From: "G. A. Edgar" <edgar@math.ohio-state.edu.nospam>
Subject: Re: Linear differential equation
Date: Mon, 10 May 1999 11:13:49 -0400
Newsgroups: sci.math
Keywords: Euler's function

In article <3735c485.6861566@nntp.service.ohio-state.edu>, Zbigniew
Fiedorowicz <fiedorow@math.ohio-state.edu> wrote:

> Has anyone studied the first order linear differential equation:
>   x^2 y' + (x-1)y + 1 = 0
> (in the real domain)? It has the following solution which is valid
> for x=0, given by
>                   -I(1,x)/(x exp(1/x)) for x>0
>           y   =   1                    for x=0
>                   -I(0,x)/{x exp(1/x)) for x<0
> where I(a,b) is the integral from a to b of [exp(1/t)]/t.
> (In the right hand branch I(1,x) could be replaced by I(a,x) for any
> fixed positive a, but the left hand branch is uniquely determined.)
> 
> The question I have is whether the solution is C^\infty: ie. is
> the function infinitely differentiable at x=0? If so, its MacLaurin
> series would be sum(n=0..\infty, n!x^n) with 0 radius of convergence.
> 
> [I am aware of other examples of C^\infty functions with this
>  MacLaurin series, such as that given in Gelbaum & Olmsted's
>  Counterexamples in Analysis, but this would be a nicer example if it
>  works.]
> 
> Zbigniew Fiedorowicz

By coincidence I used essentially this example in a couple 
of problems for my honors differential equations class last quarter.
See HW 2 and HW 3 on page
<http://www.math.ohio-state.edu/~edgar/H521/homework.html>.
I guess Euler worked with the series  n! (-1)^n x^n,
and ignored modern technicalities like convergence.
Spanier & Oldham mention "Euler's Function" and define it
(-1/x) exp(1/x) Ei(-1/x) where Ei is the exponential integral
function  integral(exp(t)/t, t = -infinity .. x).

I do not know a proof that it is C^infinity, but I always supposed
that would follow from the asymptotic expansion of Ei(x).

If you give the differential equation to Maple, it yields
a solution (in terms of some exponential integral function).
It is the expected solution on the uniquely determined side
of zero, but has nonzero imaginary part on the other side.

It might be interesting to find what Euler actually said about
this series.

-- 
Gerald A. Edgar              edgar@math.ohio-state.edu
Department of Mathematics    telephone: 614-292-0395 (Office)
The Ohio State University      614-292-4975 (Math. Dept.)
Columbus, OH 43210             614-292-1479 (Dept. Fax)