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

Use the Cauchy-Riemann equations to show that $f\left(z\right)=\stackrel{―}{z}$ is not analytic.
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Liyana Mansell
Step 1
A function is said to be analytic when Cauchy-Riemann equations are satisfied.
The Cauchy-Riemann equations are satisfied when ${u}_{x}={v}_{y}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{v}_{x}=-{u}_{y}$.
The given function is $f\left(z\right)=\stackrel{―}{z}$
Rewrite the given function as follows.
$f\left(z\right)=\stackrel{―}{z}$
=x-iy
Here u(x,y)=x and v(x,y)=-y.
Step 2
Evaluate ${u}_{x}$ as follows.
${u}_{x}=\frac{\partial }{\partial x}\left(x\right)$
=1
Thus, ${u}_{x}=1$.
Evaluate ${u}_{y}$ as follows.
${u}_{y}=\frac{\partial }{\partial y}\left(x\right)$
=0
Thus, ${u}_{y}=0$.
Step 3
Evaluate ${v}_{x}$ as follows.
${v}_{x}=\frac{\partial }{\partial x}\left(-y\right)$
=0
Thus, ${v}_{x}=0$.
Evaluate ${v}_{y}$ as follows.
${v}_{y}=\frac{\partial }{\partial y}\left(-y\right)$
=-1
Thus, ${v}_{y}=-1$.
Clearly, ${u}_{x}\ne {v}_{y}$. So, the function did not satisfy Cauchy-Riemann equations.
Therefore, the given function is not analytic.
Jeffrey Jordon