## Hyperbolic Lemniscate Function

By analogy with the Lemniscate Functions, hyperbolic lemniscate functions can also be defined

 (1)

 (2)

Let and , and write
 (3)

where is the constant obtained by setting and . Then
 (4)

where is a complete Elliptic Integral of the First Kind, and Ramanujan showed
 (5)

 (6)

and

 (7)

(Berndt 1994).