Flux in a circular circuit

The flux density, H, at the center of a circular circuit of one turn is known accurately to be 2 π i/r; in fact, it has been made the quantitative connecting link between current and magnetism and the basis of the ampere. But the H in any other part of the circle is very difficult to calculate as it involves elliptical integrals, and to integrate these for getting the total flux is almost hopelessly complicated and then at best only approximate.