Question

All the values of the multivalued complex function
\[\Large 1^{i}\] are--
(1)purely imaginary (2)real and non negetive
(3)on the unit circle (4)equal in real and imaginary parts

Answer 1

i'm going for option B
anyone agrees with me?

Answer 2

I would agree, since \(1^i=1\).

Answer 3

yeah thanks
\[1=e^{i(2n \pi)}\rightarrow 1^i=e^{-2n \pi}\]
right ?

Answer 4

Well if you look at this: \[\LARGE e^{i \theta} = \cos \theta + i \sin \theta\] We can quickly see that if we plug in theta = 0 we get: \[\LARGE e^{i 0} = \cos 0+ i \sin 0\]\[\LARGE (e^0)^i = 1+i0\]\[\LARGE 1^i=1\]

It seems to me that both 2 and 3 are correct since the value 1 lies on the unit circle and is both real and nonnegative.