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# 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 base 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 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 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

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

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

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 imaginary numbers (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 powers $\displaystyle(-1)^{{\sqrt{2}}}$ (see this entry).

6.

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## Comments

## anilped: unusual examples of limits

Anilped, you have requested unusual examples of limits of composite functions.

When x --> 0,

we have (1+1/x)^x --> 1.

Swapping the inner and the outer function,

we get 1+1/x^x --> 2.

Do you mean such cases?

Regards,

Jussi

## Re: anilped: unusual examples of limits

The answer looks correct, but what do you mean by 'Swapping the inner and the outer function' ?

## Re: anilped: unusual examples of limits

Now, when I read anew the request of anilped, I see that my examples do not answer to it.

## Re: anilped: unusual examples of limits

How about f(x)=g(x)=1/x. Then the limit as x->0 of f(x) and g(x) does not exist yet the limit as x->0 of f(g(x)) is equal to 0.