I'm currently in a complex analysis course and on one of our problem sets we were asked to find all values of $\log{\sqrt i}$. The complex logarithm, $\log$, was defined in class as: $\log(z) = \textrm{Log} |z| +i*\textrm{Arg}(z) + 2\pi in$, where $\textrm{Log}$ is the real logarithm and $n \in \mathbb Z$.
I got that the final answer was $-\textrm{Log}(2)+i\left(\frac {\pi}{4} + 4 \pi n\right)$.
The TA for the class took off points for simplification errors and told me to look at the solution guide (there is no solution guide). I tried plugging into Wolfram Alpha but Wolfram only gives the single value for the principle branch, but we need the general solution.
After looking back over my work I can't seem to find where I made an error so I was hoping someone could catch my mistake. I attached a picture of my work below. Any clarification would be much appreciated so I don't make the same mistake on my midterm! Thanks!
I see several errors.
In the middle of the page you didn't distribute the $1/2$: $$\qquad e^{1/2(i\pi/2+2\pi ni)}\\\overset?=e^{i\pi/4+2\pi ni}$$
In the blue part you miscalculated the absolute value: $$\sqrt{\left(\frac{\sqrt2}2\right)^2+\left(\frac{\sqrt2}2\right)^2}\overset?=\frac12$$
Below the blue part you combined $i2\pi n+i2\pi n=i4\pi n$. The problem is that these were two arbitrary integers, not necessarily the same. It should be something like $i2\pi n_1+i2\pi n_2=i2\pi(n_1+n_2)=i2\pi n_3$, where $n_3\in\mathbb Z$ may or may not be even. (But this is still incorrect because it's based on the 1st error.)