Asymptotic expansion by repeated integration by parts

Repeated integration by parts generates successive inverse powers of a large endpoint, with a remainder expressed by the same integral with shifted parameters.

Table of contents
- Large-lower-limit expansion of the incomplete gamma function Asymptotic expansion by repeated integration by parts
- Fixed-lower-limit sine-integral expansion Asymptotic expansion by repeated integration by parts

Large-lower-limit expansion of the incomplete gamma function

For fixed

x

and

y \to \infty

,

γ (x, y) \sim y^{x - 1} e^{- y} \sum_{n \geq 0} (\prod_{j = 1}^{n} (x - j)) y^{- n} .

Fixed-lower-limit sine-integral expansion

For

si (x) = \int_{1}^{\infty} \frac{s i n ( x t )}{t} d t,

repeated integration by parts gives

si (x) \sim cos x \sum_{n \geq 0} (- 1)^{n} (2 n)! x^{- 2 n - 1} + sin x \sum_{n \geq 0} (- 1)^{n} (2 n + 1)! x^{- 2 n - 2} .

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Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 79 / 1 / b / Solution

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