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

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

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

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

Note: If we did
\[ \begin{gather} f'(z)=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y}\\[5pt] w'=2y+ix \end{gather} \]
  • For (x, y)=(0, 0)
\[ \begin{gather} w'=2.0+i0\\ w'=0 \end{gather} \]
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