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Long story short: fourier series is taken in two subjects (for now).

One doc says that the dc component is 0 if the function is odd.

The other says that odd and even has no effect on the dc component; it is the symmetry to x-axis that matters.

Is there a function that is odd , has a dc component and periodic?

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The dc component of $x\mapsto f(-x)$ is the same as for $f$. The dc componet of $x\mapsto -f(x)$ is the negative of the dc component of $f$. Hence for odd $f$, the dc component equals its own negative and must be zero.

Also not that no function (except constant zero) is symmetric to the $x$-axis.

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If by DC component you mean the overall integral of the function (i.e. the amount of the curve above the x axis minus the amount below it) then both are correct in that an odd function is also symmetrical about the y axis. As pointed out, no function is symmetrical over the x axis except $f(x)=0$ but what I assume you meant was it is the same for the positive x axis as it is in the negative x axis.

An odd function is not necessarily periodic, but will never have a dc component.

Also this is a mathematics forum not a physics forum... dc component is not a mathematical term.

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