Re: simplify expr. involving sinh and arcsinh

In article <8iywe.114187$xm3.88469@attbi_s21>,
Stephen Montgomery-Smith <stephen@xxxxxxxxxxxxxxxxx> wrote:
>S. Badel wrote:

>> i have this expression which i'd like to simplify

>> sinh( a*arcsinh(x/a) )

>> where

>> a is rational, between 1 and 2
>> x is real, between -1 and +1

>My guess is that it is somewhat hopeless. For example, the similar
>expression

>cos(a arccos(x))

>will involve finding the root of a cubic if a=1/3, and Chebychev
>polynomials if a is a positive integer.


I think you could say

sinh(a arcsinh(x/a)) = Im(ChebyshevT(a,ix/a))/sin(a pi/2)

but I wouldn't call that simpler.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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