Hyperelliptic integrals and special functions for the spatial variational problem

The study of the properties of special functions plays an important role in solving many problems in geometric function theory. We study the properties of hyperelliptic integrals and special functions, which definition includes a parameter that depends on the dimension of the space. The appearance of these functions is associated with the solution of a specific variational problem of finding in $n$-dimensional Euclidean space a surface that has the smallest area in a given metric among the hypersurfaces formed by rotation around the polar axis of a plane curve connecting two fixed points in the upper half-plane.