Re: Multiplying two series expansion
- From: "Klueless" <klueless@xxxxxxxxxxxxxxxx>
- Date: Fri, 02 Mar 2007 17:32:03 GMT
"Claudia Mathy" <maths_freak@xxxxxxxxxxx> wrote in message news:1172855658.270664.299110@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Now we are interested in the first few terms of the Taylor expansion
of the product of f(x) and g(x).
Question: Can we simply take the first few terms of f and the terms
of g and multiply them out? Or do we need to require that the points
of expansion b and c are the same?
The points of expansion need to be the same. Given that,
it is sufficient to take the first few terms of the series for f and of
the series for g, and multiply them out, to get first few termes of
the Taylor expansion of the product of f(x) and g(x). If you do this
up to order n for f and g, then the answer is good up to order n
for f*g. The radius of convergence is at least the minimum of the
radius of convergence for the series for f and that of the series for g.
If the points of expansion aren't the same, then one *might*
in principle be able derive a new series expansion about b from
a series expansion about c, to sufficient accuracy to get good
enough approximations to the coefficients -- if say, b is within the
radius of convergence of the series about c --, and then carry out
the program of the previous paragraph to derive the first few terms
with approximately correct coefficients of the series for f*g . I would
only suggest such an undertaking as a matter of desperation, however.
.
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- Multiplying two series expansion
- From: Claudia Mathy
- Multiplying two series expansion
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