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## Main Question or Discussion Point

Hello everyone,

I know that the integral of an odd function over a symmetric interval is 0, but there's something that's bothering my mind about it.

Consider, for example, the following isosceles trapezoidal wave in the interval [0,L]:

When expressed in Fourier series, the coefficient multiplied by the cosine (a_n) drops, but to be honest, I don't see why. By looking at the graph above, the only thing I can tell is that there is reflection symmetry of the function when looked at from L/2 which clearly doesn't indicate an odd function but rather an even one. Also, the interval is symmetric only when looked at from L/2. So, I can't really see how the theorem of odd functions in a symmetric interval is applied (Unless this isn't why the coefficient a_n is zero :p).

I do, however, see by looking at the graph that the function has a sine-like structure so the Fourier series is most probably going to be consisted of sines (Not sure though if that's a way to look at it. Correct me if I'm wrong).

But even when I calculated the coefficient a_n, it didn't drop. Checked my calculations a couple of times and I'm yet to find anything wrong (if needed, I'll upload it).

I'd be really happy if someone could get this straight for me.

Thank you!

P.S: wasn't sure whether to upload here or to a more physics-related section since I'm talking about waves and strings, move it if needed.

I know that the integral of an odd function over a symmetric interval is 0, but there's something that's bothering my mind about it.

Consider, for example, the following isosceles trapezoidal wave in the interval [0,L]:

When expressed in Fourier series, the coefficient multiplied by the cosine (a_n) drops, but to be honest, I don't see why. By looking at the graph above, the only thing I can tell is that there is reflection symmetry of the function when looked at from L/2 which clearly doesn't indicate an odd function but rather an even one. Also, the interval is symmetric only when looked at from L/2. So, I can't really see how the theorem of odd functions in a symmetric interval is applied (Unless this isn't why the coefficient a_n is zero :p).

I do, however, see by looking at the graph that the function has a sine-like structure so the Fourier series is most probably going to be consisted of sines (Not sure though if that's a way to look at it. Correct me if I'm wrong).

But even when I calculated the coefficient a_n, it didn't drop. Checked my calculations a couple of times and I'm yet to find anything wrong (if needed, I'll upload it).

I'd be really happy if someone could get this straight for me.

Thank you!

P.S: wasn't sure whether to upload here or to a more physics-related section since I'm talking about waves and strings, move it if needed.