general power
The general power ${z}^{\mu}$, where $z\phantom{\rule{veryverythickmathspace}{0ex}}(\ne 0)$ and $\mu $ are arbitrary complex numbers^{}, is defined via the complex exponential function and complex logarithm (denoted here by “$\mathrm{log}$”) of the by setting
$${z}^{\mu}:={e}^{\mu \mathrm{log}z}={e}^{\mu (\mathrm{ln}z+i\mathrm{arg}z)}.$$ 
The number $z$ is the base of the power ${z}^{\mu}$ and $\mu $ is its exponent.
Splitting the exponent $\mu =\alpha +i\beta $ in its real and imaginary parts^{} one obtains
$${z}^{\mu}={e}^{\alpha \mathrm{ln}z\beta \mathrm{arg}z}\cdot {e}^{i(\beta \mathrm{ln}z+\alpha \mathrm{arg}z)},$$ 
and thus
$${z}^{\mu}={e}^{\alpha \mathrm{ln}z\beta \mathrm{arg}z},\mathrm{arg}{z}^{\mu}=\beta \mathrm{ln}z+\alpha \mathrm{arg}z.$$ 
This shows that both the modulus and the argument (http://planetmath.org/Complex) of the general power are in general multivalued. The modulus is unique only if $\beta =0$, i.e. if the exponent $\mu =\alpha $ is real; in this case we have
$${z}^{\mu}={z}^{\mu},\mathrm{arg}{z}^{\mu}=\mu \cdot \mathrm{arg}z.$$ 
Let $\beta \ne 0$. If one lets the point $z$ go round the origin anticlockwise, $\mathrm{arg}z$ gets an addition $2\pi $ and hence the ${z}^{\mu}$ has been multiplied by a having the modulus ${e}^{2\pi \beta}\ne 1$, and we may say that ${z}^{\mu}$ has come to a new branch.
Examples

1.
${z}^{\frac{1}{m}}$, where $m$ is a positive integer, coincides with the ${m}^{\mathrm{th}}$ root (http://planetmath.org/CalculatingTheNthRootsOfAComplexNumber) of $z$.

2.
${3}^{2}={e}^{2\mathrm{log}3}={e}^{2(\mathrm{ln}3+2n\pi i)}=9{({e}^{2\pi i})}^{2n}=9$ $\forall n\in \mathbb{Z}$.

3.
${i}^{i}={e}^{i\mathrm{log}i}={e}^{i(\mathrm{ln}1+\frac{\pi}{2}i2n\pi i)}={e}^{2n\pi \frac{\pi}{2}}$ (with $n=0,\pm 1,\pm 2,\mathrm{\dots}$); all these values are positive real numbers, the simplest of them is $\frac{1}{\sqrt{{e}^{\pi}}}}\approx 0.20788$.

4.
${(1)}^{i}={e}^{(2n+1)\pi}$ (with $n=0,\pm 1,\pm 2,\mathrm{\dots}$) also are situated on the positive real axis.

5.
${(1)}^{\sqrt{2}}={e}^{\sqrt{2}\mathrm{log}(1)}={e}^{\sqrt{2}i(\pi +2n\pi )}={e}^{i(2n+1)\pi \sqrt{2}}$ (with $n=0,\pm 1,\pm 2,\mathrm{\dots}$); all these are (meaning here that their imaginary parts are distinct from 0), situated on the circumference of the unit circle^{} such that all points of the circumference are accumulation points of the sequence of the ${(1)}^{\sqrt{2}}$ (see this entry (http://planetmath.org/SequenceAccumulatingEverywhereIn11)).

6.
${2}^{1i}=2{e}^{2n\pi}(\mathrm{cos}\mathrm{ln}2+i\mathrm{sin}\mathrm{ln}2)$ (with $n=0,\pm 1,\pm 2,\mathrm{\dots}$), are situated on the half line beginning from the origin with the argument $\mathrm{ln}2\approx 0.69315$ radians.
Title  general power 
Canonical name  GeneralPower 
Date of creation  20130322 14:43:17 
Last modified on  20130322 14:43:17 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  31 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 30D30 
Synonym  complex power 
Related topic  Logarithm 
Related topic  ExponentialOperation 
Related topic  GeneralizedBinomialCoefficients 
Related topic  PuiseuxSeries 
Related topic  PAdicExponentialAndPAdicLogarithm 
Related topic  FractionPower 
Related topic  SomeValuesCharacterisingI 
Related topic  UsingResidueTheoremNearBranchPoint 
Defines  base of the power 
Defines  base 
Defines  exponent 
Defines  branch 