Where is the error in this argument

[Bachelier]

It is possible to do this and it is correct:

$$\log \left[\sqrt{i}\right] = \log\left\{\exp\left[\frac{i}{2}\left(\frac{\pi}{2}+2\pi n\right)\right]\right\} = \frac{i}{2}\left(\frac{\pi}{2} + 2\pi n\right) = i\left(\frac{\pi}{4} + \pi n\right)$$

But:

$$\log \left[i^2 \right] = \log\left\{\exp\left[2i \left(\frac{\pi}{2}+2\pi n\right)\right]\right\} = 2i \left(\frac{\pi}{2} + 2\pi n\right) = i\left(\pi + 4\pi n\right)$$

yet

$\log \left[i^2 \right] = \log \left[-1 \right] = i\left(\pi + 2\pi n\right) \ for \ k \in \mathbb{Z}$ which is the correct argument.

 

[Dick]

It is possible to do this and it is correct:

$$\log \left[\sqrt{i}\right] = \log\left\{\exp\left[\frac{i}{2}\left(\frac{\pi}{2}+2\pi n\right)\right]\right\} = \frac{i}{2}\left(\frac{\pi}{2} + 2\pi n\right) = i\left(\frac{\pi}{4} + \pi n\right)$$

But:

$$\log \left[i^2 \right] = \log\left\{\exp\left[2i \left(\frac{\pi}{2}+2\pi n\right)\right]\right\} = 2i \left(\frac{\pi}{2} + 2\pi n\right) = i\left(\pi + 4\pi n\right)$$

yet

$\log \left[i^2 \right] = \log \left[-1 \right] = i\left(\pi + 2\pi n\right) \ for \ k \in \mathbb{Z}$ which is the correct argument.

I'm not sure what your argument really is. Sure (sqrt(i))^4=(i)^2=(-1). (sqrt(-i))^4 is also (-1). So?