Fourier series odd and even functions

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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:

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

It's a standard fourier serie, where n and c is positive. T
hen it is written in the solution that [tex]B_{n}c^{n} = 0[/tex] 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?!
 

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mathman
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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
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When you find the Fourier series, you are taking the function to be periodic with period [itex]2\pi[/itex].

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

If you defined the function as [itex]\sin \phi/2[/itex] over the interval [itex][-\pi, \pi][/itex], 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 [itex]c^n[/itex] (or for [itex]c_n[/itex], if there was a typo).
 

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