the Weierstrass P-function
/ VY-er-shtrahss /
The Weierstrass P-function (written with the special script symbol, often spelled 'wp') is the fundamental example of an elliptic function: a function on the complex plane that is meromorphic and DOUBLY periodic. Doubly periodic means it repeats in two independent directions at once. Fix two complex numbers omega_1 and omega_2 that do not point the same way; they generate a lattice, the grid of all points m omega_1 + n omega_2 for integers m, n. The P-function satisfies P(z + omega_1) = P(z) and P(z + omega_2) = P(z), so its entire behaviour is determined by what it does on one fundamental parallelogram — like a wallpaper pattern that tiles the plane.
Its explicit form is a sum designed to be periodic and to have a double pole at every lattice point: P(z) = 1/z^2 + sum over non-zero lattice points w of (1/(z - w)^2 - 1/w^2). The leading term 1/z^2 gives a double pole at the origin; periodicity forces a matching double pole at every other lattice point; and the correction -1/w^2 inside the sum is exactly what is needed to make the series converge. There are no other singularities and no zeros forced by the construction — it is the simplest non-constant elliptic function, since a theorem (Liouville again) shows a holomorphic doubly periodic function must be constant, so any genuine elliptic function MUST have poles.
The reason it sits in this field of triumphs is that it ties complex analysis to algebra and geometry. The P-function and its derivative satisfy a single algebraic relation, P'(z)^2 = 4 P(z)^3 - g_2 P(z) - g_3, where g_2 and g_3 are constants built from the lattice. That equation is the equation of an elliptic curve, and z -> (P(z), P'(z)) parametrises the curve by the complex plane modulo the lattice — a torus. So elliptic functions are the analytic uniformisation of elliptic curves, and they connect onward to theta functions, modular forms, and (through L-functions) back to number theory. A caution: 'elliptic function' has nothing to do with ellipses directly; the name is inherited from the elliptic integrals that arise in computing the arc length of an ellipse, which these functions invert.
Take the square lattice generated by omega_1 = 1 and omega_2 = i. The fundamental cell is the unit square, and P(z) has a double pole at each corner point m + n i. As z circles once around the origin, P(z) blows up like 1/z^2, then settles back to a finite value as z moves toward the centre of the cell. Translating z by 1 or by i leaves the value unchanged — the function literally tiles the plane with the pattern on the unit square.
A doubly periodic function tiles the plane from one parallelogram; P(z) has a double pole at every lattice point and obeys P'^2 = 4P^3 - g_2 P - g_3.
A non-constant elliptic function must have poles — Liouville's theorem rules out a bounded entire one. So you cannot build an interesting doubly periodic function with no singularities; the P-function's double poles at the lattice points are unavoidable, not a defect.