Proof of Even Harmonics Absence in Function w/ Odd Symmetry | Fourier Transform

[magnifik]

This isn't really a homework problem, but I am having trouble understanding why this is true:

A function with the following symmetry does not have any even harmonics in its spectrum.

I understand the concept based on odd/even symmetry properties, but can anyone provide a mathematical proof?

 

[vela]

I'd just try calculating the 2n-th Fourier coefficients. For example,

a_{2n} = \frac{1}{T}\int_{-T/2}^{T/2} f(t) \cos 2n\omega t\,dt = \frac{1}{T}\left(\int_{-T/2}^{0} f(t) \cos 2n\omega t\,dt + \int_{0}^{T/2} f(t) \cos 2n\omega t\,dt\right)

where ω=2π/T. Try using a change of variable like t' = t + T/2 on the first integral, then use the symmetry of f(t), and see if stuff cancels out.