Solved Problem on Cauchy-Riemann Equations
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a) \( \displaystyle w=\left(x^{2}-y^{2}-2x\right)+2iy\left(x-1\right) \)
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)
\[
\begin{array}{l}
u(x,y)=x^{2}-y^{2}-2x\\
v(x,y)=2y\left(x-1\right)
\end{array}
\]
Calculating the partial derivatives
\[
\begin{array}{l}
\dfrac{\partial u}{\partial x}=2x-2=2(x-1)\\[5pt]
\dfrac{\partial v}{\partial y}=2(x-1) \\[5pt]
\dfrac{\partial u}{\partial y}=-2y \\[5pt]
\dfrac{\partial v}{\partial x}=2y
\end{array}
\]
Condition 2: The derivatives are continuous everywhere in the complex plane.
Applying the Cauchy-Riemann Equations
\[
\begin{gather}
\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\\
2(x-1)=2(x-1)
\end{gather}
\]
\[
\begin{gather}
\frac{\partial u}{\partial y}=-{\frac{\partial v}{\partial x}}\\
-2y=-2y
\end{gather}
\]
Condition 3: The function w satisfies the Cauchy-Riemann Equations.
The function w is continuous, the derivatives are continuous, and the Cauchy-Riemann Equations are satisfied, the function w is analytic.
The function is differentiable everywhere in the complex plane (entire function).
The derivative is given by
\[ \bbox[#99CCFF,10px]
{f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y}}
\]
\[ \bbox[#FFCCCC,10px]
{w'=2(x-1)+2yi}
\]
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