In a fourier series if $a_0$ is equal to 0 is $a_n$ also equal to 0?

Question

When im trying to determine a fourier series if I determin the $a_0$ is 0 does it follow that $a_n$ must also be 0?

Answer 1

Example is the function $f(x) = \cos x$ with period $2\pi$. In this case, the coefficient $a_0 = 0$ but $a_1 = 1$.

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Maybe I can guess your reason for the question: if $f(x) \ge 0$, then $a_0 = 0$ is $$ \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\;dx= 0 $$ and (since $f(x) \ge 0$) that does, indeed, imply $f(x) = 0$ for almost all $x$. And then we conclude that all Fourier coefficients are $0$.