I am going through the Mathematics of the DFT book by Julius O. Smith III.
One of the questions ask the following:
How would you convert the complex number $\ln(i)$ into the form $x + yi$, where $x$ is the real part and $y$ is the imaginary part?
I am stumped.
Best Answer
Remember that any non-zero complex number $z = x+iy$ can be written in the form $z = r e^{i\theta}$, where $r = |z|$ and $\theta$ is the "argument" of $z$, i.e. the angle between $z$ and the positive real axis when we identify $\mathbb C$ with $\mathbb R^2$. Since $e^{i\theta} = \cos\theta + i\sin\theta$, this is just writing $z$ in polar coordinates. $\theta$ is a priori defined only up to a multiple of $2\pi$, but we can get a unique value of $\theta$ by requiring $\theta\in [0,2\pi)$.
Now, we can also write $re^{i\theta} = e^{i\theta + \ln r}$. So, it is natural to define the logarithm of $z$ as $\ln z = i\theta + \ln r$, so that it cancels the exponential.
In the case $z= i$, we have $r = |i| = 1$ and $\theta = \pi/2$, since $i$ is on the positive imaginary axis, which makes a right angle with the positive real axis. Therefore $\ln(i) = i\pi/2 + \ln 1 = i\pi/2$.