Re: Handling branch cuts for hypergeometric functions
Subject: Re: Handling branch cuts for hypergeometric functions
From: Raymond Toy
Date: Sun, 13 Feb 2005 09:06:52 -0500
Barton Willis wrote:
> A&S chapter 8 on Legendre functions is confusing. The same
> name is used for different functions--only the name of the
> independent variable (x or z) is changed. (Look at the
> two formulae in 8.6.6.) I don't trust A&S 15.4.7---15.4.26.
Yes, but in the Notation section, A&S says that z is a complex value,
and x is always a real number and -1 <= x <= 1. I would think the
difference in 8.6.6 reflects the difference branch cuts taken.
I vaguely remember reading somewhere that for complex values, (z^2-1) is
used, but for real values (1-x^2) is used, due to the branch cut. There
are similar cases where the negative of the expression is used.
Unfortunately, I don't remember where I saw that.
>
> Have you considered leaving 2F1 functions as 2F1 functions and
> not using any one of 15.4.7---15.4.26 as simplifications?
That is an option. But then you don't get the results you would find in
a table of laplace transforms.
Ray