How to integrate the following gamma type function

Question

I know how to integrate the Gamma function using various gamma formula. But, I was wondering if the following type of problems also can be solved using the gamma integral:

$$\int_0^\infty \frac 1 {y^m}\exp(-y) \, dy$$

Please tell me the method that we can use to do the above type of integral where the value of $m$ can be any positive integer. Thanks in advance.

Answer 1

$$ \int_0^\infty \frac 1 {y^m} \exp(-y) \, dy = \int_0^\infty y^n \exp(-y) \, dy \text{ where } n = -m. $$ The latter integral has a finite value if $n>-1$ and is $+\infty$ if $n\le-1.$

Therefore the former integral has a finite value if $m<1$ and is $+\infty$ if $m\ge 1.$

Answer 2

The problem happens because of the lower bound equal to $0$. $$\int_\epsilon^\infty \frac 1 {y^m} \exp(-y) \, dy =\Gamma (1-m,\epsilon )\qquad \text{if}\qquad \Im(\epsilon )\neq 0\lor \epsilon >0$$ If $\epsilon=0$, as Michael Hardy already answered, $$\int_0^\infty \frac 1 {y^m} \exp(-y) \, dy =\Gamma (1-m)\qquad \text{if}\qquad \Re(m)<1$$ Close to $m=1$, the asymptotics is $$\Gamma (1-m)=-\frac{1}{m-1}-\gamma -\frac{6 \gamma ^2+\pi ^2}{12} (m-1)+O\left((m-1)^2\right)$$