Difference between revisions of "Riemann zeta as contour integral"
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(Created page with "==Theorem== The following formula holds: $$\zeta(z)=-\dfrac{\Gamma(1-z)}{2\pi i} \displaystyle\int_C \dfrac{(-\xi)^{z-1}}{e^{\xi}-1} \mathrm{d}\xi,$$ where $\zeta$ is Rieman...") |
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Revision as of 23:38, 17 March 2017
Theorem
The following formula holds: $$\zeta(z)=-\dfrac{\Gamma(1-z)}{2\pi i} \displaystyle\int_C \dfrac{(-\xi)^{z-1}}{e^{\xi}-1} \mathrm{d}\xi,$$ where $\zeta$ is Riemann zeta, $\Gamma$ is gamma, $\pi$ is pi, $i$ is the imaginary number, $e^{\xi}$ denotes the exponential, and $C$ is a contour that begins at $\infty$ on the real axis, encircles the origin once counter-clockwise (excluding $\pm 2\pi i, \pm 4\pi i, \ldots$) and returns to the starting point.
Proof
References
- 1930: Edward Charles Titchmarsh: The Zeta-Function of Riemann ... (previous) ... (next): § Introduction $(4)$
