Applied Complex Variables for Scientists and Engineers
This introduction to complex variable methods begins by carefully defining complex numbers and analytic functions, and proceeds to give accounts of complex integration, Taylor series, singularities, residues and mappings. Both algebraic and geometric tools are employed to provide the greatest understanding, with many diagrams illustrating the concepts introduced. The emphasis is laid on understanding the use of methods, rather than on rigorous proofs. Throughout the text, many of the important theoretical results in complex function theory are followed by relevant and vivid examples in physical sciences. This second edition now contains 350 stimulating exercises of high quality, with solutions given to many of them. Material has been updated and additional proofs on some of the important theorems in complex function theory are now included, e.g. the Weierstrass–Casorati theorem. The book is highly suitable for students wishing to learn the elements of complex analysis in an applied context.
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2 Analytic Functions
3 Exponential Logarithmic and Trigonometric Functions
4 Complex Integration
5 Taylor and Laurent Series
6 Singularities and Calculus of Residues
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analytic function angle Argz bilinear transformation boundary condition branch cut branch points Cauchy integral formula Cauchy-Riemann relations circle of convergence complex function complex numbers complex plane complex potential complex variables Consider constant contour integral corresponding deduce deﬁned deﬁnition denote derivative differential domain equation Evaluate Example Figure ﬁnd ﬁnite ﬂow ﬁeld ﬂuid ﬂow ﬂux Fourier function f(z given harmonic conjugate image curve inequality inﬁnite integrand inverse isolated singularity Laplace transform Laurent series Laurent series expansion mapping modulus obtain point z0 potential ﬂow power series problem radius real axis real numbers region residue respectively Riemann sphere sequence Show simple pole sinh sinz Solution steady state temperature streamlines Suppose symmetric points Taylor series theorem unit circle upper half-plane vector velocity vertical x-axis z-plane z1 and z2 zero