Sums of trig functions...........Joe F........................that looks like the graph of the tangent fun....

Gamma function

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For the gamma function of ordinals, see Veblen function. For the gamma distribution in statistics, see Gamma distribution.

The gamma function along part of the real axis

In mathematics, the gamma function (represented by the capital Greek alphabet letter Γ) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. That is, if n is a positive integer:

${\displaystyle \Gamma (n)=(n-1)!.}$

The gamma function is defined for all complex numbers except the non-positive integers. For complex numbers with a positive real part, it is defined via a convergent improper integral:

${\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,\mathrm {d} x.}$

This integral function is extended by analytic continuation to all complex numbers except the non-positive integers (where the function has simple poles), yielding the meromorphic function we call the gamma function. In fact the gamma function corresponds to the Mellin transform of the negative exponential function: