## Functions of complex variables: an introduction |

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absolutely convergent analytic function angle antiderivative approaches zero arcsin argument assume Cauchy-Riemann equations choose circle of radius coefficients complex number complex plane complex variable conformal mapping consider constant continuous converges absolutely cosh defined Definition derivative differentiable domain evaluate example exists F plane fact Figure find the residues finite number flow region fluid formula function f(z geometric series given goes to zero harmonic functions hence infinite inside integral integrand isolated singular Laurent expansion Laurent series Lemma logarithm Maclaurin series map the upper meromorphic multiple-valued nonzero nth root obtain origin point z0 poles polygon polynomial power series problem Proof prove rational function real and imaginary real axis real number result simple closed contour sin2 singular point sinh square root streamline Suppose Taylor series Theorem Let unit circle univalent upper half plane velocity potential