Weierstrass elliptic function
/ VY-er-shtrahss /
If you want the cleanest possible doubly periodic function — one that repeats over a whole lattice of parallelograms in the complex plane — the Weierstrass elliptic function is the canonical answer. Where the Jacobi functions descend from real arc-length problems, the Weierstrass P-function is built directly from a lattice and is the natural building block for all elliptic functions.
It is defined as a lattice sum: P(z) = 1 over z^2 plus the sum over all nonzero lattice points omega of ( 1 over (z minus omega)^2 minus 1 over omega^2 ). The subtraction inside the sum is precisely what makes the series converge. P(z) is an even function with a double pole at every lattice point, and it is doubly periodic — P(z plus omega) = P(z) for every lattice vector. Its master property is the differential equation it satisfies: (P prime)^2 = 4 P^3 minus g_2 P minus g_3, where the lattice invariants g_2 and g_3 are fixed numbers. This algebraic relation is exactly the equation of an elliptic curve, which is why P parametrises elliptic curves and ties this corner of analysis to number theory.
The Weierstrass function is the standard tool for integrating any cubic or quartic under a square root in its most symmetric form, for the exact solution of certain spinning-top and geodesic problems, and as the foundation of the theory of elliptic curves used throughout modern number theory and cryptography. Jacobi's sn, cn, dn and Weierstrass's P describe the same world of doubly periodic functions from two complementary viewpoints, and explicit formulas convert between them.
Inverting the algebraic relation, the integral integral of dx over square root of (4 x^3 minus g_2 x minus g_3) is solved by x = P(z): the Weierstrass function is the inverse of that cubic-square-root integral, just as sin inverts the integral with a quadratic square root.
P turns the cubic-under-a-square-root integral into a clean inverse, the elliptic analogue of arcsine.
P is doubly periodic on a lattice, not singly periodic like sine; a non-constant doubly periodic analytic function must have poles (Liouville), so P necessarily has its double poles at the lattice points.