Recursive computation of derivatives of elliptic functions and of incomplete elliptic integrals

摘要

Presented are the recurrence formulas to compute the derivatives of a general elliptic function, Weierstrass’s ℘ function, the Jacobian elliptic functions, and the incomplete elliptic integrals in the forms of Jacobi and Legendre with respect to the argument or the amplitude. The double precision computation by the formulas is correct with 15 digits or so for the first 10 orders of differentiation at least.