A reflection relation is a functional equation relating 
to 
,
or more generally, 
to 
.
Perhaps the best known example of a reflection formula is the gamma function identity
 |
(1)
|
originally discovered by Euler (Havil 2003, pp. 58-59).
The reflection relation for the Riemann zeta function 
is given by
 |
(2)
|
where
 |
(3)
|
and 
is the gamma function, as first suggested by Euler
in 1761 (Havil 2003, p. 193).
The xi-function has the reflection relation
 |
(4)
|
(Havil 2003, p. 203).
The Barnes G-function satisfies
 |
(5)
|
The Rogers L-function satisfies
 |
(6)
|
The tau Dirichlet series 
satisfies the reflection relation
 |
(7)
|
(Hardy 1999, p. 173).
