Complex Variables and Transform Calculus
Based on a series of lectures given by the author this text is designed for undergraduate students with an understanding of vector calculus, solution techniques of ordinary and partial differential equations and elementary knowledge of integral transforms. It will also be an invaluable reference to scientists and engineers who need to know the basic mathematical development of the theory of complex variables in order to solve field problems. The theorems given are well illustrated with examples.
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analytic continuation analytic everywhere analytic function angle arctan bilinear transformation boundary Cauchy integral formula Cauchy-Goursat theorem Cauchy-Riemann conditions Cauchy's Chapter coefficients complex numbers complex plane complex potential complex variable conformal mapping conjugate consider constant corresponding cosh cosine cylinder defined denoted Determine differential equation domain double pole equipotential lines Example Exercise exponential order fi(z flow fluid following integral Fourier series function f(z given harmonic Hence image curve infinite inside the contour inverse Laplace transform Laplace's equation Laurent series Laurent series expansion obtain periodic function piecewise regular polynomial power series problem Proof quadrant radius Re(z real and imaginary real axis region residue of f(z respectively right half plane roots Schwarz-Christoffel semi-infinite Show shown in Fig simple closed contour simple pole sin0 singular point sinh sinz streamlines temperature distribution to-plane unit circle unit step function written z-plane zero