1.1.12 Complex Powers and Logarithms

It may seem odd to con­sid­er rais­ing a number to an imag­i­nary power, but it is pos­si­ble to use the al­ge­bra­ic prop­er­ties of the nat­ur­al log­a­rithm and ex­po­nents. We can recall the fol­low­ing prop­er­ties for $x\in \mathbb{R}$,

For com­plex num­bers, ex­po­nen­ti­a­tion can be more easily de­ter­mined by using the com­plex polar form of $(a + ib)$

We can state that $(a + ib) = z$, to recast the ex­pres­sion above, as

where $|z|$is the mod­u­lus. We can now define $w = u \bullet v = re^{i\theta}$, and find

The prin­ci­pal so­lu­tion for the log­a­rithm $w\in \mathbb{C}$, can now be stated as

where we are taking a branch cut for $\theta,$on the in­ter­val $( - \pi,\pi\rbrack$, since any in­te­ger mul­ti­ple of $2\pi$ could be added to $\theta$, re­sult­ing in a point in the same lo­ca­tion. Now, we ex­am­ine cal­cu­lat­ing com­plex num­bers raised to com­plex powers. For $w,z\in \mathbb{C}$, and $z = (c + id),$the number $w^{z}$, can be ex­pressed by the fol­low­ing

which ex­pands to

Taking the simple ex­am­ple $e^{i\theta}$, a rep­re­sen­ta­tion of the uni­tary group $U$, we raise $e^{i\theta}$ to the imag­i­nary power $- i\phi$, find­ing that

So, in essence, we have raised the $U(1)$ group to a neg­a­tive imag­i­nary power, trans­form­ing $U(1)$ into a real ex­po­nen­tial series com­posed of hy­per­bol­ic trigono­met­ric func­tions.