Hyperbolic Lemniscate Function

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)

(10)

and

(11)

(Berndt 1994).

REFERENCES:

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 255-258, 1994.