Volume 9, Issue 4
Tricks of the Trade
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In and Out
Asymptotic Expansion of a Sum
Q: Let be an arbitrary irrational number. How can I find an asymptotic approximation to
as approaches 1 from below?
A: The sum can be expressed as a Gaussian hypergeometric function.
There are at least three advantages of this representation:
1. Mathematica includes arbitrary-precision algorithms for computing this function. For example, here is the sum with and .
2. Asymptotic approximations can be computed via series expansion. Introducing , the following expansion is useful when .
Here is the asymptotic approximation with and .
3. The asymptotic series expansion in closed form can be obtained using functions.wolfram.com/07.23.06.0012.01,
Here is the result.
This can be used to compute the number of asymptotic terms required. For example, terms are required to get a result accurate to at least 10 decimal places with and .
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