Editing Incomplete gamma function (section)

======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.