In mathematics, a reflection formula or reflection relation for a function f is a relationship between f(a-x) and f(x). Its is a spcial case of a functional equation, and it is very common in the literature to refer to use the term "functional equation" when "reflection formula" is meant.

The even and odd functions satisfy simple reflection relations around a=0. For all even functions,

$f(-x)\; =\; f(x),$

and for all odd functions,

$f(-x)\; =\; -f(x).$

A famous relationship is Euler's reflection formula

$\backslash Gamma(z)\backslash Gamma(1-z)\; =\; \backslash frac\{\backslash pi\}\{\backslash sin\{\backslash pi\; z\}\}$

for the Gamma function $\backslash Gamma(z)$, due to Leonhard Euler. There is also a reflection formula for the general n:th order polygamma function $\backslash psi^n(z)$,

$\backslash psi^n(1-z)+(-1)^\{n+1\}\backslash psi^n(z)\; =\; (-1)^n\; \backslash pi\; \backslash frac\{d^n\}\{d\; z^n\}\; \backslash cot\{\backslash pi\; z\}.$

The Riemann zeta function $\backslash zeta(z)$ satisfies

$\backslash zeta(1-z)\; =\; 2(2\backslash pi)^\{-z\}\; \backslash cos\backslash left(\backslash frac\{\backslash pi\; z\}\{2\}\backslash right)\backslash Gamma(z)\backslash zeta(z),$

and the xi function $\backslash xi(z)$ satisfies

$\backslash xi(z)\; =\; \backslash xi(1-z).$

Reflection formulas are useful for numerical computation of special functions. In effect, an approximation that has greater accuracy or only converges on one side of a reflection point (typically in the positive half of the complex plane) can be employed for all arguments.

References

Calculus