When we dive deep into physics—whether it's quantum mechanics, classical field theory, or statistical mechanics—we often find the same set of functions showing up again and again. These aren’t just mathematical coincidences. They're special functions that naturally arise as solutions to the kinds of differential equations that describe physical systems. Their recurrence isn’t accidental; it reflects the deep symmetry and structure of the physical laws themselves.
This page brings together a set of such functions—orthogonal polynomials like Legendre, Hermite, and Laguerre; Bessel and spherical Bessel functions; and others like the Gamma, Beta, Airy, and error functions. You’ll find a quick overview of each one: what the function looks like mathematically, where it tends to show up in physics, and what makes it useful. The idea isn’t to be exhaustive, but to serve as a compact reference or a quick refresher when you're working through problems or revisiting theory.
If you're a student working through a course, or someone trying to get a feel for how all these mathematical tools connect across different areas of physics, this gallery is meant for you.