The Sine Function. Calm and dignified, it sits among the royal court of the Elementary Functions, presiding with undulating grace over the trigonometric functions, partnered with the Exponential Function, and showing forth his power over the realms of physics and mathematics.
On one of the less subtle TV networks, this installment might be called When Good Functions Go Bad.
The sine function is ubiquitous in physics. Figuring out vector components, solving differential equations in E&M and quantum mechanics, decomposing Fourier series, you name it. It’s about as well-behaved as functions get. At least along the real line, it’s periodic, finite, bounded, continuous, and continuously differentiable with derivatives of all orders. It’s hard to ask for a cleaner function. But just for fun, let’s replace the argument x with 1/x and see what happens.
It’s continuous and differentiable everywhere except the origin. Discontinuities are not exactly rare in the world of functions, but this one is of a type not often seen in physics. It’s not a point discontinuity, or an asymptote, or a jump discontinuity. Instead, the function oscillates infinitely often near the origin and so the limit completely fails to exist from either side.
When these types of functions turn up in physics, we tend to wave our hands and declare them “unphysical”. As indeed they are. What this means is usually that whatever theory has resulted in something like this is simply not valid near the point of discontinuity and needs to be extended.
Probably the most famous of this type of scenario was the ultraviolet catastrophe. Back in the early 20th century, there was a theory due to Rayleigh and Jeans that described the blackbody spectrum of objects very well for long wavelengths. But as wavelength approached 0, the emitted intensity approached infinity. Obviously this was impossible, and the theory which ended up describing the situation correctly was quantum mechanics.
Nonphysical equations popping up in physical theories are quite common even today, and remain both aggravating problems and tremendous opportunities to extend our understanding with better theories.