Reflection formula

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

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.

Known formulaEdit

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  , due to Leonhard Euler.

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

 

which springs trivially from the fact that the polygamma functions are defined as the derivatives of   and thus inherit the reflection formula.

The Riemann zeta function ζ(z) satisfies

 

and the Riemann Xi function ξ(z) satisfies

 

ReferencesEdit

  • Weisstein, Eric W. "Reflection Relation". MathWorld.
  • Weisstein, Eric W. "Polygamma Function". MathWorld.