Re: expansion of \int f(t) exp(kt) dt
From: Robert Israel (israel_at_math.ubc.ca)
Date: 09/09/04
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Date: 9 Sep 2004 16:56:34 GMT
In article <87752c88.0409090038.15745ffb@posting.google.com>,
Tom Weston <tomweston_usenet@yahoo.com> wrote:
>If you successively integrate \int f(t) exp(kt) dt by parts
>you obtain a series on powers of (1/k),
>f exp(kt)/k - f'exp(kt)/k^2 + f''exp(kt)/k^3 + ...
>I am looking for some advice on the convergence of such
>a series, which I imagine depends on the nature of the function f(t)
>(it terminates for polynomials etc). Can it be shown to be convergent
>or asymptotic for certain classes of functions?
Your series is exp(kt)/k sum_{j=0}^infinity (D/k)^j f(t)
where D is the differentiation operator. If this converges for some t,
the terms must be bounded, so there must be an estimate
|D^j f(t)| < C |k|^j where C is a constant.
Conversely, if |D^j f(t)| <= C r^j, your series converges when |k| > r.
I don't know if there is a standard name for the class of functions
satisfying an inequality of this type. They do form an algebra:
note that if |D^j f(t)| <= C_1 r_1^j and |D^j g(t)| <= C_2 r_2^j, then
|D^j (fg)(t)| <= sum_{i=0}^j (j choose i) |D^i f(t)| |D^(j-i) g(t)|
<= C_1 C_2 sum_{i=0}^j (j choose i) r_1^i r_2^(j-i)
= C_1 C_2 (r_1 + r_2)^j
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
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