Reflection formula

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

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

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

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

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

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

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

\xi (z)=\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