Set of all values of sqrt(i)

1. Feb 15, 2013

Bachelier

Why is it that set of all values of $log(√i)$ is $(\frac{1}{4} + n)\pi i$ and not $(\frac{1}{4} + 2n)\pi i$ since $√i = e^\frac{ \pi i}{4}$

and $log(z) = ln|z|+ i(Arg(z) + 2 n \pi)$

2. Feb 16, 2013

antibrane

Looking at it, I get:
$$\log \left[\sqrt{i}\right] = \log\left\{\exp\left[\frac{i}{2}\left(\frac{\pi}{2}+2\pi n\right)\right]\right\} = \frac{i}{2}\left(\frac{\pi}{2} + 2\pi n\right)\log e = i\left(\frac{\pi}{4} + \pi n\right)$$
where $n$ is an integer. It looks like you might have used a period of $\pi$ rather than $2\pi$.

3. Feb 16, 2013

jbunniii

If we exponentiate $(\frac{1}{4} + n)\pi i$ and then square it, we get
$$\left(\exp\left(\left(\frac{1}{4} + n\right)\pi i\right)\right)^2 = \left(\exp\left(\frac{i\pi}{4}\right)\exp\left(in \pi \right)\right)^2 = \exp\left(\frac{i\pi}{2}\right)\exp\left(i2n \pi\right) = i$$
so every number of the form $(\frac{1}{4} + n)\pi i$ is a solution. Your proposed answer misses all of the solutions for which $n$ is odd.

4. Feb 16, 2013

haruspex

True, but if √ is taken to denote the principal square root (which would be standard, yes?) then only the 2n solutions arise.

5. Feb 16, 2013

jbunniii

Yes, if the principal square root is intended. The "set of all values of sqrt(i)" in the subject led me to presume otherwise, but that may not have been warranted. Bachelier, what do you intend √ to mean in this context?