On-axis field approximations for a (semi-)spheroid in a uniform field

I have recently been analyzing various problems in the area of breakdown of solids and gases using a highly prolate, conducting spheroid in a uniform electric field as a model for a breakdown channel. The electric field distribution for a prolate spheroid in a uniform field can be solved in spheroidal hyperbolic coordinates. The solution is exact for a conducting prolate spheroid in an infinite, uniform field or a semiprolate conducting spheroid on the surface of a plane electrode with the opposite electrode infinitely far away. In working with these equations, I have developed several approximations, including a very simple equation for the field at the tip of the spheroid and an equation for the field on axis of the spheroid, which may be useful to others.