Editing Incomplete gamma function (section)

==Indefinite and definite integrals==

The following indefinite integrals are readily obtained using [[integration by parts]] (with the [[constant of integration]] omitted in both cases):
<math display="block">\begin{align}
\int x^{b-1} \gamma(s,x) \, dx &= \frac{1}{b} \left( x^b \gamma(s,x) - \gamma(s+b,x) \right), \\[1ex]
\int x^{b-1} \Gamma(s,x) \, dx &= \frac{1}{b} \left( x^b \Gamma(s,x) - \Gamma(s+b,x) \right).
\end{align}</math>
The lower and the upper incomplete gamma function are connected via the [[Fourier transform]]:
<math display="block">\int_{-\infty}^\infty \frac {\gamma\left(\frac s 2, z^2 \pi \right)} {(z^2 \pi)^\frac s 2} e^{-2 \pi i k z} dz = \frac {\Gamma\left(\frac {1-s} 2, k^2 \pi \right)} {(k^2 \pi)^\frac {1-s} 2}.</math>
This follows, for example, by suitable specialization of {{harv|Gradshteyn|Ryzhik|Geronimus|Tseytlin|2015|loc=§7.642}}<!-- location 7.642 does not match chapter given further below. Needs to be sorted out. -->.