i to the i power

Sidebar: What is ln(-1)?

Euler's Equation states that

eiπ + 1 = 0

Thus,

eiπ = -1

So

iπ = ln(-1)

Question

What is ii, expressed as a power of e?

Answer

Since we want to express it as a power of e, we can start by saying that

ex = ii

for some x. So

x = ln(ii)

x = i ln(i)

Now, i is the square root of -1, so

x = i ln(-11/2)

x = (i/2) ln(-1)

But as demonstrated in the sidebar,

ln(-1) = iπ

So

x = (i/2) · iπ

x = (i · i · π) /2

but i · i is -1, so

x = -π/2

so

ii = e-π/2

(Note that this is a real number! It's about 0.208.)

Thanks to Peter Hartman for showing me this elegant derivation.