From a35c58dffed76bd9991af62eb88aa1138782d0ef Mon Sep 17 00:00:00 2001 From: Chris Dams Date: Thu, 1 Feb 2007 16:15:02 +0000 Subject: [PATCH] Added a proof for our new simplification rule for powers. --- doc/powerlaws.tex | 18 +++++++++++++++--- 1 file changed, 15 insertions(+), 3 deletions(-) diff --git a/doc/powerlaws.tex b/doc/powerlaws.tex index 1943957a..ade0547d 100644 --- a/doc/powerlaws.tex +++ b/doc/powerlaws.tex @@ -151,7 +151,19 @@ Hence & = & x^{ab} \mbox{ q.e.d.} \end{eqnarray} -proof contributed by Adam Strzebonski from Wolfram Research -({\tt adams@wolfram.com}) in newsgroup {\tt sci.math.symbolic}. +Proof contributed by Adam Strzebonski ({\tt adams@wolfram.com}) from +Wolfram Research in newsgroup {\tt sci.math.symbolic}. -\end{document} \ No newline at end of file +\subsubsection{$x$ positive, $a$ real and $b$ arbitrary complex} +We have +\begin{equation} +(x^a)^b = e^{b\log e^{a\log x}}. +\end{equation} +Because $a$ is real and $x$ is positive, $a\log x$ is real. From this +it follows that $\log e^{a\log x} = a\log x$. I.e, we see that +\begin{equation} +(x^a)^b = e^{ba\log x} = x^{ab}. +\end{equation} +Qed. + +\end{document} -- 2.44.0