Argument principle for non-integer multiplicity

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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. 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. For details, visit:

 

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