Incomplete Gamma Function

$\backslash Gamma(a,x)\; =\; \backslash int\_x^\{\backslash infty\}\; t^\{a-1\}\backslash ,e^\{-t\}\backslash ,dt\; .\backslash ,\backslash !$

$\backslash gamma(a,x)\; =\; \backslash int\_0^x\; t^\{a-1\}\backslash ,e^\{-t\}\backslash ,dt\; .\backslash ,\backslash !$

$\backslash Gamma(a+1,x)\; =\; a\backslash Gamma(a,x)\; +\; x^a\; e^\{-x\}\backslash ,$

$\backslash gamma(a+1,x)\; =\; a\backslash gamma(a,x)\; -\; x^a\; e^\{-x\}.\backslash ,$

$\backslash Gamma(a)\; =\; \backslash int\_0^\{\backslash infty\}\; t^\{a-1\}\backslash ,e^\{-t\}\backslash ,dt\; \backslash ,\backslash !$

$\backslash gamma(a,x)\; +\; \backslash Gamma(a,x)\; =\; \backslash Gamma(a).\backslash ,$

$\backslash Gamma(a,0)\; =\; \backslash Gamma(a)\backslash ,$

$\backslash gamma(a,x)\; \backslash rightarrow\; \backslash Gamma(a)$

   \quad \mathrm{as\ } x \rightarrow \infty.  \, 

$\backslash Gamma(0,x)\; =\; -\backslash mbox\{Ei\}(-x)\backslash mbox\{\; for\; \}x>0\; \backslash ,$

$\backslash Gamma\backslash left(\{1\; \backslash over\; 2\},\; x\backslash right)\; =\; \backslash sqrt\backslash pi\backslash ,\backslash mbox\{erfc\}\backslash left(\backslash sqrt\; x\backslash right)\; \backslash ,$

$\backslash gamma\backslash left(\{1\; \backslash over\; 2\},\; x\backslash right)\; =\; \backslash sqrt\backslash pi\backslash ,\backslash mbox\{erf\}\backslash left(\backslash sqrt\; x\backslash right)\; \backslash ,$

$\backslash Gamma(1,x)\; =\; e^\{-x\}\; \backslash ,$

$\backslash gamma(1,x)\; =\; 1\; -\; e^\{-x\}\; \backslash ,$

References

• M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 6.)

• G. Arfken and H. Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000. (See Chapter 10.)

• W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Section 6.2.)

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