Use the Cauchy-Riemann equations to show that f(z)=bar(z) is not analytic.

Tolnaio 2021-02-26 Answered
Use the Cauchy-Riemann equations to show that f(z)=z is not analytic.
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Expert Answer

Liyana Mansell
Answered 2021-02-27 Author has 97 answers
Step 1
A function is said to be analytic when Cauchy-Riemann equations are satisfied.
The Cauchy-Riemann equations are satisfied when ux=vyandvx=uy.
The given function is f(z)=z
Rewrite the given function as follows.
f(z)=z
=x-iy
Here u(x,y)=x and v(x,y)=-y.
Step 2
Evaluate ux as follows.
ux=x(x)
=1
Thus, ux=1.
Evaluate uy as follows.
uy=y(x)
=0
Thus, uy=0.
Step 3
Evaluate vx as follows.
vx=x(y)
=0
Thus, vx=0.
Evaluate vy as follows.
vy=y(y)
=-1
Thus, vy=1.
Clearly, uxvy. So, the function did not satisfy Cauchy-Riemann equations.
Therefore, the given function is not analytic.
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Jeffrey Jordon
Answered 2021-11-03 Author has 2064 answers

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