Weierstrass's Elliptic Functions - The Case With 1 A Basic Half-period

The Case With 1 A Basic Half-period

If, much of the above theory becomes simpler; it is then conventional to write for . For a fixed τ in the upper half-plane, so that the imaginary part of τ is positive, we define the Weierstrass ℘ function by

The sum extends over the lattice {n+mτ : n and m in Z} with the origin omitted. Here we regard τ as fixed and ℘ as a function of z; fixing z and letting τ vary leads into the area of elliptic modular functions.