Download Applied Complex Variables for Scientists and Engineers, by Yue Kuen Kwok PDF

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By Yue Kuen Kwok

This creation to complicated variable equipment starts off via rigorously defining advanced numbers and analytic features, and proceeds to provide money owed of complicated integration, Taylor sequence, singularities, residues and mappings. either algebraic and geometric instruments are hired to supply the best knowing, with many diagrams illustrating the innovations brought. The emphasis is laid on realizing using tools, instead of on rigorous proofs. through the textual content, the various very important theoretical ends up in complicated functionality thought are by way of proper and brilliant examples in actual sciences. This moment variation now includes 350 stimulating routines of top of the range, with options given to lots of them. fabric has been up to date and extra proofs on a few of the very important theorems in complicated functionality idea are actually integrated, e.g. the Weierstrass-Casorati theorem. The ebook is extremely appropriate for college students wishing to profit the weather of advanced research in an utilized context.

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Extra info for Applied Complex Variables for Scientists and Engineers, Second edition

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The highlights of the chapter are the Cauchy– Riemann relations and the definition of an analytic function. Analyticity plays a central role in complex variable theory. The relations between harmonic functions and analytic functions are established. We show how to solve the Poisson equation effectively using the formulation of complex conjugate variables. In addition, the application of complex differentiation in dynamics problems and the use of complex functions in describing fluid flows and steady state heat distribution are illustrated.

2 Show that any small circle whose circumference contains the north pole N on the Riemann sphere corresponds to a straight line not passing through the origin in the complex plane. Solution A small circle whose circumference contains the north pole N on the Riemann sphere is determined by the intersection of the Riemann sphere ξ 2 + η2 + ζ 2 − ζ = 0 and a plane through N . In terms of ξ, η and ζ , a plane passing through the north pole N (0, 0, 1) has the general form Aξ + Bη + C(ζ − 1) = 0, C = 0.

Let w1 and w2 be two complex numbers. Denote w = w2 − w1 . Suppose w2 = rw1 , where r is real and non-zero. Show that 1 1 + cos(Arg w2 − Arg w) 2 = r 1 + cos(Arg w1 − Arg w) 1 when 0 < r < 1 when r > 1 . When r < 0, show that it becomes undefined. 42. For each sequence {zn }, find the corresponding limit points (if they exist): 1 (a) zn = (−1)n ; (b) zn = (−1)n ; n n nπi (d) zn = e 4 . 43. Show that the whole complex plane and the empty set are both open. Are they both closed? 44. For each of the following point sets, find the interior points, exterior points, boundary points and limit points: (a) 0 ≤ Re(iz) ≤ 3; (b) 0 ≤ Arg z < π 4 and |z| > 2.

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