Gamma function

The Gamma function (denoted as \(\Gamma(n)\)) is a complex mathematical function that extends the concept of factorial to complex and real number arguments. It is used in various areas of mathematics, including calculus, statistics, and number theory, as well as in the fields of physics and engineering.

Definition

The Gamma function is defined for all complex numbers except the non-positive integers. For any positive integer \(n\), the Gamma function is given by:

\[ \Gamma(n) = (n-1)! \]

For complex numbers with a real part greater than 0, it is defined through an improper integral:

\[ \Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt \]

Properties

The Gamma function has several important properties:

• **Recurrence Relation**: \(\Gamma(z+1) = z\Gamma(z)\). This property helps in computing the Gamma function for any argument based on its value at another point.
• **Reflection Formula**: \(\Gamma(1-z)\Gamma(z) = \frac{\pi}{\sin(\pi z)}\). This formula is useful for evaluating the Gamma function for negative arguments.
• **Euler's Reflection Formula**: Provides a way to extend the Gamma function to complex numbers with a negative real part.
• **Multiplication Theorem**: A formula that relates the Gamma function at multiple points to a product of Gamma functions at those points, scaled by a power of 2 and \(\pi\).

Applications

The Gamma function is used in various branches of mathematics and science:

• In statistics, it is used to define the gamma distribution, a two-parameter family of continuous probability distributions.
• In physics, the Gamma function appears in the solutions of certain types of differential equations and in the calculation of partition functions.
• In number theory, it is involved in the study of Riemann zeta function and other L-functions.

Special Values

Some special values of the Gamma function include:

• \(\Gamma(\frac{1}{2}) = \sqrt{\pi}\), which is related to the Gaussian integral.
• \(\Gamma(1) = 0!\) and \(\Gamma(2) = 1!\), which align with the factorial function for positive integers.

See Also

• Factorial
• Beta function
• Stirling's approximation