odd we have 1-(-1) = 1-(-1) For odd k we have 0 The sine fourier series is given by

b-1 k-odd k5 π kodd21