# Fourier series odd and even functions

## Main Question or Discussion Point

Hello

I'am a little confused. In my text book it is written that all odd function can be described by a sine series.

I have this following equation from an exercise:

$$A_{0}+\sum\limits_{n=1}^\infty (A_{n} cos(n \phi) + B_{n} sin(n \phi))c^{n} = sin(\dfrac{\phi}{2})$$

It's a standard fourier serie, where n and c is positive. T
hen it is written in the solution that $$B_{n}c^{n} = 0$$ because of symmetry reasons. And I'am confused because then the fourier serie only have cosine term and the function on the right hand side is an odd function?!

mathman
Off hand I would say you are right. For an odd function the A's should be 0, not the B's. Also if c is a constant, what is the point of cn, since the direct calculation of the coefficients doesn't give them.

AlephZero
Homework Helper
When you find the Fourier series, you are taking the function to be periodic with period $2\pi$.

I think your book is taking the function as $\sin \phi/2$ on the interval $[0, 2\pi]$ and extending it to be periodic for other values of $\phi$. That is an even function.

If you defined the function as $\sin \phi/2$ over the interval $[-\pi, \pi]$, that is a different function which is odd.

It could be that the book forgot to say which of these functions it is talking about.

I agree with #2, I don't see the purpose the $c^n$ (or for $c_n$, if there was a typo).