Complex numbers

1. Aug 9, 2012

limitkiller

1- is there any complex number, x ,such that x^x=i?

2- (-1)^($\sqrt{2}$)=?

2. Aug 9, 2012

HallsofIvy

Yes, but finding it is non-trivial, involving, I think, the Lambert W function.

We can write -1 in "polar form" as $e^{i\pi}$ and then $(-1)^{\sqrt{2}}= (e^{i\pi})^{\sqrt{2}}= e^{i\pi\sqrt{2}}= cos(\pi\sqrt{2})+ i sin(\pi\sqrt{2})$

3. Aug 9, 2012

limitkiller

Thanks.

4. Aug 15, 2012

haruspex

Writing z = re, zz = i gives θ sec(θ) eθ tan(θ) = π/2 + 2πn and r = eθ tan(θ). For n = 0, θ has a solution in (π/6, π/4), and probably infinitely many for each n.

Last edited: Aug 15, 2012