Incomplete gamma function - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the gamma function

Gamma function

In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

is defined by a definite integral

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

. The incomplete gamma function is defined as an integral function of the same integrand

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

. 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 (i.e. where the "upper" limit is fixed), and the lower incomplete gamma function can vary the upper limit of integration.

The upper incomplete gamma function is defined as:

The lower incomplete gamma function is defined as:

Properties

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

By integration by parts

Integration by parts

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

we find the recurrence relations

and conversely

Since the ordinary gamma function is defined as

we have

Continuation to complex values

The lower incomplete gamma and the upper incomplete gamma function, as defined above for real positive s and x, can be developed into holomorphic function

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

s, with respect both to x and s, defined for almost all combinations of complex x and s.. Complex analysis shows how properties of the real incomplete gamma functions extend to their holomorphic counterparts.

Holomorphic Extension

Repeated application of the recurrence relation for the lower incomplete gamma function leads to the power series expansion: http://dlmf.nist.gov/8.8.E7

Given the rapid growth in absolute value of

Gamma function

In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

when k → ∞, and the fact that the reciprocal of

is an entire function

Entire function

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...

, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all complex s and x. By a theorem of Weierstraß, the limiting function, sometimes denoted as

http://dlmf.nist.gov/8.7.E1

is entire

Entire function

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...

with respect to both z (for fixed s) and s (for fixed z) http://dlmf.nist.gov/8.2.ii, and, thus, holomorphic on ℂ×ℂ by Hartog's theoremhttp://www.math.umn.edu/~garrett/m/complex/hartogs.pdf. Hence, the following decomposition

http://dlmf.nist.gov/8.2.E6,

extends the real lower incomplete gamma function as a holomorphic

Holomorphic function

function, both jointly and separately in z and s. It follows from the properties of

and the

-function

Gamma function

In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

, that the first two factors capture the singularities

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

(z = 0 and s a non-positive integer), whereas the last factor contributes to its zeros.

Branches

In particular, the factor

causes

to be multi-valued for s not an integer. This complication is often overcome by cutting the image

Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...

, for fixed s, (usually) along the negative real axis into separate, single-valued branches

Branch point

In the mathematical field of complex analysis, a branch point of a multi-valued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point...

, and then restricting oneself to the principal branch

Principal branch

In mathematics, a principal branch is a function which selects one branch, or "slice", of a multi-valued function. Most often, this applies to functions defined on the complex plane: see branch cut....

corresponding to that of

. Values from other branches can be derived by multiplication by

http://dlmf.nist.gov/8.2.E8, k an integer. (For another view on these phenomena see Riemann surfaces

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

Behavior near Branch Point

The decomposition above further shows, that

behaves near z = 0 asymptotically like:

For positive real x, y and s,

, when (x, y) → (0, s). This seems to justify setting

for real s > 0. However, matters are somewhat different in the complex realm. Only if (a) the real part of s is positive, and (b) values from just a finite set of branches of

are taken, then

is guaranteed to converge to zero as (u, v) → (0, s), and so does

. A single branch

Branch point

naturally fulfills (b), so

for s with positive real part is a continuous limit

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

there. Also note that such a continuation is by no means an analytic one

Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...

Algebraic Relations

All algebraic relations and differential equations observed by the real

hold for its holomorphic counterpart as well. This is a consequence of the identity theorem http://planetmath.org/encyclopedia/RigidityTheoremForAnalyticFunctions.html, stating that equations between holomorphic functions valid on a real interval, hold everywhere. In particular, the recurrence relation http://dlmf.nist.gov/8.8.E1 and

http://dlmf.nist.gov/8.8.E12 are preserved on corresponding branches.

Integral Representation

The last relation tells us, that, for fixed s,

is a primitive or antiderivative of the holomorphic function

. Consequently http://planetmath.org/encyclopedia/ComplexAntiderivative.html, for any complex u, v ≠ 0,

holds, as long as the path of integration

Line integral

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

does not wind around the singular branch point 0. If the path is entirely contained in the interior of a single branch of the integrand, and the real part of s is positive, then the limit

→ 0 for u → 0 applies, finally arriving at the complex integral definition of

http://dlmf.nist.gov/8.2.E1
Any path of integration containing 0 only at its beginning, and never crossing or touching the negative real line, is valid here, for example, the straight line connecting 0 and z. If z is a negative real, some technical adjustments are required to guarantee the result is from the correct branch.

Overview

is:

entire
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...

in z for fixed, positive integral s;
multi-valued holomorphic
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

in z for fixed s not an integer, with a branch point
Branch point
In the mathematical field of complex analysis, a branch point of a multi-valued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point...

at z = 0;
on each branch meromorphic in s for fixed z ≠ 0, with simple poles at non-positive integers s.

Upper Incomplete Gamma Function

As for the upper incomplete gamma function, a holomorphic

Holomorphic function

extension, with respect to z or s, is given by

http://dlmf.nist.gov/8.2.E3
at points (s, z), where the right hand side exists. Since

is multi-valued, the same holds for

, but a restriction to principal values only yields the single-valued principal branch of

.

When s is a non-positive integer in the above equation, neither part of the difference is defined, and a limiting process

Limit of a function

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

, here developed for s → 0, fills in the missing values. Complex analysis

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

guarantees holomorphicity, because

proves to be bounded

Bounded function

In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...

in a neighbourhood

Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...

of that limit for a fixed zhttp://planetmath.org/encyclopedia/RiemannsRemovableSingularityTheorem.html.

To determine the limit, the power series of

at z = 0 turns out useful. When replacing

by its power series in the integral definition of

, one obtains (assume x,s positive reals for now):

. http://dlmf.nist.gov/8.7.E1
which, as a series representation of the entire

function, converges for all complex x (and all complex s not a non-positive integer).

With its restriction to real values lifted, the series allows the expansion:

When s → 0:

,
(

is the Euler-Mascheroni constant

Euler-Mascheroni constant

The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....

here), hence,

is the limiting function to the upper incomplete gamma function as s → 0, also known as

Exponential integral

In mathematics, the exponential integral is a special function defined on the complex plane given the symbol Ei.-Definitions:For real, nonzero values of x, the exponential integral Ei can be defined as...

.

By way of the recurrence relation, values of

for positive integers n can be derived from this result, so the upper incomplete gamma function proves to exist and be holomorphic, with respect both to z and s, for all s and z ≠ 0.

is:

entire
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...

in z for fixed, positive integral s;
multi-valued holomorphic
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

in z for fixed s not an integer, with a branch point
Branch point
In the mathematical field of complex analysis, a branch point of a multi-valued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point...

at z = 0;
= for s with positive real part and z = 0 (the limit when ), but this is a continuous extension, not an analytic one
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...

(does not hold for real s<0!);
on each branch entire
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...

in s for fixed z ≠ 0.

Special values

if s is a positive integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

,

if s is a positive integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

,

Here, Ei is the exponential integral

Exponential integral

, erf is the error function

Error function

In mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics and partial differential equations...

, and erfc is the complementary error function, erfc(x) = 1 − erf(x).

Asymptotic behavior

as and

as an asymptotic series where and .

Evaluation formulae

The lower gamma function has the straight forward expansion

where M is Kummer's confluent hypergeometric function

Confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity...

Connection with Kummer's confluent hypergeometric function

When the real part of z is positive,

where

has an infinite radius of convergence.

Again with confluent hypergeometric functions and employing Kummer's identity,

For the actual computation of numerical values, Gauss's continued fraction provides a useful expansion:

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

The upper gamma function has the continued fraction

and

Multiplication theorem

The following multiplication theorem

Multiplication theorem

In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name...

holds true:

Regularized Gamma functions and Poisson random variables

Two related functions are the regularized Gamma functions:

is the cumulative distribution function

Cumulative distribution function

In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

for Gamma random variables with shape parameter

Shape parameter

In probability theory and statistics, a shape parameter is a kind of numerical parameter of a parametric family of probability distributions.- Definition :...

and scale parameter

Scale parameter

In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions...

1.

When

is an integer,

is the cumulative distribution function for Poisson random variables: If

is a

random variable then

This formula can be derived by repeated integration by parts.

Derivatives

The derivative of the upper incomplete gamma function

with respect to x is well known. It is simply given by the integrand of its integral definition:

The derivative with respect to its first argument

is given by

and the second derivative by

where the function

is a special case of the Meijer G-function

Meijer G-Function

In mathematics, the G-function was introduced by as a very general function intended to include most of the known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's...

This particular special case has internal closure properties of its own because it can be used to express all successive derivatives. In general,

where

All such derivatives can be generated in succession from:

and

This function

can be computed from its series representation valid for

with the understanding that s is not a negative integer or zero. In such a case, one must use a limit. Results for

can be obtained by analytic continuation

Analytic continuation

. Some special cases of this function can be simplified. For example,

, where

is the Exponential integral

Exponential integral

. These derivatives and the function

provide exact solutions to a number of integrals by repeated differentiation of the integral definition of the upper incomplete gamma function.
For example,

This formula can be further inflated or generalized to a huge class of Laplace transforms and Mellin transform

Mellin transform

In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform...

s. When combined with a computer algebra system

Computer algebra system

A computer algebra system is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.-Symbolic manipulations:...

, the exploitation of special functions provides a powerful method for solving definite integrals, in particular those encountered by practical engineering applications (see Symbolic integration

Symbolic integration

In calculus symbolic integration is the problem of finding a formula for the antiderivative, or indefinite integral, of a given function f, i.e...

for more details).

Indefinite and definite integrals

The following indefinite integrals are readily obtained using integration by parts

Integration by parts

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

The lower and the upper incomplete Gamma function are connected via the Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

This follows, for example, by suitable specialization of .

Miscellaneous utilities

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