# general power

The general power$z^{\mu}$, where $z\,(\neq 0)$ and $\mu$ are arbitrary complex numbers, is defined via the complex exponential function and complex logarithm (denoted here by “$\log$”) of the by setting

 $z^{\mu}:=e^{\mu\log{z}}=e^{\mu(\ln{|z|}\!+\!i\arg{z})}.$

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

 $z^{\mu}=e^{\alpha\ln{|z|}-\beta\arg{z}}\cdot e^{i(\beta\ln{|z|}+\alpha\arg{z})},$

and thus

 $|z^{\mu}|=e^{\alpha\ln{|z|}-\beta\arg{z}},\quad\arg{z^{\mu}}=\beta\ln{|z|}\!+% \!\alpha\arg{z}.$

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

 $|z^{\mu}|=|z|^{\mu},\quad\arg{z^{\mu}}=\mu\cdot\arg{z}.$

Let  $\beta\neq 0$.  If one lets the point $z$ go round the origin anticlockwise, $\arg{z}$ gets an addition $2\pi$ and hence the $z^{\mu}$ has been multiplied by a having the modulus  $e^{-2\pi\beta}\neq 1$, and we may say that $z^{\mu}$ has come to a new branch.

Examples

1. 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. 2.

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

3. 3.

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

4. 4.

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

5. 5.

$\displaystyle(-1)^{\sqrt{2}}=e^{\sqrt{2}\log{(-1)}}=e^{\sqrt{2}i(\pi+2n\pi)}=e% ^{i(2n+1)\pi\sqrt{2}}$   (with  $n=0,\,\pm 1,\,\pm 2,\,\ldots$);  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 $\displaystyle(-1)^{\sqrt{2}}$ (see this entry (http://planetmath.org/SequenceAccumulatingEverywhereIn11)).

6. 6.

$2^{1-i}=2e^{2n\pi}(\cos\ln{2}+i\sin\ln{2})$   (with  $n=0,\pm 1,\,\pm 2,\,\ldots$), are situated on the half line beginning from the origin with the argument $\ln{2}\approx 0.69315$ radians.

 Title general power Canonical name GeneralPower Date of creation 2013-03-22 14:43:17 Last modified on 2013-03-22 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