From: "j.e.mebius" <j.e.mebius@twi.tudelft.nl>
Newsgroups: sci.math
Subject: Re: lemniscatic functions - reply
Date: Fri, 18 Sep 1998 15:33:55 +0200

Delft - September 18th 1998

Dear mr Greil,

Lemniscate functions make up a subclass of so-called elliptic functions.
To be specific: 

  Like y = sin(x) is the inverse function of x = Integral from 0 to y of
  1 / sqrt (1 - x^2),

  y = sin lemn (x) is the inverse function of x = Integral from 0 to y  
of
  1 / sqrt (1 - x^4).

This integral arises in the rectification of the lemniscate and related
geometrical problems.
Lemniscate functions turn out to be elliptic functions with modulus
k = 1/sqrt(2). It is tedious work, but try and transform the above
integral into Legendre's normal form.

The best you can do is IMO to consult almost any text on elliptic
functions.
The 19th-century literature is >very< rich; key authors are Legendre,
Gauss, Jacobi, Weierstrass, and for textbooks: Dur`ege, Weber, Cayley,
Hancock, to mention only a few. 
Key authors from the 18th century are Euler, Landen. 
Some modern textbooks exist; I do not know the authors by heart.

Good luck with this! FrGr: J.E.Mebius (j.e.mebius@twi.tudelft.nl)

============ reply to ===========

Anton Greil wrote:
> 
> Is there a monography or a comprehensive article about
> the "lemniscatic functions" ?
> 
> Where questions are treated like
>  - What is the addition theorem of the lemniscatic cosine function?
>  - What is the representation of the (double periodic) lemniscatic functions
>    by their Weierstrass \wp -function ?
> 
> Thank you & greetings,
> Anton