![Upper incomplete gamma function](https://upload.wikimedia.org/wikipedia/commons/8/8e/Upper_incomplete_gamma_function.jpg)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the upper and lower incomplete gamma functions are types of
special functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined by ...
which arise as solutions to various mathematical problems such as certain
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s.
Their respective names stem from their integral definitions, which are defined similarly to the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
but with different or "incomplete" integral limits. The gamma function is defined as an integral from zero to infinity. This contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit. Similarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity.
Definition
The upper incomplete gamma function is defined as:
whereas the lower incomplete gamma function is defined as:
In both cases is a complex parameter, such that the real part of is positive.
Properties
By
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
we find the
recurrence relations
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
and
Since the ordinary gamma function is defined as
we have
and
Continuation to complex values
The lower incomplete gamma and the upper incomplete gamma function, as defined above for real positive and , can be developed into
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s, with respect both to and , defined for almost all combinations of complex and . Complex analysis shows how properties of the real incomplete gamma functions extend to their holomorphic counterparts.
Lower incomplete gamma function
=Holomorphic extension
=
Repeated application of the recurrence relation for the lower incomplete gamma function leads to the
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
expansion
Given the
Gamma function#Approximations, rapid growth in
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of
when , and the fact that the
reciprocal of is an
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
, the coefficients in the rightmost sum are well-defined, and locally the sum
converges uniformly
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
for all complex and . By a theorem of Weierstraß, the limiting function, sometimes denoted as
is
Entire function, entire with respect to both (for fixed ) and (for fixed
and, thus, holomorphic on by
Hartog's theorem
Hence, the following ''decomposition''
:
extends the real lower incomplete gamma function as a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
, both jointly and separately in and . It follows from the properties of
and the
Gamma function, Γ-function, that the first two factors capture the
singularities of
(at or a non-positive integer), whereas the last factor contributes to its zeros.
=Multi-valuedness
=
The
complex logarithm
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
is determined up to a multiple of only, which renders it
multi-valued
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
. Functions involving the complex logarithm typically inherit this property. Among these are the
complex power, and, since appears in its decomposition, the -function, too.
The indeterminacy of multi-valued functions introduces complications, since it must be stated how to select a value. Strategies to handle this are:
* (the most general way) replace the domain of multi-valued functions by a suitable manifold in called
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
. While this removes multi-valuedness, one has to know the theory behind i
* restrict the domain such that a multi-valued function decomposes into separate single-valued
Branch point, branches, which can be handled individually.
The following set of rules can be used to interpret formulas in this section correctly. If not mentioned otherwise, the following is assumed:
Sectors
Sectors in having their vertex at often prove to be appropriate domains for complex expressions. A sector consists of all complex fulfilling and with some and . Often, can be arbitrarily chosen and is not specified then. If is not given, it is assumed to be , and the sector is in fact the whole plane , with the exception of a half-line originating at and pointing into the direction of , usually serving as a
branch cut
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, a ...
. Note: In many applications and texts, is silently taken to be 0, which centers the sector around the positive real axis.
Branches
In particular, a single-valued and holomorphic logarithm exists on any such sector D having its imaginary part bound to the range . Based on such a restricted logarithm, and the incomplete gamma functions in turn collapse to single-valued, holomorphic functions on (or ), called branches of their multi-valued counterparts on D. Adding a multiple of to yields a different set of correlated branches on the same set . However, in any given context here, is assumed fixed and all branches involved are associated to it. If , the branches are called
principal, because they equal their real analogues on the positive real axis. Note: In many applications and texts, formulas hold only for principal branches.
Relation between branches
The values of different branches of both the complex power function and the lower incomplete gamma function can be derived from each other by multiplication of