Special Functions.

Question

How do I show that

$\Gamma(v,x)=(v-1)\Gamma(v-1,x) + x^{v-1} e^{-v}$.

I know I should use Integration by parts on the formula for upper incomplete gamma function. But the result is not coming out. A hint would be helpful, Thanks.

Answer 1

Did you try the integration by parts? If $x < y$ you have $$\int_x^y t^{v-1}e^{-t} \, dt = - y^{v-1}e^{-y} + x^{v-1} e^{-x} + (v-1) \int_x^y t^{v-2} e^{-t} \, dt.$$ Now let $y \to \infty$ to conclude (since both integrals converge) that $$\int_x^\infty t^{v-1} e^{-t} \, dt = x^{v-1}e^{-x} + (v-1) \int_x^\infty t^{v-2} e^{-t} \, dt.$$