Fourier series odd and even functions

[sommerfugl]

Hello

I'am a little confused. In my textbook 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]

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).