Incomplete gamma function

Incomplete Gamma Function

The Incomplete Gamma Function is a significant concept in mathematics, particularly in the fields of calculus and complex analysis. It extends the idea of the Gamma function, which plays a crucial role in various areas of mathematics and science, including statistics, physics, and engineering. The Incomplete Gamma Function is defined for two arguments and provides a way to compute the gamma function over a limited range of integration.

Definition

The Incomplete Gamma Function is defined in two forms: the lower incomplete gamma function, denoted as \(\gamma(s, x)\), and the upper incomplete gamma function, denoted as \(\Gamma(s, x)\). Mathematically, they are defined as follows:

• The lower incomplete gamma function is defined as:

\[ \gamma(s, x) = \int_0^x t^{s-1} e^{-t} dt \] where \(s > 0\) and \(x \geq 0\).

• The upper incomplete gamma function is defined as:

\[ \Gamma(s, x) = \int_x^\infty t^{s-1} e^{-t} dt \] which complements the lower incomplete gamma function, covering the rest of the gamma function's integral from \(x\) to infinity.

Properties and Applications

The Incomplete Gamma Function has several important properties and applications:

• It is used in probability and statistics, especially in the derivation and properties of some distributions such as the Gamma distribution and the Chi-squared distribution.
• In physics, it appears in the context of processes with exponential decay and in the calculation of reaction rates.
• The relationship between the lower and upper incomplete gamma functions is given by:

\[ \gamma(s, x) + \Gamma(s, x) = \Gamma(s) \] where \(\Gamma(s)\) is the complete Gamma function.

• The functions are related to special functions such as the Exponential integral for specific values of \(s\).

Numerical Methods

Numerical computation of the Incomplete Gamma Function is essential for practical applications. Various algorithms and software libraries are available for accurately computing \(\gamma(s, x)\) and \(\Gamma(s, x)\), including methods based on continued fractions, series expansions, and iterative techniques.

See Also

• Gamma function
• Exponential integral
• Gamma distribution
• Chi-squared distribution

References

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  Upper incomplete gamma function
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  Plot of the regularized incomplete gamma function Q(2,z) in the complex plane