- #1

- 178

- 4

- Homework Statement
- Consider the function ##p(t)=\sin{(t/\tau)}## for ##0\leq t <2\pi \tau## and ##p(t)=0## for ##2\pi \tau \leq t < T##, which is periodically repeated outside the interval ##[0,T)## with period ##T## (we assume ##2\pi \tau < T##). Which restrictions do you expect for the Fourier coefficients ##a_j## and which Fourier coefficient do you expect to be largest?

- Relevant Equations
- For even functions, ##a_j=a_{-j}##. For odd functions, ##a_j=-a_{-j}##. Also, I use the complex Fourier series, i.e. ##\sum_{j=-\infty}^{\infty} a_j e^{i2\pi jt/T}##. Note that for even and odd functions the coefficients are real and imaginary respectively.

I would try to determine whether ##p(t)## is even or odd. This would be so much easier if the values of ##\tau## and ##T## would be specified, but maybe it's possible to do without it, which I'd prefer. If for example ##\tau=1/2## and ##T=2\pi##, then ##p(t)=\sin{(2t)}## for ##0\leq t <\pi ## and ##p(t)=0## for ##\pi \leq t < 2\pi##. Then ##p(\pi)=0## and ##p(-\pi)=p(-\pi+2\pi)=p(\pi)=0##. The function is even (so ##a_j=a_{-j}## and ##a_j## is real).

I am unsure which Fourier coefficient will be largest. Possibly it has something to do with the frequency ##1/\tau## and the ##2\pi j/T## in the exponent of ##e## in the Fourier series. I am unsure.

I am unsure which Fourier coefficient will be largest. Possibly it has something to do with the frequency ##1/\tau## and the ##2\pi j/T## in the exponent of ##e## in the Fourier series. I am unsure.