I am learning the Gamma function, based on some lecture notes, and I wish to ask a couple of questions regarding infinite products.

Let $z$ be a complex number except $\{0, -1, \ldots \}$.

(1) How do I show that $\prod_{m=1}^n\left(1+\frac{z}{m}\right)^{-1}$ diverges as $n\rightarrow\infty$; but

(2) $\prod_{m=1}^n\left(1+\frac{z}{m}\right)^{-1}\left(1+\frac{1}{m}\right)^z$ converges?

Any suggested readings on the "limit-definition" of the Gamma function $\Gamma(z)$? Thanks!

  • 1
    $\begingroup$ Take logarithms so you get a sum instead, and use the Taylor series for $\ln(1+x)\approx x$. Since you're dealing with complex numbers, you may need to be a bit delicate with choice of logarithms. $\endgroup$ – Samuel Jun 9 '13 at 22:04
  • $\begingroup$ @Samuel - do you think taking the principal logarithm suffices? $\endgroup$ – Eric Jun 9 '13 at 22:06
  • $\begingroup$ I think all you need is that it exists for all but finitely many $m$, so yes. $\endgroup$ – Samuel Jun 9 '13 at 22:09
  • $\begingroup$ Thanks! I already convinced myself for real numbers. I just need to do the "dirty" work now to include complex numbers. Thanks again! $\endgroup$ – Eric Jun 9 '13 at 22:11
  • 1
    $\begingroup$ @Eric For $m>|z|$ you are on the safe side the principal logarithm, i.e. you are within the radius of convergence of $\ln(1+x)=x+\ldots$. $\endgroup$ – Hagen von Eitzen Jun 9 '13 at 22:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.