## Functions of a complex variable, Volume 1 |

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8u dv absolutely convergent analytic function defined approach z0 calculate Cauchy-Riemann equations Cauchy's Integral Formula Cauchy's Theorem circle centre z0 circle of convergence closed contour closed Jordan contour complex number complex plane complex variable constant continuous at z0 continuous function Contour Integration converges absolutely cut-plane differentiable once Differential Calculus disc centre z0 domain of definition dv dv e-neighbourhood of z0 everywhere example exist extension of/0 Figure finite number function is differentiable Fundamental Theorem given by z(t Hence Identity Theorem If/is analytic implies Integral Calculus iv(x lies limit neighbourhood open disc P. J. Hilton partial derivatives point z0 power series primitive principal value prove radius of convergence real axis real casef real functions real variable remark sequence of distinct smooth path star-domain stepwise curve Suppose that/(z that/is Theorem of Contour track u+iv unique unit circle whole plane Z-Zn