# Gamma Function

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## gamma function

[′gam·ə ‚fəŋk·shən]
(mathematics)
The complex function given by the integral with respect to t from 0 to ∞ of e -t t z-1; this function helps determine the general solution of Gauss' hypergeometric equation.

## Gamma Function

Γ(x), one of the most important special functions; generalizes the concept of the factorial. For all positive n it is given by Γ(n) = (n - 1)! = 1·2 … (n - 1). It was first introduced by L. Euler in 1729. For real values of x > 0 it is defined by the equality

Another notation is

Γ(x + 1) = π(x) = x!

The principal relations for the gamma function are

Γ(x + 1) = xΓ(x) (functional equation)

Γ(x)Γ(1 - x) = π/sin πx (complementary formula)

Special values are

For large x the Stirling formula holds:

A large number of definite integrals, infinite products, and summations of series are expressed by the gamma function. The function has also been extended to complex values of the independent variable.

### REFERENCES

Janke, E., and F. Emde. Tablitsy funktsii s formulami i krivymi, 3rd ed. Moscow, 1959. (Translated from German.)
Fikhtengol’ts, G. M. Kurs differentsial’nogo i integral’nogo ischisleniia, 6th ed., vol. 2. Moscow, 1966.
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