Solved Problem on Cauchy-Riemann Equations
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\( \mathsf{b)}\;\; \displaystyle w=(\operatorname{e}^{y}+\operatorname{e}^{-y})\operatorname{sen}x+i(\operatorname{e}^{y}+\operatorname{e}^{-y})\cos x \)
\[ \mathsf{b)}\;\; \displaystyle w=(\operatorname{e}^{y}+\operatorname{e}^{-y})\operatorname{sen}x+i(\operatorname{e}^{y}+\operatorname{e}^{-y})\cos x \]
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)
\[
\begin{array}{l}
u(x,y)=(\operatorname{e}^{y}+\operatorname{e}^{-y})\operatorname{sen}x\\
v(x,y)=(\operatorname{e}^{y}+\operatorname{e}^{-y})\cos x
\end{array}
\]
Calculating the partial derivatives
\[
\begin{array}{l}
\dfrac{\partial u}{\partial x}=(\operatorname{e}^{y}+\operatorname{e}^{-y})\cos x\\[5pt]
\dfrac{\partial v}{\partial y}=(\operatorname{e}^{y}-\operatorname{e}^{-y})\cos x\\[5pt]
\dfrac{\partial u}{\partial y}=(\operatorname{e}^{y}-\operatorname{e}^{-y})\operatorname{sen}x\\[5pt]
\dfrac{\partial v}{\partial x}=-(\operatorname{e}^{y}+\operatorname{e}^{-y})\operatorname{sen}x
\end{array}
\]
Condition 2: The derivatives are continuous everywhere in the complex plane.
Applying the Cauchy-Riemann Equations
\[
\begin{gather}
\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\\
(\operatorname{e}^{y}+\operatorname{e}^{-y})\cos x\neq(\operatorname{e}^{y}-\operatorname{e}^{-y})\cos x
\end{gather}
\]
\[
\begin{gather}
\frac{\partial u}{\partial y}=-{\frac{\partial v}{\partial x}}\\
(\operatorname{e}^{y}-\operatorname{e}^{-y})\operatorname{sen}x=-[-(\operatorname{e}^{y}+\operatorname{e}^{-y})\operatorname{sen}x]\\
(\operatorname{e}^{y}-\operatorname{e}^{-y})\operatorname{sen}x\neq(\operatorname{e}^{y}+\operatorname{e}^{-y})\operatorname{sen}x
\end{gather}
\]
Condition 3: The function w does not satisfy the Cauchy-Riemann Equations.
The function w is continuous, the derivatives are continuous, but the function does not satisfy the Cauchy-Riemann Equations, the function w is not analytic.
The derivative is given by
\[ \bbox[#99CCFF,10px]
{f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y}}
\]
\[
w'=(\operatorname{e}^{y}+\operatorname{e}^{-y})\cos x-i(\operatorname{e}^{y}+\operatorname{e}^{-y})\operatorname{sen}x\neq(\operatorname{e}^{y}-\operatorname{e}^{-y})\cos x-i(\operatorname{e}^{y}-\operatorname{e}^{-y})\operatorname{sen}x
\]
The derivative is not unique.
The function w is not differentiable at any point of 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 .