Reflection Formula - Known Formulae

Known Formulae

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

and for all odd functions,

A famous relationship is Euler's reflection formula

for the Gamma function Γ(z), due to Leonhard Euler.

There is also a reflection formula for the general n-th order polygamma function ψ(n)(z),

which springs trivally from the fact that the polygamma functions are defined as the derivations of the and thus its reflection formula is inherited to them.

The Riemann zeta function ζ(z) satisfies

and the Riemann Xi function ξ(z) satisfies