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.