multiplicative to additive transformation

Submitted by mfg on Tue, 01/17/2017 - 22:19
English

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The multiplicative to additive mapping  for imaginary scators is a surjective function $\mathfrak{f}_{ma}:\mathbb{R}^{+1+n}\rightarrow\mathbb{S}^{1+n}$,

\[ \mathfrak{f}_{ma}:\left(\varphi_{0};\varphi_{1},\ldots,\varphi_{j},\ldots,\varphi_{n}\right)\longmapsto\left(f_{0};f_{1},\ldots,f_{j},\ldots,f_{n}\right), \]

where

\[ f_{0}=\varphi_{0}\prod_{k=1}^{n}\cos\left(\varphi_{k}\right),\quad f_{j}=\varphi_{0}\prod_{k\neq j}^{n}\cos\left(\varphi_{k}\right)\sin\left(\varphi_{j}\right)\textrm{ for }j\textrm{ from }1\textrm{ to }n.\label{eq:ma vars} \]

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The multiplicative to additive transformation $\mathbb{R}^{+1+n}\underset{\mathfrak{f}_{ma}}{\longmapsto}\mathbb{S}^{1+n}$.