Asymptotic Expansion - Detailed Example

Detailed Example

Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. Thus, for example, one may start with the ordinary series

The expression on the left is valid on the entire complex plane, while the right hand side converges only for . Multiplying by and integrating both sides yields

$\int_0^\infty \frac{e^{-w/t}}{1-w}\, dw = \sum_{n=0}^\infty t^{n+1} \int_0^\infty e^{-u} u^n\, du,$

after the substitution on the right hand side. The integral on the left hand side, understood as a Cauchy principal value, can be expressed in terms of the exponential integral. The integral on the right hand side may be recognized as the gamma function. Evaluating both, one obtains the asymptotic expansion

Here, the right hand side is clearly not convergent for any non-zero value of t. However, by truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of for sufficiently small t. Substituting and noting that results in the asymptotic expansion given earlier in this article.

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