Each of the formulae we found theoretically has been satisfactorily tested by means of Mathematica Ratio of a linear combination of complete elliptic integrals of the first and second kinds to the Appell hyperge- ometric function of two complex-conjugate arguments. Furthermore, starting with a certain binomial integral, we succeed in providing Founding upon a special hyperelliptic reduction formula due to Hermite,, π is obtained as a ratio of a complete elliptic integral and the four-variable Lauricella function. In any case we have focused on integrand functions having at least one couple of complex-conjugate roots.

Hypergeometric functions like those of Gauss, Appell or Lauricella. This book offers teachers and students an opportunity to learn about complex numbers in a state-of-the-art computational environment. All of them have been derived by tackling some elliptic or hyperelliptic known integral, and looking for another representation of it by means of Complex Analysis with Mathematica offers a way of learning and teaching a subject that lies at the heart of many areas of pure and applied mathematics, physics, engineering and even art. Of the Gauss hypergeometric function 2F1 in the analytic continuation. In this article we derive some new identities concerning π, algebraic radicals and some special occurrences

Mathematica complex conjugate