I am reading the following paper:


and cannot seem to figure out how Jaynes is deriving (P2) and below (specifically the log arithmetic log[f(x)/f(0)]....)

Can someone help me out?

  • $\begingroup$ Can you put some details in your question please, firstly so readers of your question are not required to go read a paper to understand the question, and secondly, to protect your question from the possibility that the link will disappear (at which point it becomes not so much a useful resource for others as junk cluttering up the site) $\endgroup$
    – Glen_b
    Commented Sep 1, 2014 at 23:01

1 Answer 1


Equation $$ f(x)f(y)=g(\sqrt{x^2+y^2}) \qquad (*) $$ holds for every $x,y$. If $y=0$, then $(*)$ gives $g(|x|)=f(x)f(0)$ for every $x$. Using this in $(*)$ to "eliminate" $g$, we have $f(x)f(y)=f(\sqrt{x^2+y^2})f(0)$. Dividing by $(f(0))^2$ and taking the log on both sides we find $$ \frac{f(x)f(y)}{f(0)f(0)}=\frac{f\left(\sqrt{x^2+y^2}\right)}{f(0)} \, , $$ $$ \log\left(\frac{f(x)}{f(0)}\right)+\log\left(\frac{f(y)}{f(0)}\right)= \log\left( \frac{f\left(\sqrt{x^2+y^2}\right)}{f(0)}\right) \, . $$


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