Convergence of integral in complex analysis textbook

manpreet Best Answer 3 years ago

In my complex analysis textbook it states that the following integral converges

∫∞1tα−1e−tdt∫1∞tα−1e−tdt

where αα is some real number such that α>1α>1

Also, it seems that t→tα−1e−tt→tα−1e−t is taken to be a real valued function of a real variable.

This integral appears in a demonstration of using a certain theorem. In the statement of the theorem, it states

"... and there exists a continuous non-negative h:[0,∞)→[0,∞)h:[0,∞)→[0,∞) such that the integral ∫∞0h(t)dt∫0∞h(t)dtconverges and ..." (Translated from Hebrew).

Is there more than one definition of convergence that could be applied to this? I do not think that my textbook previously defined such an (improper?) integral, so it may be a definition from "Calculus II" (Israeli Syllabus)

In the "most basic" or "most likely" sense of convergence, how can it be shown that this integral does in href="https://forum.tuteehub.com/tag/fact">fact converge?

Thanks!

(Note: I tagged the question with "improper-integrals", but this should not be taken to imply that this it is assumed that this is the definition the author meant)

manpreet 3 years ago