My book asks to prove:
$\text{Ln}[i \cdot (-1+i)]$ does not equal to $\text{Ln}(i) + \text{Ln}(-1+i)$ where $\text{Ln}$ gives the principal log of the complex number.
I don't see why this is true though. Both sides seem to evaluate to the same value and I don't understand why they aren't equal. Could someone please give me a hand?
If you're using the principal branch, the imaginary part of Ln is always in the interval $(-\pi, \pi]$. So $\text{Ln}(i) = i \pi/2$ since $\exp(i \pi/2) = i$ and $\pi/2$ is in the interval. Similarly, $\text{Ln}(-1+i) = \ln(2)/2 + 3 \pi i/4$. If you add these, you get $\ln(2)/2 + 5 \pi i/4$. But this can't be a value of Ln because $5 \pi/4$ is outside the interval.