How can I implement this function, logarithm (natural base) of $z$? I need either analytical or numerical approach. A textbook explanation would come in handy. I have no clue how this integral is supposed to be computed. The problem lies in the fact that the integral is from $1$ to (complex) $z$.
$$\ln(z) = \int_1^z \frac{1}{t} dt $$
The definition comes from: https://dlmf.nist.gov/4.2
Split the path from $1+0i$ to $z$ into a path along the real axis from $1$ to $|z|$ and a path along the circle or radius $|z|$. Using $z=re^{i\theta}$, the first integral will be $$\int_1^r\frac 1tdt=\ln r$$ Let $\gamma(p)$ be the parametrized curve in the complex plane where you want to calculate an integral, starting from $\gamma(a)$ to some $\gamma(b)$. Then $$\int_\gamma f(z)dz=\int_a^b f(\gamma(p))\gamma'(p) dp$$ In this case $$\gamma(p)=re^{ip}$$with $p$ varying from $0$ to $\theta$. Then $$\gamma'(p)=ire^{ip}$$and $$\int_0^\theta\frac1{re^{ip}}ire^{ip}dp=i\int_0^\theta dp=i\theta$$ Putting the two integrals together: $$\int_1^z\frac1tdt=\ln r+i\theta$$ Note that you can split any path into segments parallel to the radial directions, and constant radii arcs, and when you sum those up you will get the same answer, independent of the path. I suggest you try the simple case, moving along the unit circle from $0$ to $\theta$, then calculate the line integral along the radial direction from $e^{i\theta}$ to $re^{i\theta}$.