Difference between revisions of "Laurent series of the Riemann zeta function"
From specialfunctionswiki
(Created page with "<div class="toccolours mw-collapsible mw-collapsed"> <strong>Theorem:</strong> The following Laurent series holds: $$\zeta(...") |
|||
| Line 1: | Line 1: | ||
| − | + | ==Theorem== | |
| − | + | The following [[Laurent series]] holds: | |
$$\zeta(z)=\dfrac{1}{z-1} + \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k \lambda_k (z-1)^k}{k!},$$ | $$\zeta(z)=\dfrac{1}{z-1} + \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k \lambda_k (z-1)^k}{k!},$$ | ||
where $\zeta$ denotes the [[Riemann zeta]] function and $\lambda_k$ denotes the [[Stieltjes constants]]. | where $\zeta$ denotes the [[Riemann zeta]] function and $\lambda_k$ denotes the [[Stieltjes constants]]. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 05:02, 16 September 2016
Theorem
The following Laurent series holds: $$\zeta(z)=\dfrac{1}{z-1} + \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k \lambda_k (z-1)^k}{k!},$$ where $\zeta$ denotes the Riemann zeta function and $\lambda_k$ denotes the Stieltjes constants.
