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Using Cauchy-Riemann equations, show that the function f(z)=(z+6)^2 is differentiable everywhere

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

Chapter6: Rates Of Change

Section6.1: Velocity

Problem 12SBE

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Question

Using Cauchy-Riemann equations, show that the function f(z)=(z+6)^2 is differentiable everywhere.

Expert Solution

Step 1

The given function is

$f (z) = {(z + 6)}^{2} \Rightarrow f (z) = z^{2} + 12 z + 36 - (1)$

Substitute the value $z = x + i y$ in equation (1).

$f (z) = {(x + i y)}^{2} + 12 (x + i y) + 36 \Rightarrow f (z) = x^{2} + y^{2} + 2 i x y + 12 x + 12 i y + 36 \Rightarrow f (z) = (x^{2} + y^{2} + 12 x + 36) + i (2 x y + 12 y) - (2)$

Comparing equation (2) with the equation $f (z) = u + i v$ we get:

$u = x^{2} + y^{2} + 12 x + 36 v = 2 x y + 12 y$

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