The lemniscate functions arise in rectifying the Arc Length of the Lemniscate. The lemniscate functions
were first studied by Jakob Bernoulli and G. Fagnano. A historical account is given by
Ayoub (1984), and an extensive discussion by Siegel (1969). The lemniscate functions were the first functions defined by
inversion of an integral, which was first done by Gauß.

(1) 
Define the functions
where

(4) 
and write
There is an identity connecting and since

(7) 
so

(8) 
These functions can be written in terms of Jacobi Elliptic Functions,

(9) 
Now, if
, then
Let
so
,

(11) 

(12) 

(13) 
and

(14) 
Similarly,

(16) 

(17) 
and

(18) 
We know

(19) 
But it is true that

(20) 
so

(21) 

(22) 

(23) 
By expanding
in a Binomial Series and integrating term by term, the arcsinlemn function can be
written

(24) 
where is the Rising Factorial (Berndt 1994). Ramanujan gave the following inversion
Formula for . If

(25) 
where

(26) 
is the constant obtained by letting and , and

(27) 
then

(28) 
(Berndt 1994). Ramanujan also showed that if
, then

(29) 

(30) 

(31) 

(32) 
and

(33) 
(Berndt 1994).
A generalized version of the lemniscate function can be defined by
letting
and . Write

(34) 
where is the constant obtained by setting and . Then

(35) 
and Ramanujan showed

(36) 
(Berndt 1994).
See also Hyperbolic Lemniscate Function
References
Ayoub, R. ``The Lemniscate and Fagnano's Contributions to Elliptic Integrals.'' Arch. Hist. Exact Sci.
29, 131149, 1984.
Berndt, B. C. Ramanujan's Notebooks, Part IV.
New York: SpringerVerlag, pp. 245, and 247255, 258260, 1994.
Siegel, C. L. Topics in Complex Function Theory, Vol. 1. New York: Wiley, 1969.
© 19969 Eric W. Weisstein
19990526