# Complex numbers

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

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

HallsofIvy
Homework Helper
1- is there any complex number, x ,such that x^x=i?
Yes, but finding it is non-trivial, involving, I think, the Lambert W function.

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

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 $(-1)^{\sqrt{2}}= (e^{i\pi})^{\sqrt{2}}= e^{i\pi\sqrt{2}}= cos(\pi\sqrt{2})+ i sin(\pi\sqrt{2})$

Thanks.

haruspex