NationMaster - Encyclopedia: Incomplete gamma function

In mathematics, the gamma function is defined by a definite integral. The incomplete gamma function is defined as an integral function of the same integrand. There are two varieties of the incomplete gamma function: the upper incomplete gamma function is for the case that the lower limit of integration is variable (ie where the "upper" limit is fixed), and the lower incomplete gamma function can vary the upper limit of integration. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ... The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically... The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically...

The upper incomplete gamma function is defined as:

$Gamma(a,x) = int_x^{infty} t^{a-1},e^{-t},dt .,!$

The lower incomplete gamma function is defined as:

$gamma(a,x) = int_0^x t^{a-1},e^{-t},dt .,!$

1 Properties
2 Regularized Gamma functions
3 Connection with Kummer's confluent hypergeometric function
4 References
5 Miscellaneous utilities

Properties

In both cases a is a complex parameter, such that the real part of a is positive.

By integration by parts we can find that In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

$Gamma(a+1,x) = aGamma(a,x) + x^a e^{-x},$

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

Since the ordinary gamma function is defined as

$Gamma(a) = int_0^{infty} t^{a-1},e^{-t},dt ,!$

we have

$gamma(a,x) + Gamma(a,x) = Gamma(a).,$

Furthermore,

$Gamma(a,x)=(a-1)!e^{-x}sum_{k=0}^{a-1}frac{x^k}{k!}$ if a is an integer.^[1]

$Gamma(a,0) = Gamma(a),$

$Gamma(a) = (a-1)!,$ if a is an integer.

and The integers are commonly denoted by the above symbol. ... The integers are commonly denoted by the above symbol. ...

$gamma(a,x) rightarrow Gamma(a) quad mathrm{as } x rightarrow infty. ,$

Also

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

$Gammaleft({1 over 2}, xright) = sqrtpi,mbox{erfc}left(sqrt xright) ,$

$gammaleft({1 over 2}, xright) = sqrtpi,mbox{erf}left(sqrt xright) ,$

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

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

where Ei is the exponential integral, erf is the error function, and erfc is the complementary error function, erfc(x) = 1 − erf(x). In mathematics, the exponential integral Ei(x) is defined as Since 1/t diverges at t = 0, the above integral has to be understood in terms of the Cauchy principal value. ... Plot of the error function In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ... In mathematics, the complementary error function is a non-elementary function which occurs mainly in probability and statistics. ...

Regularized Gamma functions

Two related functions are the regularized Gamma functions: In mathematics, the gamma function is defined by a definite integral. ...

$P(a,x)=frac{gamma(a,x)}{Gamma(a)}$

$Q(a,x)=frac{Gamma(a,x)}{Gamma(a)}=1-P(a,x)$

Connection with Kummer's confluent hypergeometric function

It is easily shown that, when the real part of z is positive,

$gamma(a,z) = int_0^z e^{-t}t^{a-1} dt = a^{-1} z^a e^{-z} M(1,a+1,z),$

where M(1, a+1, z) is Kummer's confluent hypergeometric function. Since the series In mathematics, the confluent hypergeometric function is formed from hypergeometric series. ...

$M(1, a+1, z) = 1 + frac{1}{(a+1)}z + frac{1}{(a+1)(a+2)}z^2 + frac{1}{(a+1)(a+2)(a+3)}z^3 + cdots$

has an infinite radius of convergence, we may take

$gamma(a,z) = a^{-1} z^a e^{-z} M(1,a+1,z),$

as the definition of γ(a, z) for all complex z. In this light, the lower incomplete gamma function γ(a, z) is an entire function of the complex variable z. Since the gamma function Γ(z) is a meromorphic function with simple poles at {0, −1, −2, …}, we may define the meromorphic upper incomplete gamma function as In complex analysis, an entire function is a function that is holomorphic everywhere (ie complex-differentiable at every point) on the whole complex plane. ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ...

$Gamma(a, z) = Gamma(a) - gamma(a, z). ,$

For the actual computation of numerical values, the continued fraction of Gauss provides a useful expansion: In complex analysis, the continued fraction of Gauss is a particular continued fraction derived from the hypergeometric functions. ...

$frac{a e^z}{z^a} gamma(a, z) = cfrac{1}{1 - cfrac{z}{a+1 + cfrac{z}{a+2 - cfrac{(a+1)z} {a+3 + cfrac{2z}{a+4 - cfrac{(a+2)z}{a+5 + cfrac{3z}{a+6 - ddots}}}}}}}.$

This continued fraction converges for all complex z, provided only that a is not a negative integer.

References

M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See section §6.5.)

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

Armido R. DiDonato, Alfred H. Morris, Jr., Computation of the incomplete gamma function ratios and their inverse, ACM Transactions on Mathematical Software (TOMS), v.12 n.4, p.377-393, Dec. 1986.

Armido R. DiDonato, Alfred H. Morris, Jr., ALGORITHM 654: FORTRAN subroutines for computing the incomplete gamma function ratios and their inverse, ACM Transactions on Mathematical Software (TOMS), v.13 n.3, p.318-319, Sept. 1987. (See also Fortran 90 subroutines for ALGORITHM 654.)

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.)

^ Eric W. Weisstein, Incomplete Gamma Function at MathWorld. (equation 2)

Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ... Numerical Recipes is the generic term for the following books on algorithms and numerical analysis, all by William Press, Saul Teukolsky, William Vetterling and Brian Flannery: Numerical Recipes in C++. The Art of Scientific Computing, ISBN 0-521-75033-4. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

Miscellaneous utilities

Cephes - C and C++ language special functions math library
$P (a, x)$ - Incomplete Gamma Function Calculator
$Q (a, x)$ - Incomplete Gamma Function Calculator - Complemented
$γ(a, x)$ - Incomplete Gamma Function Calculator - Lower Limit of Integration
$Γ(a, x)$ - Incomplete Gamma Function Calculator - Upper Limit of Integration

Categories: Gamma and related functions | Continued fractions

Results from FactBites:

Gamma function - Wikipedia, the free encyclopedia (588 words)
	The derivatives of the Gamma function are described in terms of the polygamma function.
	The incomplete Gamma function 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 function; higher derivatives are the polygamma functions.

Incomplete gamma function - Wikipedia, the free encyclopedia (210 words)
	In 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 of integration is variable, and one for the upper limit of integration.

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Contents

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Regularized Gamma functions

Connection with Kummer's confluent hypergeometric function

References

Miscellaneous utilities

Properties