Complex Fourier Series
I've been having some trouble calculating the Complex Fourier Series of a function:
f(t)= 2sin(40,000*pi*t) from 0<t<1/20,000 and 2sin(80,000*pi*t) from 1/20,000<t<1/10000. Which repeats at an frequency of 10 kHz.
My initial equation was
c_n = 10,000* [ Integral (2 sin (40,000*pi*t)*e^(-i*n*20000*pi*t) ,evaluated from (0,1/20000)) + Integral (2 sin (80,000*pi*t)*e^(-i*n*20000*pi*t) ,evaluated from (1/20000,1/10000)) ]
My first question was that if this was the right setup.
My second question is a request if anybody could do the first part of the equation:
namely: 10,000* Integral (2 sin (40,000*pi*t)*e^(-i*n*20000*pi*t) ,evaluated from (0,1/20000))
My scanner isn't working at the moment, but my method involved decomposing the sin function into exponentials, combining the exponentials together, integrating, converting the exponentials back in terms of sin and cos using the formula:
I then used logic to see when the equation would be equal to zero, and ended up with:
for n=odd: 4/(4-n^2).
for n=even: 0
for n=0: 0
If anyone can help me out I would appreciate it. I'm not asking for the entire problem to be done, just if my initial setup was correct, and if possible, someone could check my answer for the first part of the equation.
THANK YOU SO MUCH!