Find the Fourier series defined in the interval (-π,π) and sketch its sum over several periods.
i) f(x) = 0 (-π < x < 1/2π) f(x) = 1 (1/2π < x < π)
2. Homework Equations
ao/2 + ∑(ancos(nx) + bnsin(nx))
an = 1/π ∫f(x)cos(nx) dx
bn = 1/π ∫f(x) sin(nx)
The Attempt at a Solution
I've worked out the a0 by splitting the limits and integrating individually for -π < x < 1/2π and 1/2π < x < π. When I did this i got a0 = ½
for an = 1/π ∫(upper limit π, lower limit 1/2π) cos(nx) dx
= 1/π[1/n sin(nx)]
= 1/π ((1/n. sin(πn) - 1/n.sin(πn/2))
= 1/nπ(0- sin(πn/2)
Here is where i get stuck as sin(πn/2) is 0 for even values of n and alternates between 1, -1 for odd values.
Can i leave this written in sin form of the Fourier series as every other example i've changed the value of sin/cos to either 0 or (-1)n.
Thanks for any help!