The Weierstrass elliptic functions (or Weierstrass ℘-functions, voiced "p-functions") are elliptic functions which, unlike the Jacobi elliptic functions, have a second-order pole at z = 0. To specify ℘(z) completely, its half-periods (ω_1 and ω_2) or elliptic invariants (g_2 and g_3) must be specified. These two cases are denoted ℘(z|ω_1, ω_2) and ℘(z;g_2, g_3), respectively.
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