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Complex Power Series
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analytic function argument principle branch cut Cauchy Goursat theorem Cauchy integral formula Cauchy Riemann equations complex contour integral complex numbers complex plane complex potential complex potential function complex valued function complex variable contour integral defined differentiable DR z0 equipotential curves evaluate EXAMPLE follows Fourier transform function F(z half-plane hence imaginary axis implies improper integral infinite inside integrand interior inversion formula isolated singularity jcf(z)dz Laplace transform Laurent series level curves limit logz Note point zQ pole of order polynomial power series previous problem quotient radius real number real valued function residue theorem series converges series expansion shown in Figure simple closed contour simple closed curve simple pole simply connected domain singular point sinz SOLUTION square integrable function streamlines strip Suppose tan9 Taylor series tends to infinity tends to zero uniform flow unit circle unit disc vector field vertical w-plane