Complex exponentiation

This extends real exponentiation.

For nonzero complex $z$ and complex $w$, the multivalued power is

For $w\ne 0$, different complex logarithm values differ by $2\pi ik$. Applying the exponential function gives the displayed multivalued power. Selecting a single logarithm branch gives one value. When $z=i$, successive branch values differ by the positive factor $e^{-2\pi }$ and lie on one ray of fixed argument $\ln |w|$. If this factor is not one, the nonzero values accumulate at zero in one direction of the integer index.

For $z\ne 0$, every logarithmic value is $\log |z|+i(\arg z+2\pi k)$ with integer $k$. Multiplying by $z$ and taking the real part proves the displayed formula for the modulus of each value of complex exponentiation. For nonreal $z$, these values form a geometric progression with nonunit positive ratio $e^{-2\pi \mathrm{Im}z}$. For real nonzero $z$, changing the logarithm branch changes only the phase and leaves the modulus fixed. A fixed initial argument changes only the indexing of the set of values.

Using principal complex logarithms, the inner power is $i^i=e^{-\pi /2}$. Consequentlywhich are different. Thus parentheses in nested complex exponentiation cannot be dropped or moved. There is an additional issue when all complex logarithm branches are allowed: each nested operation must retain its own branch choices. In the first expression, the complete set of values is $-i{e}^{-2\pi m}$, $m\in \mathbb{Z}$; in the second, it is $\exp [i(\pi /2+2\pi m)e^{-(\pi /2+2\pi k)}]$, $k,m\in \mathbb{Z}$. The principal-value comparison already establishes nonassociativity; the multivalued sets make the role of branching explicit.

The principal cube root is $z^{1/3}=\exp (\mathrm{Log}z/3)$ using the principal complex logarithm. In the right half-plane, its complex argument lies between $-\pi /6$ and $\pi /6$. Consequently the roots of $r^3=-z$ consist of one root with negative real part, $-z^{1/3}$, and two with positive real part, $z^{1/3}{e}^{\pm i\pi /3}$.

For every integer $n$ and real $\theta$,It converts powers of a complex number in polar form into multiplication of its complex argument.