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Benedict Irwin edited Explaination.tex
over 9 years ago
Commit id: 5ac4fd4b578649335a53fcd61aab488e2219a886
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index a34e48a..aa914cb 100644
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\int_0^\infty x^n\frac{d^n}{dx^n}\bigg(\frac{x^{s-1}}{e^x-1}\bigg)\;dx=(-1)^nn!\zeta(s)\Gamma(s)+\sum_{k=1}^n\frac{(-1)^{k+n}n!}{k!}\bigg[x^k\frac{d^{k-1}}{dx^{k-1}}\bigg(\frac{x^{s-1}}{e^x-1}\bigg)\bigg]_0^\infty
\end{equation}
Or rearranging \begin{equation}
\zeta(s)=\frac{(-1)^n}{n!\Gamma(s)}\Bigg[\int_0^\infty x^n\frac{d^n}{dx^n}\bigg(\frac{x^{s-1}}{e^x-1}\bigg)\;dx-\sum_{k=1}^n\frac{(-1)^{k+n}n!}{k!}\bigg[x^k\frac{d^{k-1}}{dx^{k-1}}\bigg(\frac{x^{s-1}}{e^x-1}\bigg)\bigg]_0^\infty\Bigg]
\end{equation}