
Elliptic Integrals and Elliptic Functions in Calculus Beyond (UMAP)
Yves Nievergelt, Jacqueline Coomes

Mathematics Topic: Calculus 
Application Areas: Engineering and physics 
Prerequisites: For elliptic integrals of the first kind: the concepts of limit, derivative, and integral; mathematical induction; Rolle's Theorem, the MeanValue Theorem for Derivatives, the Fundamental Theorem of Calculus, the InverseFunction Theorem with one variable; algebraic and trigonometric changes of variables in integrals. For elliptic integrals of the second kind: also infinite series. 

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 ARITHMETICGEOMETRIC 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
