Re: A day of a CAS super-hero

From: Robert Israel (israel_at_math.ubc.ca)
Date: 03/20/05 

Date: 20 Mar 2005 21:53:36 GMT

In article <Rtj%d.1115$tD1.113@news02.roc.ny>,
Alec Mihailovs <alec@mihailovs.com> wrote:

>> Could you please show how you got to your representation?

>OK. The hypergeom([1/2,-3/2],[],x) is not a function - it is only a series
>1-3/4*x+... If this series was convergent in some neighborhood of 0, then we
>could extend it to complex plane using either analytical continuation, or
>some functional equations. It is not the case here. The series is convergent
>only for x=0. In this case, the function corresponding to this series can be
>determined through differential equations.

>series(hypergeom([1/2, -3/2], [], x),x,100):
>gfun[seriestodiffeq](%,y(x));

> / 2 \
> /d \ 2 |d |
> [{y(0) = 1, D(y)(0) = -3/4, -3 y(x) - 4 |-- y(x)| + 4 x |--- y(x)|
> \dx / | 2 |
> \dx /
>
> }, ogf]

>dsolve(%[1]);

> 1/2 / 1 1 \
> y(x) = 1/2 Pi |(1 - x) BesselI(1, ---) + BesselI(0, ---)|
> \ 2 x 2 x /

> 1 / 1/2
> exp(- ---) / x +
> 2 x /

> 1 / 1 1 \
> _C2 exp(- ---) |BesselK(0, ---) + (-1 + x) BesselK(1, ---)|
> 2 x \ 2 x 2 x /
> -----------------------------------------------------------
> 1/2
> x

This should be treated with much caution: the DE has an irregular singular point
at x=0, so initial conditions there are quite iffy. If I'm not mistaken, for
positive real x,

sqrt(Pi)/2/sqrt(x)*exp(-1/(2*x))*((1-x)*BesselI(1,1/(2*x))+BesselI(0,1/(2*x)))
= 1 - 3/4*x + O(x^2)

while

1/sqrt(x)*exp(-1/(2*x))*(BesselK(0,1/(2*x))+(x-1)*BesselK(1,1/(2*x)))
= O(x^2*exp(-1/x)).

So for _C2 = 0 you have a removable singularity at x=0; for any _C2 you have
the initial conditions satisfied in the limit as x -> 0 on the positive real axis,
but probably not as x -> 0 from other directions in the complex plane.
I don't know why previous Maple versions would take a nonzero value of _C2.

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