Editing Incomplete gamma function (section)

== Derivatives ==

Using the integral representation above, the derivative of the upper incomplete gamma function <math> \Gamma (s,x) </math> with respect to {{mvar|x}} is
<math display="block"> \frac{\partial \Gamma (s,x) }{\partial x} = - x^{s-1} e^{-x}</math>
The derivative with respect to its first argument <math>s</math> is given by<ref>[[Keith Geddes|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.&nbsp;149–165, [https://doi.org/10.1007%2FBF01810298] 
</ref>
<math display="block">\frac{\partial \Gamma (s,x) }{\partial s} = \ln x \Gamma (s,x) + x\,T(3,s,x)</math>
and the second derivative by
<math display="block">\frac{\partial^2 \Gamma (s,x) }{\partial s^2} = \ln^2 x \Gamma (s,x) + 2 x \left[\ln x\,T(3,s,x) + T(4,s,x) \right]</math>
where the function <math>T(m,s,x)</math> is a special case of the [[Meijer G-function]]
<math display="block">T(m,s,x) = G_{m-1,\,m}^{\,m,\,0} \!\left( \left. \begin{matrix} 0, 0, \dots, 0 \\ s-1, -1, \dots, -1 \end{matrix} \; \right| \, x \right).</math>
This particular special case has internal ''closure'' properties of its own because it can be used to express ''all'' successive derivatives. In general, 
<math display="block">\frac{\partial^m \Gamma (s,x) }{\partial s^m} = \ln^m x \Gamma (s,x) + m x\,\sum_{n=0}^{m-1} P_n^{m-1} \ln^{m-n-1} x\,T(3+n,s,x)</math>
where <math> P_j^n </math> is the [[permutation]] defined by the [[Pochhammer symbol]]:
<math display="block">P_j^n = \binom{n}{j} j! = \frac{n!}{(n-j)!}.</math>
All such derivatives can be generated in succession from:
<math display="block">\frac{\partial T (m,s,x) }{\partial s} = \ln x ~ T(m,s,x) + (m-1) T(m+1,s,x)</math>
and
<math display="block">\frac{\partial T (m,s,x) }{\partial x} = -\frac{T(m-1,s,x) + T(m,s,x)}{x}</math>
This function <math>T(m,s,x)</math> can be computed from its series representation valid for <math> |z| < 1 </math>, 
<math display="block">T(m,s,z) = - \frac{\left(-1\right)^{m-1} }{(m-2)! } \left.\frac{d^{m-2} }{dt^{m-2} } \left[\Gamma (s-t) z^{t-1}\right]\right|_{t=0} + \sum_{n=0}^{\infty} \frac{\left(-1\right)^n z^{s-1+n}}{n! \left(-s-n\right)^{m-1} }</math>
with the understanding that {{mvar|s}} is not a negative integer or zero.  In such a case, one must use a limit. Results for <math> |z| \ge 1 </math> can be obtained by [[analytic continuation]]. Some special cases of this function can be simplified. For example, <math>T(2,s,x)=\Gamma(s,x)/x</math>, <math>x\,T(3,1,x) = \mathrm{E}_1(x)</math>, where <math>\mathrm{E}_1(x)</math> is the [[Exponential integral]]. These derivatives and the function <math>T(m,s,x)</math> provide exact solutions to a number of integrals by repeated differentiation of the integral definition of the upper incomplete gamma function.<ref>{{cite journal | first1=M. S. |last1=Milgram|title=The generalized integro-exponential function|journal=Math. Comp.| year=1985|volume=44| issue=170| pages=443–458|mr=0777276| doi=10.1090/S0025-5718-1985-0777276-4|doi-access=free}}</ref><ref>{{cite arXiv| eprint=0912.3844| author1=Mathar|title=Numerical Evaluation of the Oscillatory Integral over exp(i*pi*x)*x^(1/x) between 1 and infinity| class=math.CA|year=2009}}, App B</ref>
For example,
<math display="block"> \int_{x}^{\infty} \frac{t^{s-1} \ln^m t}{e^t} dt= \frac{\partial^m}{\partial s^m} \int_{x}^{\infty} \frac{t^{s-1}}{e^t}  dt = \frac{\partial^m}{\partial s^m} \Gamma (s,x)</math>
This formula can be further ''inflated'' or generalized to a huge class of [[Laplace transform]]s and [[Mellin transform]]s. When combined with a [[computer algebra system]], the exploitation of special functions provides a powerful method for solving definite integrals, in particular those encountered by practical engineering applications (see [[Symbolic integration]] for more details).