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Means . Note
Note
context | $S\subseteq X$ |
context | $G$ … group |
context | $w:S\to G$ |
context | $I:(S\to G)\to G$ |
definiendum | $M:(S\to G)\to G$ |
definiendum | $\langle f\rangle:=I(f\cdot w)\cdot I(w)^{-1}$ |
e.g. $\langle f\rangle_{[a,b]}:=\dfrac{\int_a^bf(x)\,{\mathrm dx}}{b-a}$
where $[a,b]\subseteq{\mathbb R}$ and $w(x):=1$.
Note
I use this in the context of Minus twelve . Note. For $z\in(0,1)$, we find
$\sum_{k=0}^\infty \langle q\mapsto q\,z^q\rangle_{[k,k+1]}=\dfrac{1}{\ln(z)^2}$,
i.e. (see Natural logarithm of complex numbers)
$\sum_{k=0}^\infty \left(k\,z^k-\langle q\mapsto q\,z^q\rangle_{[k,k+1]}\right)=\dfrac{z}{(z-1)^2}-\dfrac{1}{\ln(z)^2}=-\dfrac{1}{12}+{\mathcal O}\left((z-1)^1\right)$