[Math] Real world use of even and odd functions

Question

What is the real world use of knowing whether a function is odd or even? Any practical examples?

For example, quadratic equations, differential equations and calculus in General is used for, among other things, determining motion of bodies. Just knowing that a function is symmetrical or not, help how? In terms of solving practical problems with the knowledge.

Answer 1

For a more physics-y answer, you can turn to quantum mechanics. For example,

• If a wavefunction of a particle is an even function (symmetric, centered at the origin), then the expected location of the particle (assuming it exists) is the origin, because $|\psi(x)|^2$ is even, so $x |\psi(x)|^2$ is odd, so $\mathbb E (x) \equiv \int x|\psi(x)| \mathrm d x=0$. More generally, the odd moments of an even probability distribution vanish.
• A wavefunction has definite parity if it is odd or even. (It is an eigenvector with eigenvalue $\pm 1$ of the parity operator $\mathrm P$.) One can show that generically, for a symmetric (even) potential, the energy eigenstates have parity even, odd, even, odd, ... This leads to simple numerical strategies for trying to calculate the lowest two eigenstates by restricting to even (for ground state) and odd (for lowest excited state) functions.

There are plenty of other generic properties, like

• Odd derivatives of even functions and even derivatives of odd functions vanish, so ...
• ... analytic even functions have power series containing only even powers of $x$...
• ... and in fact also Fourier series containing only $\cos$ terms.