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angle Argand diagram bilinear transformation boundary bounded branch-point Cauchy-Riemann equations Cauchy's theorem circle of centre closed contour complex numbers complex variable theory conformal transformation consider const constant contour integral cosec cosh cot ttz cross-ratio define definition denote differentiable domain dv dv dx dy ellipse essential singularity example finite follows function f(z given Hence infinite number integral round interior point inverse isogonal Laplace's equation lemma limit point magnification many-valued function maps Mobius modulus neighbourhood obtain one-one parabola plane point at infinity point z0 pole of order polygon positive real axis power series prove quadrant radius real and imaginary real numbers real variable rectangle region regular function residue Riemann surface roots satisfied semicircle Show Similarly simple poles sinh small circle square straight line Taylor Taylor's theorem tends to zero to-plane corresponding u+iv unit circle upper half upper half-plane write x+iy z-plane corresponding