Question

Let f (x) be a periodic function on R with period 21. On the interval (-11,), f(x) is given by f(x)=sin(x) 0<x51, = Let F(x) be the Fourier series of f(x). Select all correct statements from below. The Fourier series of -f (x) is -F(x). F(-1) = 0.

Answer 1

F(x) = f(1) if f is continuous at f(x")+f(+) if f is discontinuous at

The following are TRUE

f(0-) + f(0+) . ; because f is 27 periodic. F14 )_ (47) + f(47) 1+0 1 L2 2

$F\left(\frac{\pi}{2} \right )=f\left(\frac{\pi}{2} \right )$

The fourier series of
(2)/-
is
-F(1)

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Note that remaining options are false. Reasons are given below

f(3) = f(T) = 0; f is 27 periodic.

F(-) = 2 f(--) + f(-+) 0 + (-1) TN