# Physical Interpretation of the Residue

In my complex analysis course (taught by the physics department no less) we've obviously paid close attention to the residue for solving our problems, but there has been no attempt to try and attach any kind of concrete way to think of the residue, a physical description of the residue in a system. Is there a way of thinking of this, beyond just being a mathematical tool? And similarly, what would be the physical equivalent of our contour integrations?

My intuition is telling me that we could think of the residue as some kind of "behavior source" that buried in some kind of phase space that projects it's behavior into the real axis. Our contour integration is then basically just trying calculate the flux of this dominant behavior over our interval of interest (the projection of the real phenomena in phase space). Even as I say this though, it still feels very vague since I don't have a physical model I can compare this against. Several other answers have spoken of the electrostatic potential but I don't recall any ever really describing what the residue means with respect to our (real-valued) function of interest.

• In what context are you talking about the residue here? Honestly, it is a mathematical tool that is used for a variety of different things in physics, and its physical meaning (if any) can be different depending on what it's being used for. – David Z Feb 8 '15 at 8:47
• I'd like any kind of example that allows for an effective (and accurate)physical description of a residue. – Skyler Feb 8 '15 at 8:49
• I hope the Q&A below shed some light on the residue: physics.stackexchange.com/questions/29532/… – noel_lapin Feb 8 '15 at 13:01
• I think this Math.SE will help you out. – DanielSank Sep 11 '16 at 23:13

## 2 Answers

the residue is very similar to div(electric field) , it is like a 1-d counterpart. We know that $\nabla.(r^n)\hat{r}=nr^{n-1}$ at r=0 is 0 for all r$\neq$-2. This is because the $r^{-2}$ 'compensates' the surface $\alpha r^{2}$ and integral is $4\pi$

Similarly when integrating $(z-z_0)^n$, only the $r^-{1}$ term is nonzero and is $2\pi$

You have something similar happening with magnetic fields in situations where the symmetry reduces the problem from 3D to 2D, e.g. an infinitely long straight wire. Then the flux of the magnetic field through any surface pierced by the wire does not depend on the choice of said surface. Moreover, by Stokes theorem, the flux integral becomes a line integral for the curl of the field, and since the wire is a singularity in this case, this is where you have a physical analogue of residues.

This is not the only case where the reduction of the physical dimensions by symmetries has a link with the theory of functions of complex values. Another important example, which has application in electrostatic as well as fluid dynamics, is the theory of harmonic functions. The typical examples is discussed in one of Maxwell's treaties and is an exact solution of the boundary effect of a capacitor made of half-infinite parallel plates.