# Category:Gamma Function

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This category contains results about **the Gamma function**.

Definitions specific to this category can be found in Definitions/Gamma Function.

The **gamma function** $\Gamma: \C \setminus \Z_{\le 0} \to \C$ is defined, for the open right half-plane, as:

- $\ds \map \Gamma z = \map {\MM \set {e^{-t} } } z = \int_0^{\to \infty} t^{z - 1} e^{-t} \rd t$

where $\MM$ is the Mellin transform.

For all other values of $z$ except the non-positive integers, $\map \Gamma z$ is defined as:

- $\map \Gamma {z + 1} = z \map \Gamma z$

## Subcategories

This category has the following 21 subcategories, out of 21 total.

### D

### E

### G

- Gamma Difference Equation (3 P)

### I

### K

### P

- Pfaff's Transformation (2 P)
- Pfaff-Saalschütz Theorem (3 P)

### R

### T

- Thomae's Transformation (2 P)

### W

## Pages in category "Gamma Function"

The following 65 pages are in this category, out of 65 total.

### A

### B

### D

- Definite Integral to Infinity of Exponential of -x by Logarithm of x
- Definite Integral to Infinity of Power of x over Hyperbolic Sine of a x
- Derivative of Gamma Function
- Derivative of Gamma Function at 1
- Dixon's Hypergeometric Theorem
- Dougall's Hypergeometric Theorem
- Dougall's Hypergeometric Theorem/Corollary 1
- Dougall's Hypergeometric Theorem/Corollary 2
- Dougall's Hypergeometric Theorem/Corollary 3
- Dougall's Hypergeometric Theorem/Corollary 4
- Dougall's Hypergeometric Theorem/Corollary 5
- Dougall's Hypergeometric Theorem/Corollary 6
- Dougall-Ramanujan Identity

### E

### G

- Gamma Difference Equation
- Gamma Function as Integral of Natural Logarithm
- Gamma Function Extends Factorial
- Gamma Function for Non-Negative Integer Argument
- Gamma Function is Continuous on Positive Reals
- Gamma Function is Smooth on Positive Reals
- Gamma Function is Unique Extension of Factorial
- Gamma Function of Negative Half-Integer
- Gamma Function of Positive Half-Integer
- Gauss Multiplication Formula
- Gauss's Hypergeometric Theorem

### I

- Infinite Product of Product of Sequence of n plus alpha over Sequence of n plus beta
- Integral Form of Gamma Function equivalent to Euler Form
- Integral Representation of Dirichlet Beta Function in terms of Gamma Function
- Integral Representation of Dirichlet Eta Function in terms of Gamma Function
- Integral Representation of Riemann Zeta Function in terms of Gamma Function
- Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function