Complex integration :
Cauchy-Riemann formulas

Cauchy-Riemann formulas

Let a function f(z) = u(x,y) + i v(x,y) analytic. Cauchy-Riemann formulas relate a connextion between u(x,y) and v(x,y).

$\bf\color{black}{ \text {Let L = } \lim_{\Delta \textit z \to 0} [\frac{f(z + \Delta z) - f(z)}{\Delta z} ] \\ z = x + iy \\ \Delta z = \Delta x + i \Delta y \\ f(z) = u(x,y) + i v(x,y) \\ f(z + \Delta z) = u(x + \Delta x, y + \Delta y) + iv(x + \Delta x, y + \Delta y) \\ \text { } \\ \Rightarrow \text {L = } \\ \lim_{\Delta \textit x \to 0, \Delta \textit y \to 0} [ \frac{ u(x + \Delta x, y + \Delta y) + iv(x + \Delta x, y + \Delta y) - ( u(x,y) + i v(x,y) )} {\Delta x + i \Delta y } ] \\ \text { } \\ \\ \text { } \\ \bullet If \Delta \text { is real } \Rightarrow \Delta y = 0. So \\ \text {L = } \lim_{\Delta \textit x \to 0 } [ \frac{ u(x + \Delta x, y) + iv(x + \Delta x, y) - (u(x,y) + i v(x,y)) } {\Delta x } ] = \\ \lim_{\Delta \textit x \to 0 } [ \frac{ u(x + \Delta x, y) - u(x,y)}{\Delta x } + i \frac{v(x + \Delta x, y) - v(x,y)) }{\Delta x } {\Delta x } ] = \\ \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} \\ \text { } \\ \\ \text { } \\ \bullet If \Delta \text {z is imaginary} \Rightarrow \Delta x = 0. So \\ \text {L = } \lim_{\Delta \textit y \to 0 } [ \frac{ u(x , y + \Delta y) + iv(x, y + \Delta y) - (u(x,y) + i v(x,y)) } {i \Delta y } ] = \\ \lim_{\Delta \textit y \to 0 } [ \frac{ u(x, y + \Delta y) - u(x,y)}{i \Delta y } + i \frac{v(x, y + \Delta y) - v(x,y)) }{i \Delta y } {i \Delta y } ] = \\ - i \frac{\partial u}{\partial y} + \frac{\partial v}{\partial y} \\ \text { } \\ \text {Therfore } \\ \text { } \\ L = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = \frac{- i \partial u}{\partial y} + \frac{\partial v}{\partial y} }$

$$\Large\bf\color{red}{ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} } \small\color{black}{ \; \text{ and } \; } \Large\bf\color{red}{ \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} } \\ \text { } \\ \\ \text { } \\ \bf\color{green}{ \textit {These are the Cauchy-Riemann formulas. } }$$