PlanetMath: general power

(more info)

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general power

(Definition)

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 base by setting

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

The number

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

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

and thus

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

This shows that both the modulus and the argument 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

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

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

Examples

$z^{\frac{1}{m}}$ , where is a positive integer, coincides with the $m^\mathrm{th}$ root of .
$\displaystyle 3^2 = e^{2\log{3}} = e^{2(\ln{3}+2n\pi i)} = 9(e^{2\pi i})^{2n} = 9$ $\forall n\in\mathbb{Z}$ .
$\displaystyle i^i = e^{i\log{i}} = e^{i(\ln{1}+\frac{\pi}{2}i-2n\pi i)} = e^{2n\pi-\frac{\pi}{2}}$ (with $n = 0,\,\pm1,\,\pm2,\,\ldots$ ); all these values are positive real numbers, the simplest of them is $\displaystyle\frac{1}{\sqrt{e^\pi}} \approx 0.20788$ .
$(-1)^i = e^{(2n+1)\pi}$ (with $n = 0,\,\pm1,\,\pm2,\,\ldots$ ) also are situated on the positive real axis.
$\displaystyle (-1)^{\sqrt{2}} = e^{\sqrt{2}\log{(-1)}} = e^{\sqrt{2}i(\pi+2n\pi)} = e^{i(2n+1)\pi\sqrt{2}}$ (with $n = 0,\,\pm1,\,\pm2,\,\ldots$ ); all these are imaginary numbers (meaning here that their imaginary parts are distinct from 0), situated on the circumference of the unit circle.
$2^{1-i} = 2e^{2n\pi}(\cos\ln{2}+i\sin\ln{2})$ (with $n = 0,\pm1,\,\pm2,\,\ldots$ ), are situated on the half line beginning from the origin with the argument $\ln{2} \approx 0.69315$ radians.