general power

The general power  $z^{\mu }$, where $z(\ne 0)$ 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

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

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

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

Let  $\beta \ne 0$.  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  $e^{-2\pi \beta }\ne 1$, 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. $3^2=e^{2\log 3}=e^{2(\ln 3+2n\pi i)}=9{(e^{2\pi i})}^{2n}=9$   $\forall n\in \mathbb{Z}$.

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

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. ${(-1)}^{\sqrt{2}}=e^{\sqrt{2}\log (-1)}=e^{\sqrt{2}i(\pi +2n\pi )}=e^{i(2n+1)\pi \sqrt{2}}$   (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. $2^{1-i}=2e^{2n\pi }(\cos \ln 2+i\sin \ln 2)$   (with  $n=0,\pm 1,\pm 2,\mathrm{\dots }$), 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