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Repeated application of the recurrence relation for the lower incomplete gamma function leads to the power series expansion: [2] (,) = = (+) (+) = = (+ +). Given the rapid growth in absolute value of Γ(z + k) when k → ∞, and the fact that the reciprocal of Γ(z) is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all ...
In mathematics, the gamma function (represented by Γ, capital Greek letter gamma) is the most common extension of the factorial function to complex numbers.Derived by Daniel Bernoulli, the gamma function () is defined for all complex numbers except non-positive integers, and for every positive integer =, () = ()!.
The gamma function is an important special function in mathematics.Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general.
In mathematics, a pseudogamma function is a function that interpolates the factorial. The gamma function is the most famous solution to the problem of extending the notion of the factorial beyond the positive integers only. However, it is clearly not the only solution, as, for any set of points, an infinite number of curves can be drawn through ...
A meromorphic function may have infinitely many zeros and poles. This is the case for the gamma function (see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer. The Riemann zeta function is also meromorphic in the whole complex plane, with a single pole of order 1 at ...
For all positive integers, ! = (+), where Γ denotes the gamma function. However, the gamma function, unlike the factorial, is more broadly defined for all complex numbers other than non-positive integers; nevertheless, Stirling's formula may still be applied.
Usually, the inverse gamma function refers to the principal branch with domain on the real interval [, +) and image on the real interval [, +), where = … [2] is the minimum value of the gamma function on the positive real axis and = = … [3] is the location of that minimum.
One property of the gamma function, distinguishing it from other continuous interpolations of the factorials, is given by the Bohr–Mollerup theorem, which states that the gamma function (offset by one) is the only log-convex function on the positive real numbers that interpolates the factorials and obeys the same functional equation.