From: israel@math.ubc.ca (Robert Israel)
Subject: Re: analytic continuation
Date: 9 May 2001 10:00:02 -0500
Newsgroups: sci.math.research
Summary: "Any" two analytic functions could be related via analytic continuation

In article <9daibm$m17$1@agate.berkeley.edu>,
Paul R. Chernoff <chernoff@math.berkeley.edu> wrote:

>Suppose that f(z) = Sum_n=0^{infty} a_n z^n  and
>             g(z) = Sum_n=0^{infty} b_n z^n

>are two convergent power series in the open unit disk in the complex
>plane. Is there an effective method (algorithm, whatever) to determine
>if g is obtainable from f via analytic continuation.


I'm not sure in what "effective" sense you are given the two power series
in this context, but let's say you can approximate f(z) and g(z) 
by polynomials, uniformly, arbitrarily well on any given compact set.
In particular you can determine any finite number of the coefficients
with arbitrary accuracy.
Then the answer is no, this is never sufficient to rule out 
f and g being analytic continuations of each other. 

If D be a compact set, b a real number so (-infinity, b] and D are
disjoint, let R = Log(D-b) (where Log is the 
principal value of the natural logarithm) and S = R + 2 pi i.
Let p(z) and q(z) be polynomials with |p(z) - f(z)| < epsilon and
|q(z) - g(z)| < epsilon on D.  Take M so |p(z)| + |q(z)| < M on D.
There is a polynomial h(z) such that |h(z)-1| < epsilon/M for z in R and
|h(z)| < epsilon/M for z in S.  Consider the function 
A(z) = h(Log(z-b)) p(z) + (1-h(Log(z-b))) q(z), analytic in a 
neighbourhood of D.  This satisfies |A(z) - f(z)| < 3 epsilon for z in D.  
But analytic continuation on a loop around the branch point b produces 
B(z) = h(Log(z+2)+2 pi i) p(z) + (1-h(Log(z+2)+2 pi i)) q(z) which 
satisfies |B(z) - g(z)| < 3 epsilon for z in D.

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2