Elliptic Integrals and Elliptic Functions in Calculus Beyond (UMAP)

This module demonstrates the use of the main concepts and theorems from calculus in the solution of real problems, here the computation of the arc length of an ellipse and the swinging time of a pendulum.

Table of Contents:

INTRODCUTION

PHYSICAL ORIGINS OF ELLIPTIC INTEGRALS
Elliptic Integral of the Second Kind for Ellipses
Elliptic Integrals of the First Kind for the Circular Pendulum

ELLIPTIC INTEGRALS OF THE FIRST AND SECOND KINDS
Definition and Features of Elliptic Integrals of the First Kind
Definition and Features of Elliptic Integrals of the Second Kind

THE ARITHMETIC-GEOMETRIC MEAN ALGORITHM

LANDEN'S TRANSFORMATION
Landen's Transformation on [0, Pi/2]
Landen's Transformation on R

ALGORITHM TO COMPUTE ELLIPTIC INTEGRALS OF THE FIRST KIND

ALGORITHM TO COMPUTE ELLIPTIC INTEGRALS OF THE SECOND KIND

ALGORITHM TO COMPUTE JACOBI'S ELLIPTIC FUNCTIONS

RATE OF CONVERGENCE FOR NUMERICAL COMPUTATIONS

CONCLUSION

SOLUTIONS TO THE EXERCISES