We prove that the quotient of abelian integrals associated to an elliptic surface is bounded and strictly increasing by first determining the Picard-Fuchs equation satisfied by the a,belian integrals and the Riccati equation satisfied by the quotient of the abelian integrals.

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 differentiat...

7 Quadratically converging approximations to π 27 7.1 Some representative elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 7.1.1 The elliptic integral of the first kind . . . . . . . . . . . . . . . . . . . . . . . 27 7.1.2 The elliptic integral of the second kind . . . . . . . . . . . . . . . . . . . . . . 30 7.1.3 More representations for K(k) and E(k) . . . . . . ....