The elliptic gamma function is the meromorphic function defined by the infinite product
 |
(1)
|
where 
,
,
and 
(Rains 2018). The definition is symmetric in 
and 
and gives the identity
 |
(2)
|
For 
,
a branch of its natural
logarithm is given by the absolutely convergent series
 |
(3)
|
This series follows by expanding the natural logarithm of each factor in the infinite product and summing the resulting geometric series (Rains 2018, p. 10).
An elliptic gamma function value expresses the definite integral occurring in the mean tangent diameter of an oloid.
