Fourier series coefficient (half range)

[jmher0403]

Homework Statement

f(x) = 1, 0<x<1

Extend f(x) t generate an even function P(x) and find Fourier coefficients

Homework Equations

a_n = 2/T ∫ P(x)cos(2nx/T) dx

The Attempt at a Solution

P(x) = 1, -1<x<1
0, -2<x<-1 , 1<x<2

even function so b₀ = 0

Average of P(x) over T = 0.5

a_n = 2/n∏ sin (n∏x)

I got right upto here...

answer for the exercise says

a_n = 0 when n even
2/n∏ when n=1,5,9,13...
-2/n∏ when n = 3,7,11,15...

I am confused because isn't all mutiples of pi in a sine function all equal to 0?

Please help :(

 

[LCKurtz]

Homework Statement

f(x) = 1, 0<x<1

Extend f(x) t generate an even function P(x) and find Fourier coefficients

Homework Equations

a_n = 2/T ∫ P(x)cos(2nx/T) dx

The Attempt at a Solution

P(x) = 1, -1<x<1
0, -2<x<-1 , 1<x<2

What does T represent and what is it in this problem? Is it the full period? If so, with your choice of P(x), apparently T= 4? Is that what you used in your formulas? If you are going to extend f(x) = 1 on (0,1) to an even function, why not just use f(x) = 1 on (-1,1) for your function? I don't think you have given us a complete statement of the problem.