Difference between revisions of "Stieltjes constants"
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The Stieltjes constants are defined by | The Stieltjes constants are defined by | ||
$$\gamma_n = \displaystyle\lim_{m \rightarrow \infty} \left[ \displaystyle\sum_{k=1}^m \dfrac{\log^n(k)}{k} - \dfrac{\log^{n+1}(m)}{n+1} \right]$$ | $$\gamma_n = \displaystyle\lim_{m \rightarrow \infty} \left[ \displaystyle\sum_{k=1}^m \dfrac{\log^n(k)}{k} - \dfrac{\log^{n+1}(m)}{n+1} \right]$$ | ||
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| + | =Properties= | ||
| + | {{:Laurent series of the Riemann zeta function}} | ||
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Revision as of 06:13, 1 June 2016
The Stieltjes constants are defined by $$\gamma_n = \displaystyle\lim_{m \rightarrow \infty} \left[ \displaystyle\sum_{k=1}^m \dfrac{\log^n(k)}{k} - \dfrac{\log^{n+1}(m)}{n+1} \right]$$
Contents
Properties
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.
