Let f : U → C be a differentiable function on an open set U containing 0, and assumethat f(0) = 0. Find a function v such that f(x+iy) = u(x, y) + iv(x, y) with u(x, y) =4y³x – 4x³y – x.Hint: Set up and solve the Cauchy-Riemann equations, not forgetting any integration*constants'. For example, if you find that%3D-du(r, y) = ó(x, y),thenu(x, y) = | $(x, y) dx + g(y).

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Let f : U → C be a differentiable function on an open set U containing 0, and assume that f(0) = 0. Find a function v such that f(r+iy) = u(x, y)+iv(x, y) with u(x, y) 4y³x – 4x³y – x. %3D - Hint: Set up and solve the Cauchy-Riemann equations, not forgetting any integration 'constants'. For example, if you find that du (x, y) = ¢(x, y), then u(x, y) = | $(x, y) dx + g(y). %3D

Let f : U → C be a differentiable function on an open set U containing 0, and assume that f(0) = 0. Find a function v such that f(r+iy) = u(x, y)+iv(x, y) with u(x, y) 4y³x – 4x³y – x. %3D - Hint: Set up and solve the Cauchy-Riemann equations, not forgetting any integration 'constants'. For example, if you find that du (x, y) = ¢(x, y), then u(x, y) = | $(x, y) dx + g(y). %3D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let f : U → C be a differentiable function on an open set U containing 0, and assume
that f(0) = 0. Find a function v such that f(x+iy) = u(x, y) + iv(x, y) with u(x, y) =
4y³x – 4x³y – x.
Hint: Set up and solve the Cauchy-Riemann equations, not forgetting any integration
*constants'. For example, if you find that
%3D
-
du
(r, y) = ó(x, y),
then
u(x, y) = | $(x, y) dx + g(y).

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