# How to visually identify poles, zeros, singularities and branch cuts/points of a complex vector field?

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:

1. 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 $$z$$.
2. Is the origin an essential singularity because the peak diverges through positive and negative values? Otherwise, how can I visually identify an essential singularity?
3. 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?
4. From these definitions, it seems as if the origin here is a branch point or pole. Is it either, neither or just one?
5. Are those lines of discontinuity branch cuts? Is it just a matter of specifying an $$(x,y)$$ domain that excludes these cuts?
6. 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.
7. 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]

"To my understanding, roughly speaking, a zero is when f(x,y)=0, a pole is when f(x,y)≈∞, and an essential singularity is both a zero and a pole". Not quite. A pole is an isolated singularity where $$f(x,y) \to \infty$$ as $$(x,y)$$ approaches your point. A removable singularity is an isolated singularity where $$f(x,y)$$ approaches a finite limit. An essential singularity is an isolated singularity that is neither a pole nor an isolated singularity. It turns out that in a neighbourhood of an essential singularity, the functions takes on all complex numbers as values with at most one exception (this is Picard's "great" theorem).