Incomplete gamma function

The upper incomplete gamma function is

γ (x, y) = \int_{y}^{\infty} t^{x - 1} e^{- t} d t

under the convention used here.

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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.

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} .

Fixed-lower-limit incomplete gamma asymptotic

For fixed finite

y

and

x \to \infty

, the omitted integral from zero to

y

is negligible, so

γ (x, y) \sim Γ (x)

and Stirling's formula applies.

Endpoint expansion of the incomplete gamma function

When

y = λ x

with

λ > 1

, the integral is dominated by its lower endpoint. Setting

t = y + s

and applying Watson's lemma gives coefficients rational in

λ - 1

.

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Incomplete gamma function by

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The incomplete gamma function is a mathematical function that generalizes the gamma function, which itself is a fundamental function in mathematics, particularly in the fields of statistics and probability theory. The incomplete gamma function is useful in various applications, including statistical distributions and hypothesis testing. The incomplete gamma function is defined in two forms: the lower incomplete gamma function and the upper incomplete gamma function.

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