Solved Problem on Cauchy-Riemann Equations
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d)   \( \displaystyle w=z\;\text{Re}\;z \)

Writing the function as
\[ \begin{gather} w=(x+iy)\text{Re}(x+iy)\\ w=(x+iy)x\\ w=x^{2}+i xy \end{gather} \]
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)
\[ \begin{array}{l} u(x,y)=x^{2}\\ v(x,y)=xy \end{array} \]
Calculating the partial derivatives
\[ \begin{array}{l} \dfrac{\partial u}{\partial x}=2x\\[5pt] \dfrac{\partial v}{\partial y}=x\\[5pt] \dfrac{\partial u}{\partial y}=0\\[5pt] \dfrac{\partial v}{\partial x}=y \end{array} \]
Condition 2: The derivatives are continuous everywhere in the complex plane.

Applying the Cauchy-Riemann Equations
\[ \begin{gather} \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\\ 2x=x \end{gather} \]
\[ \begin{gather} \frac{\partial u}{\partial y}=-{\frac{\partial v}{\partial x}}\\ 0=y \end{gather} \]
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.

The Cauchy-Riemann Equations are not satisfiedm but in the first condition, if we do
\[ \begin{gather} 2x=x\\ 2x-x=0\\ x=0 \end{gather} \]
the function is differentiable at the point
\[ (x,y)=(0,0) \]
The derivative is given by
\[ \bbox[#99CCFF,10px] {f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y}} \]
\[ \begin{gather} f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}\\[5pt] w'=2x+iy \end{gather} \]
  • For (x, y)=(0, 0)
\[ w'=2.0+i0 \]
\[ \bbox[#FFCCCC,10px] {w'=0} \]

Note: If we did
\[ \begin{gather} f'(z)=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y}\\[5pt] w'=x+i0 \end{gather} \]
  • For x = 0
\[ \begin{gather} w'=0 \end{gather} \]
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