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angle Argand diagram bilinear transformation boundary bounded branch-point Cauchy-Riemann equations circle of centre circle of convergence closed contour complex numbers complex variable theory confocal conformal transformation consider const constant contour integral cosec cosh defined definition denote differentiable domain dv dv ellipse essential singularity example f f(z)dz 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 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's theorem tends to zero to-plane u+iv unit circle upper half upper half-plane whole z-plane write x+iy z-plane z-plane correspond