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means_._note [2015/06/20 17:03] nikolaj |
means_._note [2016/03/09 10:34] nikolaj |
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===== Means . Note === | ===== Means . Note === | ||
==== Note ==== | ==== Note ==== | ||
- | | @#55CCEE: context | @#55CCEE: $S\subseteq X$ | | + | | @#55CCEE: context | @#55CCEE: $S$ ... set | |
| @#55CCEE: context | @#55CCEE: $G$ ... group | | | @#55CCEE: context | @#55CCEE: $G$ ... group | | ||
| @#55CCEE: context | @#55CCEE: $w:S\to G$ | | | @#55CCEE: context | @#55CCEE: $w:S\to G$ | | ||
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| @#FFBB00: definiendum | @#FFBB00: $\langle f\rangle:=I(f\cdot w)\cdot I(w)^{-1}$ | | | @#FFBB00: definiendum | @#FFBB00: $\langle f\rangle:=I(f\cdot w)\cdot I(w)^{-1}$ | | ||
- | e.g. $\langle f\rangle_{[a,b]}:=\dfrac{\int_a^bf(x)\,{\mathrm dx}}{b-a}$ | + | Here $(f\cdot w)(s):=f(s)*w(s)$ where $*$ is the group Operation. |
+ | |||
+ | == Real functions == | ||
+ | |||
+ | E.g. $\langle f\rangle_{[a,b]}:=\dfrac{\int_a^bf(x)\,{\mathrm dx}}{b-a}$ | ||
where $[a,b]\subseteq{\mathbb R}$ and $w(x):=1$. | where $[a,b]\subseteq{\mathbb R}$ and $w(x):=1$. | ||
- | == Note == | + | == Minus twelve == |
+ | For $z\in(0,1)$, we find | ||
+ | |||
+ | $\sum_{k=0}^\infty \langle q\mapsto q\,z^q\rangle_{[k,k+1]}=\dfrac{1}{\ln(z)^2}$, | ||
- | I use this in the context of [[Minus twelve . Note]]. For $z\in(0,1)$, we find | + | i.e. (see [[Natural logarithm of complex numbers]]) |
- | $\sum_{k=0}^\infty \left(k\,z^k-\langle k\mapsto k\,z^k\rangle_{[k,k+1]}\right)=\dfrac{z}{(z-1)^2}-\dfrac{1}{\ln(z)^2}=-\dfrac{1}{12}+{\mathcal O}\left((z-1)^1\right)$ | + | $\sum_{k=0}^\infty \left(k\,z^k-\langle q\mapsto q\,z^q\rangle_{[k,k+1]}\right)=\dfrac{z}{(z-1)^2}-\dfrac{1}{\ln(z)^2}=-\dfrac{1}{12}+{\mathcal O}\left((z-1)^1\right)$ |
- | See also [[Natural logarithm of complex numbers]]. | + | See [[Minus twelve . Note]]. |
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=== Requirements === | === Requirements === | ||
- | [[Function integral]] | + | [[Function integral]], [[Classical probability density function]] |