## COMPLEX VARIBALES AND APPLICATIONS |

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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 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 imaginary inequality infinite integral formula integrand inverse Laurent series limit line segment linear fractional transformation Maclaurin series maps neighborhood partial derivatives point z0 pole polygon positive number positive sense power series problem prove quadrant real axis real numbers region represents residue Riemann surface satisfy sheet shown in Fig single-valued singular point sinh strip Taylor's series tends to infinity tion upper half vanishes vector velocity write written xy plane