Question

(a) Find all the possible values of i^i

(b) Find all the possible values of -1^(-i)

(c) Is 1 to every power (real or complex) necessarily equal to 1?

Answer 1

a. Since
`i=e^(i\pi/2)`, we have

`i^i=(e^(i\pi/2))^i=e^(i^2\pi/2)=e^(-\pi/2)`

b. Since
`1=e^0`, we have

`-1^(-i)=-(e^0)^(-i)=-(e^0)=-1`

c. Yes, for the reason illustrated in part b.
`1=e^0`, and raising this to any power
`z\in\mathbb C` results in
`e^(0z)=e^0=1`.