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

Question
Equations asked 2021-02-26
Use the Cauchy-Riemann equations to show that $$\displaystyle{f{{\left({z}\right)}}}=\overline{{{z}}}$$ is not analytic.

## Answers (1) 2021-02-27
Step 1
A function is said to be analytic when Cauchy-Riemann equations are satisfied.
The Cauchy-Riemann equations are satisfied when $$\displaystyle{u}_{{x}}={v}_{{y}}{\quad\text{and}\quad}{v}_{{x}}=-{u}_{{y}}$$.
The given function is $$\displaystyle{f{{\left({z}\right)}}}=\overline{{{z}}}$$
Rewrite the given function as follows.
$$\displaystyle{f{{\left({z}\right)}}}=\overline{{{z}}}$$
=x-iy
Here u(x,y)=x and v(x,y)=-y.
Step 2
Evaluate $$\displaystyle{u}_{{x}}$$ as follows.
$$\displaystyle{u}_{{x}}=\frac{{\partial}}{{\partial{x}}}{\left({x}\right)}$$
=1
Thus, $$\displaystyle{u}_{{x}}={1}$$.
Evaluate $$\displaystyle{u}_{{y}}$$ as follows.
$$\displaystyle{u}_{{y}}=\frac{\partial}{{\partial{y}}}{\left({x}\right)}$$
=0
Thus, $$\displaystyle{u}_{{y}}={0}$$.
Step 3
Evaluate $$\displaystyle{v}_{{x}}$$ as follows.
$$\displaystyle{v}_{{x}}=\frac{\partial}{{\partial{x}}}{\left(-{y}\right)}$$
=0
Thus, $$\displaystyle{v}_{{x}}={0}$$.
Evaluate $$\displaystyle{v}_{{y}}$$ as follows.
$$\displaystyle{v}_{{y}}=\frac{\partial}{{\partial{y}}}{\left(-{y}\right)}$$
=-1
Thus, $$\displaystyle{v}_{{y}}=-{1}$$.
Clearly, $$\displaystyle{u}_{{x}}\ne{v}_{{y}}$$. So, the function did not satisfy Cauchy-Riemann equations.
Therefore, the given function is not analytic.

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