Elliptic Integrals of the First and Second Kind
- Ryan Hirst
Proposition: Develop a set of formulas for efficient calculation of the Complete and Incomplete Elliptic Integrals of the First and Second kind. Briefly, we take the following approach:
- Begin with the Integral of the First Kind. - From Gauss, we have a method for transforming the Complete integral to a more rapidly convergent integral in the same form. - Generalize this method to the Incomplete integral - We may iterate as many times as we please.
- We now address the integral of the second kind - From Legendre and Cayley, we have a set of equations relating the Complete and Incomplete integrals of the First and Second kind. - Introduce the iterated integral into these equations, and carry out the indicated operations.
- Develop a reverse integral - By bringing both parameters (k, sinθ) near 1, we can make the forward integral nearly as inconvenient as we please. We expend an ever-increasing number of calculations on a vanishing arc. Hence, we - For the Complete integrals, we begin with Cayley's formula for k' very small, and apply steps 1 and 2 above. - For the Incomplete integrals, we rewrite the integrals centered at the other end of the curve (θ = π /2), and choose a substitution of variables which allows the new equation to be expanded and then integrated. We then repeat steps 1 and 2. - the result is a set of integrals which converge more and more rapidly as the two parameters approach 1.