Elliptic integrals and the Schwarz-Christoffel transformation

The real elliptic integrals of the first and second kind in Jacobiʹs normal form are computed efficiently, using the convolution number in conjunction with the method of Frobenius. For this purpose certain treatments of the Laurent series are included. Different regions of convergence on the real axis are determined, and for each one a different series is developed. The real elliptic integral of the third kind is solved within a limited parameter plane by the same method. The integral of the Schwarz-Christoffel transformation is solved in the complex variable by complex convolution number algebra, using the unit disk as mapping region. Different regions of convergence of Frobenius, Laurent, and Taylor series are determined to cover the whole disk. The complex evaluation of the elliptic integral of the third kind is included. A Schwarz-Christoffel formula for an infinite periodic mapping is given. The solutions for exterior, interior, periodic, and cyclic polygons are separately treated. Examples of several polygon mappings are presented graphically, and compared with previous numerically integrated solutions. The parameter problem is solved by the Newton-Raphson method, using a quotient matrix as approximation for the Jacobian matrix. The coordinate relations are simplified by using an overdetermined system. An exact analytical Jacobian matrix is computed, solving Leibnizʹ derivative of the Schwarz-Christoffel integral, and results are compared with the approximate quotient matrix method.