Calculation of an integral with Gamma function

Question

Can help me to calculate this integral: $$ \int_{z}^{\infty} \exp(-l\cdot x) \Gamma(m,x) dx, \qquad $$ with $$\Gamma(m,x) = \int_x^{\infty} \exp(-y) y^{m-1} dy$$ and where w,l,z positives

Answer 1

Hint

$$I=\int e^{-kx}\,\Gamma(m,x)\,dx$$ Using integration by parts $$du=e^{-kx}\,dx\quad \implies \quad u=-\frac 1 k e^{-k x}$$ $$v=\Gamma(m,x)\quad \implies \quad dv=-e^{- x} \, x^{m-1}$$ $$I=-\frac 1 k e^{-k x}\,\Gamma(m,x)-\frac 1k \int e^{-(k+1) x} \,x^{m-1}\,dx$$

Let $(k+1)x=y$ and look again at the definition of $\Gamma(m,x)$.