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Incomplete Gamma Function
The upper incomplete gamma function for some values of s: 0 (blue), 1 (red), 2 (green), 3 (orange), 4 (purple).
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.
The upper incomplete gamma function is defined as:
whereas the lower incomplete gamma function is defined as:
In both cases s is a complex parameter, such that the real part of s is positive.
The lower incomplete gamma and the upper incomplete gamma function, as defined above for real positive s and x, can be developed into holomorphic functions, 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.
Lower incomplete Gamma function
Repeated application of the recurrence relation for the lower incomplete gamma function leads to the power series expansion: 
extends the real lower incomplete gamma function as a holomorphic function, both jointly and separately in z and s. It follows from the properties of and the ?-function, that the first two factors capture the singularities of (at z = 0 or s a non-positive integer), whereas the last factor contributes to its zeros.
The complex logarithm log z = log |z| + i arg z is determined up to a multiple of 2?i only, which renders it multi-valued. Functions involving the complex logarithm typically inherit this property. Among these are the complex power, and, since zs 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 C of multi-valued functions by a suitable manifold in C×C 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 in C having their vertex at z = 0 often prove to be appropriate domains for complex expressions. A sector D consists of all complex z fulfilling z ? 0 and ? - ? < arg z < ? + ? with some ? and 0 < ? ?. 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 C, with the exception of a half-line originating at z = 0 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.
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, zs and the incomplete gamma functions in turn collapse to single-valued, holomorphic functions on D (or C×D), called branches of their multi-valued counterparts on D. Adding a multiple of 2? to ? yields a different set of correlated branches on the same set D. 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 analogons 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 k a suitable integer.
Behavior near branch point
The decomposition above further shows, that ? behaves near z = 0 asymptotically like:
For positive real x, y and s, xy/y -> 0, when (x, y) -> (0, s). This seems to justify setting ?(s, 0) = 0 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 uv are taken from just a finite set of branches, they are guaranteed to converge to zero as (u, v) -> (0, s), and so does ?(u, v). On a single branch of ?(b) is naturally fulfilled, so there?(s, 0) = 0 for s with positive real part is a continuous limit. Also note that such a continuation is by no means an analytic one.
All algebraic relations and differential equations observed by the real ?(s, z) 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 (s,z)/?z = zs-1e-z are preserved on corresponding branches.
The last relation tells us, that, for fixed s, ? is a primitive or antiderivative of the holomorphic function zs-1e-z. Consequently, , for any complex u, v ? 0,
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 s is positive, then the limit ?(s, u) -> 0 for u -> 0 applies, finally arriving at the complex integral definition of ?
has been used in the middle. Since the final integral becomes arbitrarily small if only a is large enough, ?(s, x) converges uniformly for x -> ? on the strip 1 towards a holomorphic function, which must be ?(s) because of the identity theorem . Taking the limit in the recurrence relation ?(s,x) = (s - 1)?(s - 1,x) - xs
-1e-x and noting, that lim xne-x = 0 for x -> ? and all n, shows, that ?(s,x) converges outside the strip, too, towards a function obeying the recurrence relation of the ?-function. It follows
for all complex s not a non-positive integer, x real and ? principal.
Now let u be from the sector |arg z| < ? < ?/2 with some fixed ? (? = 0), ? be the principal branch on this sector, and look at
As shown above, the first difference can be made arbitrarily small, if |u| is sufficiently large. The second difference allows for following estimation:
where we made use of the integral representation of ? and the formula about |zs| above. If we integrate along the arc with radius R = |u| around 0 connecting u and |u|, then the last integral is
where M = ?(cos ?)-Re seIm s? is a constant independent of u or R. Again referring to the behavior of xne-x for large x, we see that the last expression approaches 0 as R increases towards ?.
In total we now have:
if s is not a non-negative integer, 0 < ? < ?/2 is arbitrarily small, but fixed, and ? denotes the principal branch on this domain.
Even if unavailable directly, however, incomplete function values can be calculated using functions commonly included in spreadsheets (and computer algebra packages). In Excel, for example, these can be calculated using the Gamma function combined with the Gamma distribution function.
The lower incomplete function: = EXP(GAMMALN(s))*GAMMA.DIST(x,s,1,TRUE)
The upper incomplete function: = EXP(GAMMALN(s))*(1-GAMMA.DIST(x,s,1,TRUE)).
All such derivatives can be generated in succession from:
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. Some special cases of this function can be simplified. For example, , , where is the 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.
^K.O. Geddes, M.L. Glasser, R.A. Moore and T.C. Scott, Evaluation of Classes of Definite Integrals Involving Elementary Functions via Differentiation of Special Functions, AAECC (Applicable Algebra in Engineering, Communication and Computing), vol. 1, (1990), pp. 149-165, 
G. Arfken and H. Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000. (See Chapter 10.)
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Früchtl, H.; Otto, P. (1994). "A new algorithm for the evaluation of the incomplete Gamma Function on vector computers". ACM Trans. Math. Softw. 20 (4): 436-446. doi:10.1145/198429.198432. S2CID16737306.
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