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The simplest operations
Multiplication division integral powers and roots
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Accordingly analytic continuation analytic function angle annulus arbitrary argz bilinear mapping boundary Cauchy-Riemann Cauchy's circle clear coefficient complex numbers complex potential conformal mapping consider const constant contains contour corresponding curve defined definition deleted neighbourhood denotes derivative differentiable disc edge equal equation essential singularity Example exists exterior finite number finite plane finite point flow follows function F(z given gives a one-one half-strip harmonic imaginary infinite interior inverse Laurent expansion logarithmic many-valued negative real axis obtained one-one conformal mapping open plane path plane field point at infinity point ZQ pole polygon principal problem real analysis regular branch regular function removable singularity represented residue residue theorem result Riemann surface segment sequence shown in Fig simply-connected simply-connected domain single-valued strip suppose theorem transformation upper half-plane variable vector velocity whence z-domain z-plane zero