1 \documentclass{article}
7 \subsection{Definitions}
9 Definitions for power and log:
10 \begin{equation}\label{powerdef}
11 x^a \equiv e^{a \ln x}
14 \ln x \equiv \ln |x| + i \arg(x) \mbox{ where } -\pi < \arg(x) \le \pi
17 \subsection{General rules}
22 for arbitrary complex \(x\) and \(y\) (with~(\ref{powerdef}) we obtain
23 the rule \(x^ax^b=x^{a+b}\) since \(x^ax^b\equiv e^{a\ln x}e^{b\ln x} =
24 e^{(a+b)\ln x}\equiv x^{a+b}\) for arbitrary complex \(a,b,x\))
27 x^{-a} = \frac{1}{x^a}
29 for arbitrary complex \(x\) and \(a\)
31 \subsection{\((ax)^b=a^b x^b\)}
33 \subsubsection{\(b\) integer, \(x\) and \(a\) arbitrary complex}
38 (ax)^b & = & \underbrace{(ax) \cdots (ax)}_{b \times}
40 & = & \underbrace{a \cdots a}_{b \times}
41 \underbrace{x \cdots x}_{b \times}
43 & = & a^b x^b \mbox{ q.e.d.}
46 if \(b<0\) (so \(b=-|b|\))
48 (ax)^b & = & \frac{1}{(ax)^{|b|}}
50 & = & \frac{1}{a^{|b|} x^{|b|}}
52 & = & a^{-|b|} x^{-|b|}
57 \subsubsection{\(a>0\), \(x\) and \(b\) arbitrary complex}
60 (ax)^b & = & e^{b \ln(ax)}
62 & = & e^{b (\ln |ax| + i \arg(ax))}
65 if \(a\) is real and positive:
67 \ln |ax| = \ln |a| + \ln |x| = \ln a + \ln |x|
76 e^{b (\ln |ax| + i \arg(ax))} & = &
77 e^{b (\ln a + \ln |x| + i \arg(x))}
79 & = & e^{b (\ln a + \ln x)}
81 & = & e^{b \ln a} e^{b \ln x}
83 & = & a^b x^b \mbox{ q.e.d.}
86 \subsection{\((x^a)^b = x^{ab}\)}
88 \subsubsection{\(b\) integer, \(x\) and \(a\) arbitrary complex}
93 (x^a)^b & = & \underbrace{(x^a) \cdots (x^a)}_{b \times}
95 & = & \underbrace{e^{a \ln x} \cdots e^{a \ln x}}_{b \times}
97 & = & e^{\underbrace{\scriptstyle a \ln x + \dots + a \ln x}_{b \times}}
101 & = & x^{ab} \mbox{ q.e.d.}
104 if \(b<0\) (so \(b=-|b|\))
106 (x^a)^b & = & \frac{1}{(x^a)^{|b|}}
108 & = & \frac{1}{x^{a|b|}}
115 \subsubsection{\(-1 < a \le 1\), \(x\) and \(b\) arbitrary complex}
119 x^a=e^{a \ln|x| + ia\arg(x)}
127 \arg(x^a)-a\arg(x)=2k\pi
129 now if \(-1 < a \le 1\), then \(-\pi < a\arg(x) \le \pi\),
134 (Note that for \(a=-1\) this may not be true, as \(-1 \arg(x)\) may be equal to \(-\pi\).)
137 \ln(x^a) & = & \ln|x^a| + i\arg(x^a)
139 & = & \ln (e^{a\ln|x|})+ia\arg(x)
141 & = & a \ln |x| + ia\arg(x) \mbox{ (because \(a\ln|x|\) is real)}
147 (x^a)^b & = & e^{b\ln x^a}
151 & = & x^{ab} \mbox{ q.e.d.}
154 proof contributed by Adam Strzebonski from Wolfram Research
155 ({\tt adams@wolfram.com}) in newsgroup {\tt sci.math.symbolic}.