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analytic continuation analytic function angle arctan boundary conditions bounded branch cut Cauchy integral formula Cauchy-Goursat theorem Cauchy-Riemann conditions closed contour coefficients complex number complex variable conformal mapping conjugate harmonic functions continuous function corresponding cosh curve defined definite integrals denote Dirichlet problem entire function essential singular point example Exercise exists exp i6 finite number flow fluid follows function f half plane harmonic function Hence inequality infinite integral formula integrand inverse Laurent series limit line segment linear fractional transformation Maclaurin series maps neighborhood partial derivatives pole polygon positive number positive sense power series problem prove quadrant real axis real numbers region represents residue satisfy shown in Fig singular point sinh strip Taylor's series tends to infinity theorem tion upper half vanishes vector velocity view of equation write written xy plane ziz2