Re: How to determine f(x), given f(x)*f(-x) = -exp(kx^2) ?

From: Robert Israel (israel_at_math.ubc.ca)
Date: 11/19/04 

Date: 19 Nov 2004 07:45:01 -0500

In article <cniqjd$7j5$1@dizzy.math.ohio-state.edu>,
Nischal Piratla <nischal@engr.colostate.edu> wrote:

>How to determine f(x), given f(x)*f(-x) = -exp(kx^2) ? where x is a
>complex variable.
>Is there a method that I could use?

I assume you're looking for solutions that are entire functions.
Note that f(x) can't ever be 0, so it must have an analytic logarithm g(x).
Thus exp(g(x) + g(-x)) = -exp(k x^2), so g(x) + g(-x) = (2 n + 1) i pi + k x^2
for some integer n (which by continuity must be constant). Of course one
solution is g(x) = (n + 1/2) i pi + k x^2/2, and any other will differ from
this by an arbitrary odd function. So the general solution is

f(x) = exp((n+1/2) i pi + k x^2/2 + h(x)) = (+/-) i exp(k x^2/2 + h(x))

where h(x) is an arbitrary odd entire function.

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