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.

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.