Reflection Relation
A reflection relation is a functional equation relating f(-x) to f(x), or more generally, f(a-x) to f(x).
Perhaps the best known example of a reflection formula is the gamma function identity
originally discovered by Euler (Havil 2003, pp. 58-59).
The reflection relation for the Riemann zeta function zeta(z) is given by
| zeta(1-z)=chi(z)zeta(z), |
(2)
|
where
| chi(z)=2(2pi)^(-z)cos(1/2piz)Gamma(z) |
(3)
|
and Gamma(z) is the gamma function, as first suggested by Euler in 1761 (Havil 2003, p. 193).
The xi-function has the reflection relation
| xi(z)=xi(1-z) |
(4)
|
(Havil 2003, p. 203).
The Barnes G-function satisfies
| G(z+1)=Gamma(z)G(z). |
(5)
|
The Rogers L-function satisfies
| L(x)+L(1-x)=1. |
(6)
|
The tau Dirichlet series f(s) satisfies the reflection relation
(Hardy 1999, p. 173).
See also
Argument Addition Relation, Argument Multiplication Relation, Functional Equation, Gamma Function, Recurrence Relation, Riemann Zeta Function, Translation RelationExplore with Wolfram|Alpha
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References
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Referenced on Wolfram|Alpha
Reflection RelationCite this as:
Weisstein, Eric W. "Reflection Relation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ReflectionRelation.html