The gamma function, one of the special functions introduced by L. Euler (1927), is the extension of the factorial into fractional and complex values of the argument and can be obtained as a solution of the equation
The function was first difined (Euler) by the integral
for the values of a complex argument z with positive real part (Re z > 0). It is widely used in analytical solutions of equations of mathematical physics by the integral transformation method, in particular, when applying the Laplace transform to the function written (approximated) as a power series in time.
For a whole value of an argument
For a fractional value of an argument 0 ≤ x ≤ 1
where a0 = 1; a1 = –.57486; a2 = –.95124; a3 = –.69986; a4 = –.42455; a5 = –.10107; |ε(x)| ≤ 5×10−5.
Other useful properties of the Gamma function, which reflect various expansions into series, asymptotic and approximating expressions, etc. are given in handbooks on special functions.
Out of the related special functions we shall note a polygamma function
and an incomplete gamma function
which have a number of known asymptotic and other properties.
Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions, National Bureau of Standards, Appl. Math. Series-55.