By analogy with the lemniscate functions, hyperbolic lemniscate functions can also be defined
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
is a hypergeometric function.
Let 
and 
,
and write
 |  |  |
(5)
|
 |  |  |
(6)
|
where 
is the constant obtained by setting 
and 
, which is given by
 |  |  |
(7)
|
 |  |  |
(8)
|
with 
is a complete elliptic integral
of the first kind. Ramanujan showed that
 |
(9)
|
![1/8pi-1/2tan^(-1)(v^2)=sum_(n=0)^infty((-1)^ncos[(2n+1)theta])/((2n+1)cosh[1/2(2n+1)pi])](/images/equations/HyperbolicLemniscateFunction/NumberedEquation2.svg) |
(10)
|
and
![ln((1+v)/(1-v))=ln[tan(1/4pi+1/2theta)]+4sum_(n=0)^infty((-1)^nsin[(2n+1)theta])/((2n+1)[e^((2n+1)pi)-1])](/images/equations/HyperbolicLemniscateFunction/NumberedEquation3.svg) |
(11)
|
(Berndt 1994).
