generally the function having zero points or poles with non-integer order such as f(z) = (z-a)^(1.5+i0.3) must be dealt with on appropriate Riemann surface. In the following link I tried to extend the argument principle for such functions on a single sheet of Riemann surface and got a formula similar to that of ordinary argument principle. Using that formula the winding number of f(z) = (z-a)^(1.5+i0.3) around the origin is expressed as 1.5+i0.3.

It is well know that the quintic equation is not solved by radicals generally. Although the solution for quintic equation is expressed explicitly by using Jacobi theta functions, in the following site I tried to find a differential equation as a solution for inverse function of general quintic polynomial, and found it is expressed with the 5-th degree nonlinear differential equation.

http://hecoaustralia.fortunecity.com/stbruno/stbruno-s.htm