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absolute value According to equation analytic continuation analytic function angle Appendix approaches zero arctan branch cut Cauchy-Goursat theorem Cauchy-Riemann conditions closed curve coefficients complex number complex potential complex variable conformal mapping continuous function corresponding cosh defined definite integral denote essential singular point example EXERCISES exists finite number flow fluid follows function F(z function is analytic half plane harmonic function Hence inequality infinite integrand interior inverse isolated singular point Laurent series limit line integral linear fractional transformation Maclaurin series maps neighborhood partial derivatives path point z0 polygon positive number power series quadrant real axis real constant real numbers real variables represents residue Riemann surface satisfied sheet shown in Fig single-valued and analytic single-valued function sinh steady temperatures strip tends to infinity unit circle upper half vanishes vector velocity write xy plane ZiZ2