Solved Problem on Cauchy-Riemann Equations

\( \mathsf{b)}\;\; \displaystyle w=(\operatorname e^y+\operatorname e^{-y})\sin x+i(\operatorname e^y+\operatorname e^{-y})\cos x \)

\[ \mathsf{b)}\;\; \displaystyle w=(\operatorname e^y+\operatorname e^{-y})\sin x+i(\operatorname e^y+\operatorname e^{-y})\cos x \]

Condition 1: The function w, given in the problem, 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), real part, and v(x, y), imaginary part

\[ \begin{array}{l} u(x,y)=(\operatorname e^y+\operatorname e^{-y})\sin x\\[5pt] v(x,y)=(\operatorname e^y+\operatorname e^{-y})\cos x \end{array} \]

Calculating the partial derivatives

\[ \begin{align} & \dfrac{\partial u}{\partial x}=(\operatorname e^y+\operatorname e^{-y})\cos x \tag{I} \\[5pt] & \dfrac{\partial v}{\partial y}=(\operatorname e^y-\operatorname e^{-y})\cos x \tag{II} \\[5pt] & \dfrac{\partial u}{\partial y}=(\operatorname e^y-\operatorname e^{-y})\sin x \tag{III} \\[5pt] & \dfrac{\partial v}{\partial x}=-(\operatorname e^y+\operatorname e^{-y})\sin x \tag{IV} \end{align} \]

Condition 2: The derivatives (I), (II), (III) and (IV) are continuous everywhere in the complex plane.

Applying the Cauchy-Riemann Equations

\[ \begin{gather} \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\\[5pt] (\operatorname e^y+\operatorname e^{-y})\cos x\neq(\operatorname e^y-\operatorname e^{-y})\cos x \end{gather} \]

\[ \begin{gather} \frac{\partial u}{\partial y}=-{\frac{\partial v}{\partial x}}\\[5pt] (\operatorname e^y-\operatorname e^{-y})\sin x=-[-(\operatorname e^y+\operatorname e^{-y})\sin x]\\[5pt] (\operatorname e^y-\operatorname e^{-y})\sin x\neq(\operatorname e^y+\operatorname e^{-y})\sin x \end{gather} \]

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 in the complex plane .

The derivative is given by

\[ \begin{gather} \bbox[#99CCFF,10px] {f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y}} \end{gather} \]

\[ \begin{gather} w'=(\operatorname e^y+\operatorname e^{-y})\cos x-i(\operatorname e^y+\operatorname e^{-y})\sin x\neq(\operatorname e^y-\operatorname e^{-y})\cos x-i(\operatorname e^y-\operatorname e^{-y})\sin x \end{gather} \]

The derivative is not unique.

The function w is not differentiable at any point of the complex plane .

Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .