Complex numbers

limitkiller

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

2- (-1)^([itex]\sqrt{2}[/itex])=?

HallsofIvy

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

We can write -1 in "polar form" as [itex]e^{i\pi}[/itex] and then [itex](-1)^{\sqrt{2}}= (e^{i\pi})^{\sqrt{2}}= e^{i\pi\sqrt{2}}= cos(\pi\sqrt{2})+ i sin(\pi\sqrt{2})[/itex]
or about .99+ .077i.

limitkiller

Thanks.

haruspex

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