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FUNCTIONS OF A COMPLEX VARIABLE
2 The CauchyRiemann Equations 1 i
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absolutely convergent analytic continuation analytic everywhere analytic function analytic throughout Answer boundary bounded branch cut branch point Cauchy-Riemann equations circle centered complex number constant converges uniformly cosh defined direct analytic continuation disc diverges entire function Evaluate exists a positive Find finite number function f(z fundamental regions harmonic Hence infinite infinity inside integer integral interior intersection inverse Laurent series lf(z lim f(z limit point maps nonzero nth root partial derivatives partial sum polygon polynomial positive integer positively-sensed circle power series Prob PROBLEM Prove rational number real number removable singularity Riemann surface satisfying sequence series converges Show shown in Fig simple closed contour simple closed curve simply connected single-valued sinh small positive number Solution straight line subset theorem transformation unit circle upper half upper half-plane vanishes variable vector x-axis z-plane zero