Unwinding paths on the Riemann sphere for continuous integrals of rational functions

Rational functions with real poles and poles in the complex lower half-plane, orthogonal on the real line, are well known. Quadrature formulas similar to the Gauss formulas for orthogonal polynomials have been studied. We generalize to the case of ...

An incomplete Riemann Zeta function Z"1(@a,x) is examined, along with a complementary incomplete Riemann Zeta function Z"2(@a,x). These functions are defined by Z"1(@a,x)={(1-2^1^-^@a)@C(@a)}^-^1@!"0^xt^@a^-^1(e^t+1)^-^1dt and Z"2(@a,x)=@z(@a)-Z"1(@a,x),...

In this paper we present formulas expressing the orthogonal rational functions associated with a rational modification of a positive bounded Borel measure on the unit circle, in terms of the orthogonal rational functions associated with the initial ...