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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 1

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 2 part 1', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Result 8.3.1 If f z u iv is an analytic function then u and v are harmonic functions. That is the Laplacians of u and v vanish Au Av 0. The Laplacian in Cartesian and polar coordinates is A Ỡ2 d2 dx2 dy2 A 1 d r dr 1 d2 r2 dB2 Given a harmonic function u in a simply connected domain there exists a harmonic function v unique up to an additive constant such that f z u iv is analytic in the domain. One can construct v by solving the Cauchy-Riemann equations. Example 8.3.1 Is x2 the real part of an analytic function The Laplacian of x2 is A x2 2 0 x2 is not harmonic and thus is not the real part of an analytic function. Example 8.3.2 Show that u e x x sin y y cos y is harmonic. f u e x sin y dx y ex x sin y y cos y e x sin y x e x sin y y e x cos y ỡ2u dx2 e x sin y e x sin y x e x sin y y e x cos y 2e- x siny xe- x siny y e- x cosy du By e x x cos y cos y y sin y 374 e x x sin y sin y y cos y sin y dy2 x e-x sin y 2 e-x sin y y e-x cos y Thus we see that Ou o 0 and u is harmonic. dx2 dy2 Example 8.3.3 Consider u cos x cosh y. This function is harmonic. uxx uyy cos x cosh y cos x cosh y 0 Thus it is the real part of an analytic function f z . We find the harmonic conjugate v with the Cauchy-Riemann equations. We integrate the first Cauchy-Riemann equation. vy ux sin x cosh y v sin x sinh y a x Here a x is a constant of integration. We substitute this into the second Cauchy-Riemann equation to determine a x . vx uy cos x sinh y a x cos x sinh y az x 0 a x c Here c is a real constant. Thus the harmonic conjugate is v sin x sinh y c. The analytic function is f z cos x cosh y 1 sin x sinh y ic We recognize this as f z cos z ic. .