I want to ask:How $ \hat{f}(0) + \sum_{n=1}^{\infty}2i\hat{f}(n)\sin(n\Theta)$ is a series of sines? How can I deal with $$\hat{f}(0)?$$
Do anyone have an idea?
2026-04-07 16:14:05.1775578445
Series Of Sines.
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If the function is odd, then after substituting with $n =0$ in$$\hat{f}(0) =\frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) e^{-int}dt$$ we get an integration over an odd function in a symmetric interval, which equals 0.