Complex numbers powers and logs

[Liquidxlax]

Homework Statement

(-e)^iπ answer is -e^-π²

not sure how to describe this one, but i need to find the roots.

Homework Equations

(r^n)e^(itheta)n = (r^n)cos(thetan) + isin(thetan) n is an element of the reals

The Attempt at a Solution

i'm not sure what to do with this, it is the most weird question on the page. it seems circular to me.

 

[CalcYouLater]

I'm not sure exactly what you are asking here. Here is my attempt at some help.

When dealing with complex numbers, we can use De Moivre's formula to find roots.

$$\alpha={r}{e}^{i\theta}$$

$$\alpha^{\frac{p}{q}}={r}^{\frac{p}{q}}{exp(ip({\theta}+2n{\pi})/q)}$$

Example: Find the roots of

$$i^{\frac{1}{3}}$$

Using:

$$i=e^{\frac{i{\pi}}{2}}$$

And the above equation gives the roots as:

$$e^{\frac{i{\pi}}{6}}$$

$$e^{\frac{5i{\pi}}{6}}$$

$$e^{\frac{9i{\pi}}{6}}$$

I hope that helps.

 

[Liquidxlax]

not really because applying that i still don't get the right answer. I've done many of these types of questions, its just that this one is really weird.

like if i ln and then put it to the e, it is the same thing...

 

[CalcYouLater]

You want to find the roots of a function, right? For which function do you want to find the roots?

 

[Liquidxlax]

You want to find the roots of a function, right? For which function do you want to find the roots?

-e^ipi i know the answer is -e^(-ipi^2)