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The notation Γ(z) is due to Adrien-Marie Legendre. If the real part of the complex number z is positive, then the integral
converges absolutely. Using integration by parts, one can show that
Because Γ(1) = 1, this relation implies that
for all natural numbers n. It can further be used to extend Γ(z) to a meromorphic function defined for all complex numbers z except z = 0, −1, −2, −3, ... by analytic continuation. It is this extended version that is commonly referred to as the gamma function.
Another important functional equation for the gamma function is Euler's reflection formula
An alternative notation which is sometimes used is the Pi function, which in terms of the gamma function is
We also sometimes find
which is an entire function, defined for every complex number. That π(z) is entire entails it has no poles, so Γ(z) has no zeros.
Perhaps the most well-known value of the gamma function at a non-integer argument is
The gamma function has a pole of order 1 at z = −n for every natural number n; the residue there is given by
The following infinite product for the gamma function, due to Weierstrass, is valid for all complex numbers z which are not non-positive integers:
where γ is the Euler-Mascheroni constantThe Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: : Intriguingly, the constant is also given by the integral: : Its valu.
The Bohr-Mollerup theoremIn mathematical analysis, the Bohr-Mollerup theorem named after the Danish mathematicians Harald Bohr and Johannes Mollerup, who proved it, characterizes the gamma function, defined for x > 0 by : as the only function f on the interval x > 0 that simultan states that among all functions extending the factorial functions to the positive real numbers, only the gamma function is log-convex.
In the first integral above, which defines the gamma function, the limits of integration are fixed. The incomplete gamma functionIn mathematics, the gamma function is defined by a definite integral. The incomplete gamma function is defined by an indefinite integral of the same integrand. There are two varieties of the incomplete gamma function, one for the case that the lower limit is the function obtained by allowing either the upper or lower limit of integration to be variable.
The derivative of the logarithm of the gamma function is called the digamma functionIn mathematics, the digamma function is defined by : where D is the differential operator. The digamma function, often denoted also ψ x or even ψ0 x , is related to the harmonic numbers in that : where H is the n minus;1)th harmonic number, and &g.