Editing Incomplete gamma function (section)

=====Multi-valuedness=====
The [[complex logarithm]] {{math|1=log ''z'' = log {{abs|''z''}} + ''i'' arg ''z''}} is determined up to a multiple of {{math|2''πi''}} only, which renders it [[Multi-valued function|multi-valued]]. Functions involving the complex logarithm typically inherit this property. Among these are the [[Exponentiation#nth roots of a complex number|complex power]], and, since {{math|''z''<sup>''s''</sup>}} appears in its decomposition, the {{math|γ}}-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 {{math|'''C'''}} of multi-valued functions by a suitable manifold in {{math|'''C''' × '''C'''}} called [[Riemann surface]]. While this removes multi-valuedness, one has to know the theory behind it;<ref>{{cite web|author=C. Teleman
|url=http://math.berkeley.edu/~teleman/math/Riemann.pdf|title=Riemann Surfaces|website=berkeley.edu|access-date=21 December 2023}}</ref>  
* 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 {{math|'''C'''}} having their vertex at {{math|1=''z'' = 0}} often prove to be appropriate domains for complex expressions. A sector {{mvar|D}} consists of all complex {{mvar|z}} fulfilling {{math|''z'' ≠ 0}} and {{math|''α'' − ''δ'' < arg ''z'' < ''α'' + ''δ''}} with some {{mvar|α}} and {{math|0 < ''δ'' ≤ ''π''}}. Often, {{mvar|α}} can be arbitrarily chosen and is not specified then. If {{mvar|δ}} is not given, it is assumed to be {{pi}}, and the sector is in fact the whole plane {{math|'''C'''}}, with the exception of a half-line originating at {{math|1=''z'' = 0}} and pointing into the direction of {{math|−''α''}}, usually serving as a [[Branch cut#Branch cuts|branch cut]]. Note: In many applications and texts, {{mvar|α}} 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 {{open-open|''α'' − ''δ'', ''α'' + ''δ''}}. Based on such a restricted logarithm, {{math|''z''<sup>''s''</sup>}} and the incomplete gamma functions in turn collapse to single-valued, holomorphic functions on {{mvar|D}} (or {{math|'''C'''×''D''}}), called branches of their multi-valued counterparts on D. Adding a multiple of {{math|2''π''}} to {{mvar|α}} yields a different set of correlated branches on the same set {{mvar|D}}. However, in any given context here, {{mvar|α}} is assumed fixed and all branches involved are associated to it. If {{math|{{abs|''α''}} < ''δ''}}, the branches are called [[principal branch|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 <math>e^{2\pi iks}</math>,<ref name="auto3"/> for {{mvar|k}} a suitable integer.