mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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-orie ...

is a relationship between and . 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 ...

. It is common in mathematical literature to use the term "functional equation" for what are specifically reflection formulae. Reflection formulae 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 t ...

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 by ...

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 (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

) can be employed for all arguments.

Known formulae

The

even and odd functions In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain. They are named for the parity of the powers of the ...

satisfy by definition simple reflection relations around . 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 Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

\Gamma(z)

, due to

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

. There is also a reflection formula for the general -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) ...

\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

dilogarithm In mathematics, the dilogarithm (or Spence's function), denoted as , is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself: :\operatorname_2(z) = -\int_0^z\, du \tex ...

also satisfies a reflection formula,

\operatorname_2(z)+\operatorname_2(1-z)=\zeta(2)-\ln (z)\ln(1-z)

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 and its analytic c ...

satisfies

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

and the Riemann Xi function satisfies

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

References

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