Science Fair Project Encyclopedia
converges absolutely. Using integration by parts, one can show that
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
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.
The duplication formula is a special case of the multiplication theorem
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.
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
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
- 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.
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details