- #51

Infrared

Science Advisor

Gold Member

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Nitpick: Not sure if I agree with this. Translating ##f(x)\to f(x)-c## does change the integral ##\int_0^{2\pi} |f(x)|^2 dx##, and anyway ##\int_0^{2\pi}f(x) dx## wouldn't be zero anymore. Did you have something else in mind?In general, any ##2\pi## periodic function can be translated to be zero at the end points, with the same integral and derivative. The result holds in general, therefore, with equality iff:

Edit: Nevermind, I assume you mean a translation ##f(x)\to f(x-c)##. Your solution is fine :)