We know that the Gamma function is
$$\Gamma(r)=\int_0^{\infty} e^{-x} x^{r-1} dx \ \ (r>0).$$
My Question is: What can we say about the integral
$$\int_{\mathbb R} e^{-x} x^{r-1} dx \ \ (r>0)?$$
Is it convergent or so?
2026-03-29 22:14:38.1774822478
$\int_0^{\infty} e^{-x} x^{r-1} dx<\infty$?
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No, because the integrand has exponential growth at $-\infty$. (There is also an ambiguity about what $x^{r-1}$ even means when $x<0$ and $r$ is just positive.)