MATH 316 Study Guide - Final Guide: Ilog, Inverse Trigonometric Functions, Atan2

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Fall 2012: let f (z) = y 2xy + i( x + x2 y2) + z2 where z = x + iy is a complex variable de ned in the whole complex plane. Solution: our plan is to identify the real and imaginary parts of f , and then check if the cauchy-riemann equations hold for them. We have f (z) = y 2xy + i( x + x2 y2) + x2 y2 + 2ixy. = x2 2xy + y y2 + i( x + 2xy + x2 y2), and so u(x, y) = x2 2xy + y y2, v(x, y) = x + 2xy + x2 y2. We compute the partial derivatives of u and v as ux(x, y) = 2x 2y, uy(x, y) = 2x + 1 2y, vx(x, y) = 1 + 2y + 2x, vy(x, y) = 2x 2y.