Question

Find all the values of:

a. (-i)^i

b. (-1)^i

please help we with this complex variable problem.

Answer 1

Answer with Step-by-step explanation:

Part 1)

we know that

`e^(i\theta )=cos(\theta )+isin(\theta )`

thus
`-i=e^{(-i* (4n-1)\pi )/(2)}`

thus
`(-i)^i=(e^{(-i* (4n-1)\pi )/(2)})^i\\\\(-i)^i=e^{(-i^2* (4n-1)\pi )/(2)}=e^{((4n-1)\pi )/(2)}\\\\\therefore (-i)^i=e^{((4n-1)\pi )/(2)}` where 'n' is any integer

Part 2)

We have
`-1=e^((2n+1)\pi )\\\\\therefore (-1)^(i)=(e^(i(2n+1)\pi ))^(i)\\\\(-1)^i=(e^(i^2(2n+1)\pi ))\\\\(-1)^i=e^(-(2n+1)\pi )` where 'n' is any integer