Editing Exponentiation (section)

====Computation====
The ''canonical form'' <math>x+iy</math> of <math>z^w</math> can be computed from the canonical form of {{mvar|z}} and {{mvar|w}}. Although this can be described by a single formula, it is clearer to split the computation in several steps.
* ''[[Polar form]] of {{mvar|z}}''. If <math>z=a+ib</math> is the canonical form of {{mvar|z}} ({{mvar|a}} and {{mvar|b}} being real), then its polar form is <math display=block>z=\rho e^{i\theta}= \rho (\cos\theta + i \sin\theta),</math> with <math display=inline>\rho=\sqrt{a^2+b^2}</math> and <math>\theta=\operatorname{atan2}(b,a)</math>, where {{tmath|\operatorname{atan2} }} is the [[atan2|two-argument arctangent]] function.
* ''[[complex logarithm|Logarithm]] of {{mvar|z}}''. The [[principal value]] of this logarithm is <math>\log z=\ln \rho+i\theta,</math> where <math>\ln</math> denotes the [[natural logarithm]]. The other values of the logarithm are obtained by adding <math>2ik\pi</math> for any integer {{mvar|k}}.
* ''Canonical form of <math>w\log z.</math>'' If <math>w=c+di</math> with {{mvar|c}} and {{mvar|d}} real, the values of <math>w\log z</math> are <math display=block>w\log z = (c\ln \rho - d\theta-2dk\pi) +i (d\ln \rho + c\theta+2ck\pi),</math> the principal value corresponding to <math>k=0.</math>
* ''Final result''. Using the identities <math>e^{x+y}=e^xe^y</math> and <math>e^{y\ln x} = x^y,</math> one gets <math DISPLAY=block>z^w=\rho^c e^{-d(\theta+2k\pi)} \left(\cos (d\ln \rho + c\theta+2ck\pi) +i\sin(d\ln \rho + c\theta+2ck\pi)\right),</math> with <math>k=0</math> for the principal value.

=====Examples=====
* <math>i^i</math> <br>The polar form of {{mvar|i}} is <math>i=e^{i\pi/2},</math> and the values of <math>\log i</math> are thus <math DISPLAY=block>\log i=i\left(\frac \pi 2 +2k\pi\right).</math> It follows that <math DISPLAY=block>i^i=e^{i\log i}=e^{-\frac \pi 2} e^{-2k\pi}.</math>So, all values of <math>i^i</math> are real, the principal one being <math DISPLAY=block> e^{-\frac \pi 2} \approx 0.2079.</math>
* <math>(-2)^{3+4i}</math><br>Similarly, the polar form of {{math|−2}} is <math>-2 = 2e^{i \pi}.</math> So, the above described method gives the values <math DISPLAY=block>\begin{align}
(-2)^{3 + 4i} &= 2^3 e^{-4(\pi+2k\pi)} (\cos(4\ln 2 + 3(\pi +2k\pi)) +i\sin(4\ln 2 + 3(\pi+2k\pi)))\\
&=-2^3 e^{-4(\pi+2k\pi)}(\cos(4\ln 2) +i\sin(4\ln 2)).
\end{align}</math>In this case, all the values have the same argument <math>4\ln 2,</math> and different absolute values.

In both examples, all values of <math>z^w</math> have the same argument. More generally, this is true if and only if the [[real part]] of {{mvar|w}} is an integer.