I would like to know whether the following relation is correct or not?

$\frac{d}{dz}\Gamma(w,\mu z)= -\mu^wz^{w-1}e^{-\mu z}$,

where $\Gamma(w,\mu z)$ is the upper incomplete gamma integral.

Can anyone provide me a reference for the above relation if it is correct?


I assume $w, \mu$ do not depend on $z$. From the definition as an integral or as an explicit reference e.g. http://functions.wolfram.com/GammaBetaErf/Gamma2/20/01/02/0001/ we have $$\frac{\partial}{\partial z} \Gamma(a,z) = -e^{-z}z^{a-1}$$ and therefore with the chain-rule $$\frac{\partial}{\partial z}\Gamma(w,\mu z)= -e^{-\mu z}(\mu z)^{w-1}\mu = -\frac{e^{-\mu z}}{z}(\mu z)^w $$ The difference between this an your expression is $$ -\frac{e^{-\mu z}}{z}((\mu z)^w - \mu^w z^w) $$ which vanishes if $(\mu z)^w= \mu^w z^w.\;$ Thus the validity of the relation depends on the domains of $\mu, z, w;\,$ it is correct e.g. for real positive values.


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