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- Academia - Gamma Function
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- Kansas State University - Mathematics Department - The Gamma Function
- Mathematics LibreTexts - Gamma Function
- MIT OpenCourseWare - The sine product formula and the gamma function
- Williams College - The Gamma Function and Related Distributions
- NIST Digital Library of Mathematical Functions - Gamma Function
- University of Washington - Department of Mathematics - On the Gamma function and its applications
- Math is Fun - Gamma Function
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While every effort has been made to follow citation style rules, there may be some discrepancies.Please refer to the appropriate style manual or other sources if you have any questions.
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Our editors will review what you’ve submitted and determine whether to revise the article.
External Websites
- Academia - Gamma Function
- Wolfram MathWorld - Gamma Function
- Kansas State University - Mathematics Department - The Gamma Function
- Mathematics LibreTexts - Gamma Function
- MIT OpenCourseWare - The sine product formula and the gamma function
- Williams College - The Gamma Function and Related Distributions
- NIST Digital Library of Mathematical Functions - Gamma Function
- University of Washington - Department of Mathematics - On the Gamma function and its applications
- Math is Fun - Gamma Function
Also known as: Euler’s integral of the second kind
Written and fact-checked by
The Editors of Encyclopaedia Britannica
Last Updated: •Article History
gamma function, generalization of the factorial function to nonintegral values, introduced by the Swiss mathematician Leonhard Euler in the 18th century.
For a positive whole number n, the factorial (written as n!) is defined by n! = 1 × 2 × 3 ×⋯× (n − 1) × n. For example, 5! = 1 × 2 × 3 × 4 × 5 = 120. But this formula is meaningless if n is not an integer.
Britannica Quiz
Numbers and Mathematics
To extend the factorial to any real number x > 0 (whether or not x is a whole number), the gamma function is defined as Γ(x) = Integral on the interval [0, ∞ ] of∫0∞t x −1 e−t dt.
Using techniques of integration, it can be shown that Γ(1)=1. Similarly, using a technique from calculus known as integration by parts, it can be proved that the gamma function has the following recursive property: if x>0, then Γ(x+1)=xΓ(x). From this it follows that Γ(2)=1Γ(1)=1; Γ(3)=2Γ(2)=2×1=2!; Γ(4)=3Γ(3)=3×2×1=3!; and so on. Generally, if x is a natural number (1, 2, 3,…), then Γ(x)=(x−1)! The function can be extended to negative non-integer real numbers and to complex numbers as long as the real part is greater than or equal to 1. While the gamma function behaves like a factorial for natural numbers (a discrete set), its extension to the positive real numbers (a continuous set) makes it useful for modeling situations involving continuous change, with important applications to calculus, differential equations, complex analysis, and statistics.