What is the polar form of a complex number raised to a fractional power?

[shemer77]

Homework Statement

I have some trouble understanding some of my math homework maybe somebody can help me out?

1) find all the values of
ln(e)
(-1)^i

I know that i am going to have to use eulers formula in some way i believe but I am not really sure what the question is asking, what does it mean all the values?

2)let z= x+iy, where both x and y are real. find the real and imaginary parts for e^(1/z).
i figure i would have e^(1/x+iy) but where do i go from there?

3) rewrite in the polar form
(√(z))^(1/n), now this i don't really understand. I can really easily change something like 4 +6i to polar form but this i don't get.

 

[genericusrnme]

When things say all values they generally mean the 'general solution'
Like $ArcCos(1)=2n \pi$

For part two, you could try multiplying 1/z by 1 in such a way that you end up with a real denominator (you should know how to do this)

For part three you just need to know about how exponents work, what happens when you take the square root of a^x, what happens when you take the nth power of a^x?

 

[shemer77]

still don't really understand part 1,

thanks i got part 2, was just a stupid mistake on my part
and for part 3 i know how exponents work in the sense that i am taking the nth root of z but how does one convert that to a polar form.

 

[A. Bahat]

The exponential function is periodic, with period $2\pi i$ (that is, $e^z=e^{z+2\pi i}$ for every $z\in\mathbb{C}$), and the logarithm is defined as the inverse of the exponential. Since the exponential is not one-to-one, the logarithm of a number is not uniquely determined; it's not an actual function on $\mathbb{C}$. The best you can do to find $\log(z)$ is to give a set of complex numbers (differing by integer multiples of $2\pi i$) whose exponential is $z$.

Likewise, since $a^z$ is defined as $e^{z\log(a)}$, it is also multivalued.