mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a reflection formula or reflection relation for a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

''f'' is a relationship between ''f''(''a'' − ''x'') and ''f''(''x''). It is a special case of a

functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...

, 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 Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...

special function Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined b ...

s. 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 In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

) can be employed for all arguments.

Known formulae

The

even and odd functions In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power ser ...

satisfy by definition 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, \qquad z \not\in \mathbb Z

for the

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...

\Gamma(z)

, due to

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

. There is also a reflection formula for the general ''n''-th order

polygamma function In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function: :\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z). Thus :\psi^(z) ...

''ψ''^(''n'')(''z''), :

\psi^ (1-z)+(-1)^\psi^ (z) = (-1)^n \pi \frac \cot

which springs trivially from the fact that the polygamma functions are defined as the derivatives of

\ln \Gamma

and thus inherit the reflection formula. The

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...

''ζ''(''z'') satisfies :

\frac = \frac \cos\left(\frac\right),

and the

Riemann Xi function In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann. Definition Riemann's original lower-ca ...

''ξ''(''z'') satisfies :

\xi(z) = \xi(1-z).

References

* * {{MathWorld, urlname=PolygammaFunction, title=Polygamma Function Calculus