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2.2 Use the Cauchy-Riemann equation to determine if the function f(2) = r³ – i(2- y)³ is analytic or not. Provide all the sufficient conditions and the domain of analyticity and then find the derivative if it exists.

2.2 Use the Cauchy-Riemann equation to determine if the function f(2) = r³ – i(2- y)³ is analytic or not. Provide all the sufficient conditions and the domain of analyticity and then find the derivative if it exists.

Advanced Engineering Mathematics
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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2.2

2.2 Use the Cauchy-Riemann equation to determine if the function f(z) = x³ – i(2 – y)3 is analytic or not. Provide all the sufficient conditions and the domain of analyticity and then find the derivative if it exists.
Transcribed Image Text:2.2 Use the Cauchy-Riemann equation to determine if the function f(z) = x³ – i(2 – y)3 is analytic or not. Provide all the sufficient conditions and the domain of analyticity and then find the derivative if it exists.
Expert Solution
Step 1

Cauchy-Riemann equation: Let fz=ux, y+ιvx, y be a complex function then

the Cauchy-Riemann equation is given by ux=vy and uy=-vx.

We have to check whether the given complex function is analytic or not. Comparing

it with fz=ux, y+ιvx, y we get, ux, y=x3 and vx, y=-2-y3.

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