Incomplete gamma function


In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals.
Their respective names stem from their integral definitions, which are defined similarly to the gamma function 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 we find the recurrence relations
Since the ordinary gamma function is defined as
we also have

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 functions, 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 expansion:
Given the rapid growth in absolute value of when, and the fact that the reciprocal of is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all complex and. By a theorem of Weierstrass, the limiting function, sometimes denoted as
is entire with respect to both and , and, thus, holomorphic on by Hartogs' theorem. Hence, the following decomposition
extends the real lower incomplete gamma function as a holomorphic function, both jointly and separately in and. It follows from the properties of and the Γ-function, that the first two factors capture the singularities of , whereas the last factor contributes to its zeros.
Multi-valuedness
The complex logarithm is determined up to a multiple of only, which renders it multi-valued. 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:
  • replace the domain of multi-valued functions by a suitable manifold in called Riemann surface. While this removes multi-valuedness, one has to know the theory behind it;
  • restrict the domain such that a multi-valued function decomposes into separate single-valued 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. 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 , 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, for a suitable integer.
Behavior near branch point
The decomposition above further shows, that γ behaves near asymptotically like:
For positive real, and,, when. This seems to justify setting for real. However, matters are somewhat different in the complex realm. Only if the real part of is positive, and values are taken from just a finite set of branches, they are guaranteed to converge to zero as, and so does. On a single branch of is naturally fulfilled, so there for with positive real part is a continuous limit. Also note that such a continuation is by no means an analytic one.
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, stating that equations between holomorphic functions valid on a real interval, hold everywhere. In particular, the recurrence relation and are preserved on corresponding branches.
Integral representation
The last relation tells us, that, for fixed, is a primitive or antiderivative of the holomorphic function. Consequently, for any complex,
holds, as long as the path of integration is entirely contained in the domain of a branch of the integrand. If, additionally, the real part of is positive, then the limit for applies, finally arriving at the complex integral definition of
Any path of integration containing 0 only at its beginning, otherwise restricted to the domain of a branch of the integrand, is valid here, for example, the straight line connecting and.
Limit for
Real values
Given the integral representation of a principal branch of, the following equation holds for all positive real, :
''s'' complex
This result extends to complex. Assume first and. Then
where
has been used in the middle. Since the final integral becomes arbitrarily small if only is large enough, converges uniformly for on the strip towards a holomorphic function, which must be Γ because of the identity theorem. Taking the limit in the recurrence relation and noting, that lim for and all, shows, that converges outside the strip, too, towards a function obeying the recurrence relation of the Γ-function. It follows
for all complex not a non-positive integer, real and principal.
Sectorwise convergence
Now let be from the sector with some fixed , be the principal branch on this sector, and look at
As shown above, the first difference can be made arbitrarily small, if is sufficiently large. The second difference allows for following estimation:
where we made use of the integral representation of and the formula about above. If we integrate along the arc with radius around 0 connecting and, then the last integral is
where is a constant independent of or. Again referring to the behavior of for large, we see that the last expression approaches 0 as increases towards.
In total we now have:
if is not a non-negative integer, is arbitrarily small, but fixed, and denotes the principal branch on this domain.
Overview
is:
  • entire in for fixed, positive integer ;
  • multi-valued holomorphic in for fixed not an integer, with a branch point at ;
  • on each branch meromorphic in for fixed, with simple poles at non-positive integers s.

    Upper incomplete gamma function

As for the upper incomplete gamma function, a holomorphic extension, with respect to or, is given by
at points, 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 is a non-positive integer in the above equation, neither part of the difference is defined, and a limiting process, here developed for, fills in the missing values. Complex analysis guarantees holomorphicity, because proves to be bounded in a neighbourhood of that limit for a fixed.
To determine the limit, the power series of at is useful. When replacing by its power series in the integral definition of, one obtains :
or
which, as a series representation of the entire function, converges for all complex .
With its restriction to real values lifted, the series allows the expansion:
When :
, hence,
is the limiting function to the upper incomplete gamma function as, also known as the exponential integral
By way of the recurrence relation, values of for positive integers can be derived from this result,
so the upper incomplete gamma function proves to exist and be holomorphic, with respect both to and, for all and.
is:
  • entire in for fixed, positive integral ;
  • multi-valued holomorphic in for fixed non zero and not a positive integer, with a branch point at ;
  • equal to for with positive real part and , but this is a continuous extension, not an analytic one ;
  • on each branch entire in for fixed.

    Special values

  • if is a positive integer,
  • if is a positive integer,
  • ,
  • ,
  • ,
  • for,
  • ,
  • ,
  • .
Here, is the exponential integral, is the generalized exponential integral, is the error function, and is the complementary error function,.

Asymptotic behavior