# Characterize Fourier coefficients

• schniefen
In summary, the conversation discusses determining whether a given function is even or odd, and the possibility of doing so without specifying certain values. It is suggested that the largest Fourier coefficient may have something to do with the frequency and exponent in the Fourier series. The function in question is found to be neither even nor odd, and its properties are explored further. The conversation concludes with questions about the effect of certain values on the coefficients.

#### schniefen

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.

You are on the right track. With the ##e{}## in your Fourier series you will have a periodic ##\delta## function.

$p$ is neither even nor odd: it consists of a single complete period of $\sin(t/\tau)$ over $0 \leq t \leq 2\pi \tau$ followed by a constant zero over $2\pi \tau < t < T$. Thus $|f(-t)| = 0 \neq |f(t)|$ for $0 \leq t \leq 2\pi \tau$. The function is real, so $a_{{-}j}$ and $a_j$ are complex conjugates. The average is zero, so $a_0$ is zero.

What happens if $T$ is an integer multiple of $2\pi \tau$? What happens if this only approxiamtely true?

In this case it is easy to compute the coefficients $a_j$ expressly in order to confirm your hypotheses.