Solved Problem on Cauchy-Riemann Equations

e) \( \displaystyle w=|z| \)

Writing the function as

\[ w=\sqrt{x^{2}+y^{2}\;} \]

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)

\[ \begin{array}{l} u(x,y)=\sqrt{x^{2}+y^{2}}\\ v(x,y)=0 \end{array} \]

Calculating the partial derivatives

\[ \begin{array}{l} \dfrac{\partial u}{\partial x}=\dfrac{1}{2}\dfrac{2x}{\sqrt{x^{2}+y^{2}\;}}=\dfrac{x}{\sqrt{x^{2}+y^{2}\;}}\\[5pt] \dfrac{\partial v}{\partial y}=0\\[5pt] \dfrac{\partial u}{\partial y}=\dfrac{1}{2}\dfrac{2y}{\sqrt{x^{2}+y^{2}\;}}= \dfrac{y}{\sqrt{x^{2}+y^{2}\;}}\\[5pt] \dfrac{\partial v}{\partial x}=0 \end{array} \]

Condition 2: The derivatives are not continuous at point z = 0, where (x, y) = (0, 0).

Applying the Cauchy-Riemann Equations

\[ \begin{gather} \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\\ \frac{x}{\sqrt{x^{2}+y^{2}\;}}\neq 0 \end{gather} \]

\[ \begin{gather} \frac{\partial u}{\partial y}=-{\frac{\partial v}{\partial x}}\\ \frac{y}{\sqrt{x^{2}+y^{2}\;}}\neq 0 \end{gather} \]

Condition 3: The function w does not satisfy the Cauchy-Riemann Equations.

The function w is constinuous and the derivatives are not continuous, and the function does not satisfy the Cauchy-Riemann Equations, the function w is not analytic.

The Cauchy-Riemann Equations are not satisfiedm but in the first condition, if we do

\[ \begin{gather} \frac{x}{\sqrt{x^{2}+y^{2}\;}}=0\\ x=0.\sqrt{x^{2}+y^{2}\;}\\ x=0 \end{gather} \]

and in the second equation

\[ \begin{gather} \frac{y}{\sqrt{x^{2}+y^{2}\;}}=0\\ y=0.\sqrt{x^{2}+y^{2}\;}\\ y=0 \end{gather} \]

but we cannot have (x, y)=(0, 0)

The function w is not differenciable in the complex plane.

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