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Opalg · #4 December 27th, 2008, 05:39 AM

Quote:

Originally Posted by chiph588@

I have a couple question regarding the Riemann zeta function and its extension onto the complex plane. I know the function is given by the reflection formula $\zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)$ .

My first question is how is this proved?

My second question is how does this extend $\zeta(s)$ to the complex plane?

The zeta function is initially defined as $\zeta(s) = \sum_{n=1}^\infty n^{-s}$ . This series converges for all complex numbers s whose real part is greater than 1. To extend the domain of definition, the usual procedure (as far as I understand it—I'm not an expert in this area) is to use analytic continuation to define it for all s with real part greater than 0 (except for the pole at s=1). An intricate argument using contour integration (as in the previous comments) then establishes the functional equation for s in the strip 0<re(s)<1. The functional equation is then used to define $\zeta(s)$ in the remainder of the complex plane.

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