Editing Incomplete gamma function (section)

=====Limit for {{math|''z'' → +∞}}=====

======Real values======
Given the integral representation of a principal branch of {{math|''γ''}}, the following equation holds for all positive real {{mvar|s}}, {{mvar|x}}:<ref>{{Cite web|url=https://dlmf.nist.gov/5.2|title=DLMF: §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function|website=dlmf.nist.gov}}</ref>
<math display="block">\Gamma(s) = \int_0^\infty t^{s-1}\,e^{-t}\, dt = \lim_{x \to \infty} \gamma(s, x)</math>

======''s'' complex======
This result extends to complex {{mvar|s}}. Assume first {{math|1 ≤ Re(''s'') ≤ 2}} and {{math|1 < ''a'' < ''b''}}. Then
<math display="block">\left|\gamma(s, b) - \gamma(s, a)\right| \le \int_a^b \left|t^{s-1}\right| e^{-t}\, dt = \int_a^b t^{\Re s-1} e^{-t}\, dt \le \int_a^b t e^{-t}\, dt</math>
where<ref>{{Cite web|url=https://dlmf.nist.gov/4.4|title=DLMF: §4.4 Special Values and Limits ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions|website=dlmf.nist.gov}}</ref>
<math display="block">\left|z^s\right| = \left|z\right|^{\Re s} \, e^{-\Im s\arg z}</math>
has been used in the middle. Since the final integral becomes arbitrarily small if only {{mvar|a}} is large enough, {{math|''γ''(''s'', ''x'')}} converges uniformly for {{math|''x'' → ∞}} on the strip {{math|1 ≤ Re(s) ≤ 2}} towards a holomorphic function,<ref name="class notes" /> which must be Γ(s) because of the identity theorem. Taking the limit in the recurrence relation {{math|1=''γ''(''s'', ''x'') = (''s'' − 1) ''γ''(''s'' − 1, ''x'') − ''x''<sup>''s'' − 1</sup> ''e''<sup>−''x''</sup>}} and noting, that lim {{math|1=''x''<sup>''n''</sup> ''e''<sup>−''x''</sup> = 0}} for {{math|''x'' → ∞}} and all {{mvar|n}}, shows, that {{math|''γ''(''s'', ''x'')}} converges outside the strip, too, towards a function obeying the recurrence relation of the Γ-function. It follows
<math display="block">\Gamma(s) = \lim_{x \to \infty} \gamma(s, x)</math>
for all complex {{mvar|s}} not a non-positive integer, {{mvar|x}} real and {{math|''γ''}} principal.

======Sectorwise convergence======
Now let {{mvar|u}} be from the sector {{math|{{abs|arg ''z''}} < ''δ'' < ''π''/2}} with some fixed {{mvar|δ}} ({{math|1=''α'' = 0}}), {{math|''γ''}} be the principal branch on this sector, and look at
<math display="block">\Gamma(s) - \gamma(s, u) = \Gamma(s) - \gamma(s, |u|) + \gamma(s, |u|) - \gamma(s, u).</math>

As shown above, the first difference can be made arbitrarily small, if {{math|{{abs|''u''}}}} is sufficiently large. The second difference allows for following estimation:
<math display="block">\left|\gamma(s, |u|) - \gamma(s, u)\right| \le \int_u^{|u|} \left|z^{s-1} e^{-z}\right| dz = \int_u^{|u|} \left|z\right|^{\Re s - 1} \, e^{-\Im s\,\arg z} \, e^{-\Re z} \, dz,</math>
where we made use of the integral representation of {{math|''γ''}} and the formula about {{math|{{abs|''z''<sup>''s''</sup>}}}} above. If we integrate along the arc with radius {{math|1=''R'' = {{abs|''u''}}}} around 0 connecting {{mvar|u}} and {{math|{{abs|''u''}}}}, then the last integral is
<math display="block">\le R \left|\arg u\right| R^{\Re s - 1}\, e^{\Im s\,|\arg u|}\,e^{-R\cos\arg u} \le \delta\,R^{\Re s}\,e^{\Im s\,\delta}\,e^{-R\cos\delta} = M\,(R\,\cos\delta)^{\Re s}\,e^{-R\cos\delta}</math>
where {{math|1=''M'' = ''δ''(cos ''δ'')<sup>−Re ''s''</sup> ''e''<sup>Im ''sδ''</sup>}} is a constant independent of {{mvar|u}} or {{mvar|R}}. Again referring to the behavior of {{math|''x''<sup>''n''</sup> ''e''<sup>−''x''</sup>}} for large {{mvar|x}}, we see that the last expression approaches 0 as {{mvar|R}} increases towards {{math|∞}}.
In total we now have:
<math display="block">\Gamma(s) = \lim_{|z| \to \infty} \gamma(s, z), \quad \left|\arg z\right| < \pi/2 - \epsilon,</math>
if {{mvar|s}} is not a non-negative integer, {{math|0 < ''ε'' < ''π''/2}} is arbitrarily small, but fixed, and {{math|''γ''}} denotes the principal branch on this domain.