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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 Cauchy's integral formula closed curve coefficients complex number complex potential complex variable conformal mapping continuous function coordinates corresponding cosh defined definition denote example EXERCISES exists finite number flow fluid follows function f(z function is analytic half plane harmonic function Hence indefinite integral inequality infinite integrand interior inverse 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 series expansion 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