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means_._note [2015/06/20 17:05] nikolaj |
means_._note [2015/06/20 17:06] nikolaj |
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| == Minus twelve == | == Minus twelve == | ||
| - | + | For $z\in(0,1)$, we find | |
| - | I use this in the context of [[Minus twelve . Note]]. For $z\in(0,1)$, we find | + | |
| $\sum_{k=0}^\infty \langle q\mapsto q\,z^q\rangle_{[k,k+1]}=\dfrac{1}{\ln(z)^2}$, | $\sum_{k=0}^\infty \langle q\mapsto q\,z^q\rangle_{[k,k+1]}=\dfrac{1}{\ln(z)^2}$, | ||
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| $\sum_{k=0}^\infty \left(k\,z^k-\langle q\mapsto q\,z^q\rangle_{[k,k+1]}\right)=\dfrac{z}{(z-1)^2}-\dfrac{1}{\ln(z)^2}=-\dfrac{1}{12}+{\mathcal O}\left((z-1)^1\right)$ | $\sum_{k=0}^\infty \left(k\,z^k-\langle q\mapsto q\,z^q\rangle_{[k,k+1]}\right)=\dfrac{z}{(z-1)^2}-\dfrac{1}{\ln(z)^2}=-\dfrac{1}{12}+{\mathcal O}\left((z-1)^1\right)$ | ||
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| + | See [[Minus twelve . Note]]. | ||
| ----- | ----- | ||
| === Requirements === | === Requirements === | ||
| [[Function integral]] | [[Function integral]] | ||
