Question

If u = e* (x cos y - ysin y), th​

Answer 1

Answer:

Let u = u (x, y) = e^x(x cos y - y sin y). Show that u = u(x, y) is a harmonic function. Find the harmonic conjugate of u = u(x, y) Write the analytic function f(z) = u(x, y) + v(x, y)i in the terms of z. we need to show that u_xx + u_yy = 0. So we have: u_x = partial differential u (x, y)/partial differential x = e^x (x cos y - y sin y) + e^x cos y implies u_xx = partial differential^2 u (x, y)/partial differential x^2 = e^x (x cos y - y sin y) + e^x cos y + e^x cos y u_y = partial differential u (x, y)/partial differential y = e^x (-x sin y - sin y - y cos y) implies u_yy = partial differential^2 u (x, y)/partial differential y^2 = e^x (-x cos y - cos y - cos y + y sin y) u_xx + u_yy = e^x (x cos y - y sin y) + e^x cos y + e^x cos y + e^x (-x cos y - cos y - cos y + y sin y) = e^x (x cos y - y sin y + cos y + cos y - x cos y - cos y - cos y + y sin y) = 0

Step-by-step explanation:

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