Solved Problem on Cauchy-Riemann Equations
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a) \( \displaystyle w=\bar{z} \)
Writing the function as
\[
w=\bar{z}=x-iy
\]
Condition 1: The function w is continuous everywhere in the complex plane.
The Cauchy-Riemann Equations are given by
\[ \bbox[#99CCFF,10px]
{\begin{gather}
\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\\[5pt]
\frac{\partial u}{\partial y}=-{\frac{\partial v}{\partial x}}
\end{gather}}
\]
Identifying the functions u(x, y) and v(x, y)
\[
\begin{array}{l}
u(x,y)=x\\
v(x,y)=-y
\end{array}
\]
Calculating the partial derivatives
\[
\begin{array}{l}
\dfrac{\partial u}{\partial x}=1\\[5pt]
\dfrac{\partial v}{\partial y}=-1\\[5pt]
\dfrac{\partial u}{\partial y}=0\\[5pt]
\dfrac{\partial v}{\partial x}=0
\end{array}
\]
Condition 2: The derivatives are continuous everywhere in the complex plane.
Applying the Cauchy-Riemann Equations
\[
\begin{gather}
\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\\
1\neq -1
\end{gather}
\]
\[
\begin{gather}
\frac{\partial u}{\partial y}=-{\frac{\partial v}{\partial x}}\\
0=0
\end{gather}
\]
Condition 3: The function w does not satisfy the Cauchy-Riemann Equations.
The function w and the derivatives are continuous, but the function does not satisfy the Cauchy-Riemann Equations, the function w is not analytic and is not differentiable in the complex plane.
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .