Cauchy-Riemann Equations

Theorem

Let $G \subseteq \mathbb{C}$ be an open subset. Let the function $f = u + i v$ be differentiable at a point $z = x+iy \in G$ . Then all partial derivatives of $u$ and $v$ exist at $\left( x,y \right)$ and the following Cauchy-Riemann equations hold:

$\dfrac{\partial u}{\partial x} \left( x,y \right) = \dfrac{\partial v}{\partial y} \left( x,y \right)$

$\dfrac{\partial u}{\partial y} \left( x,y \right) = - \dfrac{\partial v}{\partial x} \left( x,y \right)$

In this case, the derivative of $f$ at $z$ can be represented by the formula

$f'\left( z \right) = \dfrac{\partial u}{\partial x} \left( x,y \right) - i \dfrac{\partial u}{\partial y} \left( x,y \right) = \dfrac{\partial v}{\partial y} \left( x,y \right) + i \dfrac{\partial v}{\partial x} \left( x,y \right)$

Proof

Let $h := k+i0 \left( k \in \mathbb{R} \right)$ . Then

$\begin{array}{rcl} f'\left( z \right) & = & \lim\limits_{h \to 0} \dfrac{f\left( z+h \right) - f\left( z \right)}{h} \\ & = &\lim\limits_{k \to 0} \dfrac{u\left( x+k,y \right) + iv\left(x+k,y \right) -u\left( x,y \right) -iv\left( x,y \right) }{k} \\ & = &\lim\limits_{k \to 0} \dfrac{u\left( x+k,y \right) - u\left( x,y \right) }{k} +i\dfrac{v\left( x+k,y \right) - v\left( x,y \right) }{k} \\ & = & \dfrac{\partial u}{\partial x} \left( x,y \right) + i\dfrac{\partial v}{\partial x} \left( x,y \right) \end{array}$

Let $h := 0+il \left( l \in \mathbb{R} \right)$ . Then

$\begin{array}{rcl} f'\left( z \right) & = & \lim\limits_{h \to 0} \dfrac{f\left( z+h \right) - f\left( z \right)}{h}\\ & = &\lim\limits_{l \to 0} \dfrac{u\left( x,y+l \right) + iv\left(x,y+l \right) -u\left( x,y \right) -iv\left( x,y \right) }{il}\\ & = &\lim\limits_{l \to 0} \dfrac{1}{i}\dfrac{u\left( x,y+l \right) - u\left( x,y \right) }{l} +\dfrac{v\left( x,y+l \right) - v\left( x,y \right) }{l}\\ & = & \dfrac{\partial v}{\partial y} \left( x,y \right) - i\dfrac{\partial u}{\partial y} \left( x,y \right) \end{array}$

Hence:

$f'\left( z \right) =\dfrac{\partial u}{\partial x} \left( x,y \right) + i\dfrac{\partial v}{\partial x} \left( x,y \right) = \dfrac{\partial v}{\partial y} \left( x,y \right) - i\dfrac{\partial u}{\partial y} \left( x,y \right)$

Equating the real and imaginary parts, we get the Cauchy-Riemann equations. The representation formula follows from the above line and the Cauchy-Riemann equations.

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