real world applications of Mathematics which use functions with singularities, not just as a matter of mathematical taste but for conceptual reasons

Question

EDIT [for aptness of this question to this site, read 'real world applications' as 'applications in Physics']

The concept of function (of the form $f : \mathbb{R} \to \mathbb{R}$ ) has been used in various real world applications of Mathematics. In almost all of the applications the general consideration such as approximating a given function with another function is viewed through a lens of mean square error or one of several other such very similar metrics.

My question is concerning the use of functions which have singularities. I've heard from some Physicists that the use of such functions is just a matter of mathematical taste as such functions could be easily approximated with smooth functions and hence the choice of using such functions is just a matter of mathematical taste and it wouldn't make any conceptual difference. I tend not to agree with such an argument because it all depends on the lens of approximation that is being used. If we do not use the metrics like the ones mentioned above then it could be different. A function which contains singularities is fundamentally conceptually different from a smooth function which approximates it no matter how good the approximation is.

So my question is, are there any applications of Mathematics in which functions with singularities are used not just as a matter of mathematical taste but for conceptual reasons ?

Answer 1

[I understand you are looking for more of a pure mathematical take on this, but there are many more examples of these sorts of approximations in applied mathematics (a couple I will document below). Which of course must have a pure mathematical basis as many are to do with the differentiability of the underlying functions.]

Yes there are functions in which singularities are approximated by piece-wise continuous functions. I will assume for the sake of argument that you are not limiting this discussion to relativistic-type "singularities" which are understood as a point at which a given mathematical object is not defined, but also to those of a point of an exceptional set which fail to be well-behaved in some particular way, such as differentiability. In the latter case there are many examples in computational mathematics and mathematical physics where the discontinuity or singularity can cause mathematical instability. An example of this to name one would be the modelling of a shock wave or "jump discontinuity" in computational fluid dynamics. A sudden step-change in any given variable can cause the numerical scheme to become unstable and "blow-up" in this instance the evolution equations are smoothed using artificial dissipation to avoid this behaviour.

In fact there are many examples of such modelling in computational relativistic astrophysics. Not only in the modelling of shock waves etc. but also in the modelling of flow moving across entities such as event horizons when modelling flows from compact objects.

I am sure there are many more instances but these are a few I have thought of.

I hope this is of some use.

Answer 2

One example from physics that springs to mind is the Operator Product Expansion (OPE) particularly used in conformal field theory and string theory. The singular part of the OPE is the most important, while the regular part rarely plays a significant role. Conceptually, the singular part (often called the contraction) measures the non-commutativity of the operators.

Answer 3

I agree the asker's opinion, an equation has its own category for its solution, a unconcerned breaking point or singularity is deadly to ruin the meaning and sometime far away off the real solution.

Answer 4

To simplify the computations in N-body simulations it is used to do $\frac{1}{(d+\epsilon)^2}\:(\epsilon>0)$ instead of $\frac{1}{d^2}$ to avoid collisions. In my opinion the singularity is mainly due to the discretisation method. Other way to deal with the problem is the use of a much shorter time unit (adaptative) in the region where stars get closer. In the real world, AFAIK, there are no reports of star collisions, nor when galaxies are merging neither when they are heavily packed as it happens with clusters.

Answer 5

There are functions with singularities that arise all the time in physics. For example, an undamped (no friction) harmonic oscillator of mass $m$ driven by $F(t)=F_0\sin \omega t$ will have infinite amplitude at the resonance frequency, for the amplitude is given by $$ A(\omega)=\frac{F_0/m}{\omega^2-\omega_0^2}, $$ where $\omega_0=\sqrt{k/m}$ is the resonance frequency. Therefore, $\omega=\omega_0$ is a singularity of $A(\omega)$.

It's true that there are no undamped oscillators in nature, and the above equation is modified: $$ A(\omega)=\frac{F_0/m}{\sqrt{(\omega^2-\omega_0^2)^2+\left(\frac{b\omega}{m}\right)^2}}, $$ where $b$ is a constant that comes in with the friction term.

However, when, for example, mechanics build suspension systems for cars, they make sure that the typical frequency of oscillation of the system is far away from the resonance frequency, so that the car does not start oscillating violently when the driver goes over a bump.

Another example (this involves no idealization) comes from quantum field theory, and it is that physical masses of particles appear as poles of the Fourier-transformed two-point function of the field corresponding to the particle. So there is a function there that has a pole (singularity) when its variable becomes equal to the mass of the physical particle.

Answer 6

I can recall solving -1D differential equations in the 70's using the Galerkin method. If solutions that have singularities are present it is possible to incorporate the form of the singularity into one or more basis functions. Of course the weight function has to be choosen carefully, so that for instance all neccessary integrals are finite (usually the intgral of two basis functions times the weight function). As long as the positions and form of the singularities are known beforehand this seemed to work well.