Solved: Cauchy-Riemann equations In the advanced subject

Chapter 12, Problem 95

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Cauchy-Riemann equations In the advanced subject of complex variables, a function typically has the form f 1x, y2 = u1x, y2 + iv1x, y2, where u and v are real-valued functions and i = 1-1 is the imaginary unit. A function f = u + iv is said to be analytic (analogous to differentiable) if it satisfies the Cauchy-Riemann equations: ux = vy and uy = -vx. a. Show that f 1x, y2 = 1x2 - y22 + i12xy2 is analytic. b. Show that f 1x, y2 = x1x2 - 3y22 + iy13x2 - y22 is analytic. c. Show that if f = u + iv is analytic, then uxx + uyy = 0 and vxx + vyy = 0. Assume u and v satisfy the conditions in Theorem 12.4.