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In mathematics, the Gamma function extends the factorial function to complex and non natural numbers (where it is defined). The factorial function of an integer n is written n! and is equal to the product n! = 1 × 2 × 3 × ... × n. The Gamma function "fills in" the factorial function for fractional values of n and for complex values of n. If z is a complex variable, then for integer values only, we have
but for fractional and complex values of z, the above equation does not apply, since the factorial function is not defined.
Because Γ(1) = 1, this relation implies that
It is this extended version that is commonly referred to as the Gamma function.
where γ is the Euler-Mascheroni constant.
Other important functional equations for the Gamma function are Euler's reflection formula
and the duplication formula
The duplication formula is a special case of the multiplication theorem
Perhaps the most well-known value of the Gamma function at a non-integer argument is
which can be found by setting z=1/2 in the reflection formula or by noticing the beta function for (1/2, 1/2), which is .
The derivatives of the Gamma function are described in terms of the polygamma function. For example:
The Bohr-Mollerup theorem states that among all functions extending the factorial functions to the positive real numbers, only the Gamma function is log-convex, that is, its natural logarithm is convex.
An alternative notation which was originally introduced by Gauss and which is sometimes used is the Pi function, which in terms of the Gamma function is
Using the Pi function the reflection formula takes on the form
where sincN is the normalized Sinc function, while the multiplication theorem takes on the form
We also sometimes find
Relation to other functions Edit
In the first integral above, which defines the Gamma function, the limits of integration are fixed. The incomplete Gamma function is the function obtained by allowing either the upper or lower limit of integration to be variable.
The Gamma function is related to the Beta function by the formula
Particular values Edit
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- Beta function
- Bohr-Mollerup theorem
- Digamma function
- Gamma distribution
- Gauss's constant
- Multivariate Gamma function
- Polygamma function
- Stirling's approximation
- Trigamma function
- Elliptic gamma function
- Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 6)
- G. Arfken and H. Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000. (See Chapter 10.)
- Harry Hochstadt. The Functions of Mathematical Physics. New York: Dover, 1986 (See Chapter 3.)
- W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Section 6.1.)
- Examples of problems involving the Gamma function can be found at Exampleproblems.com.
- P. Sebah, X. Gourdon. Introduction to the Gamma Function. In PostScript and HTML formats.de:Gammafunktion
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