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Moduli and Conjugates
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analytic function angle antiderivative arctan calculus Cauchy Cauchy-Goursat theorem Cauchy-Riemann equations coefficients complex variable continuous converges corresponding cosh curve defined denote differentiable Dirichlet problem disk domain of definition essential singular point evaluate EXAMPLE Exercise exists expression FIGURE finite number first-order partial derivatives flow follows function f half line harmonic conjugate harmonic function Hence imaginary improper integrals inequality integral formula integrand inverse Laurent series limit line segment linear fractional transformation logz Maclaurin series mapping nonzero complex number nonzero point Note nth roots obtained origin point zq points interior pole of order polygon polynomial positive number power series quadrant real axis real numbers region residue result Riemann surface satisfies series representation shown in Fig simple closed contour sinh sinz smooth arc square strip Suppose Taylor series temperature theorem in Sec vector verify write xy plane zero