For a computational model of a physics problem, I have numerical values of a complex function from physics over a 2D grid (for example: $f(x,y)= z \in \mathbb{C}$). I numerically calculate the winding number $W$ of a closed contour $C$ in this 2D grid, where $W=\oint_C d\arg(z)$. While I am able to calculate this $W$ numerically, I am now trying to understand why it takes the values it does. So, I visualized $Re(z)$, $Im(z)$, $\arg(z)$ and $d\arg(z)$ (pictured below) and noticed that several lines of discontinuity arise from the same point. $\arg(z)$ seems to jump between $+\pi,-\pi$ and $0$, while $d\arg(z)$ jumps between $\pm 2\pi,+\pi$ and $0$.
To my understanding, roughly speaking, a zero is when $f(x,y)= 0$, a pole is when $f(x,y)\approx \infty$, and an essential singularity is both a zero and a pole. I have included quick definitions of branch point and branch cut below the pictures.
Now, my questions:
- Am I able to learn more about how $z$ influences $W$ based off these pictures? Besides the fact that $W$ counts the number of poles (and zeros?) of $z$.
- Is the origin an essential singularity because the peak diverges through positive and negative values? Otherwise, how can I visually identify an essential singularity?
- If there aren't zeros or poles of $z$ inside the loop, by the argument principle, I expect $W=0$, but it doesn't seem to be the case (and I know with certainty that my code is correct). Is there any other function of $z$ I can plot to unveil features that could possibly explain this?
- From these definitions, it seems as if the origin here is a branch point or pole. Is it either, neither or just one?
- Are those lines of discontinuity branch cuts? Is it just a matter of specifying an $(x,y)$ domain that excludes these cuts?
- Is a numerical integration of a contour around the origin valid despite these jumps? I am guessing it is because the Cauchy integral formula for $W$ should hold.
- Should I instead be thinking about Riemann surfaces and ramification loci/indices?
I posted all these tiny questions together because they deal with closely-related concepts in a specific example. I apologize if this should have instead gone on Math Overflow (or elsewhere).
According to Wikipedia:
Roughly speaking, branch points are the points where the various sheets of a multiple valued function come together. The branches of the function are the various sheets of the function. ... A branch cut is a curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued function on the plane minus that curve. Branch cuts are usually, but not always, taken between pairs of branch points.
[A (logarithmic) branch point is a point (usually a pole) around which a function does not return to its initial value after traversing a closed loop.]
[A branch cut can be thought of infinitely many branch points with infinitely many residues]