 Even function

A function on a domain invariant under $x\mapsto -x$ is even when $f(-x)=f(x)$. Its graph is symmetric about the vertical axis. Examples include $x^2$, $\cos x$ and the centered Gaussian function $e^{-x^2}$. On a symmetric finite interval, an integrable even function satisfies ${\int }_{-a}^{a}f=2{\int }_0^{a}f$.

When a Taylor series about zero represents the function, its odd coefficients vanish: substituting $-x$ and comparing coefficients gives $(-1{)}^n{a}_n=a_n$. The analogous statement holds for a Laurent series on an annulus centered at zero. Parity does not itself guarantee such a series representation. For instance, the even smooth function $e^{-1/x^2}$, extended by zero at zero, has every Taylor coefficient zero there while being positive away from zero.