# proof of the Cauchy-Riemann equations

## Existence of complex derivative implies the Cauchy-Riemann equations.

Suppose that the complex derivative

 $f^{\prime}(z)=\lim_{\zeta\rightarrow 0}\frac{f(z+\zeta)-f(z)}{\zeta}$ (1)

exists for some $z\in\mathbb{C}$. This means that for all $\epsilon>0$, there exists a $\rho>0$, such that for all complex $\zeta$ with $|\zeta|<\rho$, we have

 $\left|f^{\prime}(z)-\frac{f(z+\zeta)-f(z)}{\zeta}\right|<\epsilon.$

Henceforth, set

 $f=u+iv,\quad z=x+iy.$

If $\zeta$ is real, then the above limit reduces to a partial derivative in $x$, i.e.

 $f^{\prime}(z)=\frac{\partial f}{\partial x}=\frac{\partial u}{\partial x}+i% \frac{\partial v}{\partial x},$

Taking the limit with an imaginary $\zeta$ we deduce that

 $f^{\prime}(z)=-i\frac{\partial f}{\partial y}=-i\frac{\partial u}{\partial y}+% \frac{\partial v}{\partial y}.$

Therefore

 $\frac{\partial f}{\partial x}=-i\frac{\partial f}{\partial y},$

and breaking this relation up into its real and imaginary parts gives the Cauchy-Riemann equations.

## The Cauchy-Riemann equations imply the existence of a complex derivative.

Suppose that the Cauchy-Riemann equations

 $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\quad\frac{% \partial u}{\partial y}=-\frac{\partial v}{\partial x},$

hold for a fixed $(x,y)\in\mathbb{R}^{2}$, and that all the partial derivatives are continuous at $(x,y)$ as well. The continuity implies that all directional derivatives exist as well. In other words, for $\xi,\eta\in\mathbb{R}$ and $\rho=\sqrt{\xi^{2}+\eta^{2}}$ we have

 $\frac{u(x+\xi,y+\eta)-u(x,y)-(\xi\frac{\partial u}{\partial x}+\eta\frac{% \partial u}{\partial y})}{\rho}\rightarrow 0,\;\mbox{as }\rho\rightarrow 0,$

with a similar relation holding for $v(x,y)$. Combining the two scalar relations into a vector relation we obtain

 $\rho^{-1}\left\|\begin{pmatrix}u(x+\xi,y+\eta)\\ v(x+\xi,y+\eta)\end{pmatrix}-\begin{pmatrix}u(x,y)\\ v(x,y)\end{pmatrix}-\begin{pmatrix}\frac{\partial u}{\partial x}&\frac{% \partial u}{\partial y}\\ \frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}\\ \end{pmatrix}\begin{pmatrix}\xi\\ \eta\end{pmatrix}\right\|\rightarrow 0,\;\mbox{as }\rho\rightarrow 0.$

Note that the Cauchy-Riemann equations imply that the matrix-vector product above is equivalent to the product of two complex numbers, namely

 $\left(\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}\right)(\xi+% i\eta).$

Setting

 $\displaystyle f(z)$ $\displaystyle=$ $\displaystyle u(x,y)+iv(x,y),$ $\displaystyle f^{\prime}(z)$ $\displaystyle=$ $\displaystyle\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}$ $\displaystyle\zeta$ $\displaystyle=$ $\displaystyle\xi+i\eta$

we can therefore rewrite the above limit relation as

 $\left|\frac{f(z+\zeta)-f(z)-f^{\prime}(z)\zeta}{\zeta}\right|\rightarrow 0,\;% \mbox{as }\rho\rightarrow 0,$

which is the complex limit definition of $f^{\prime}(z)$ shown in (1).

Title proof of the Cauchy-Riemann equations ProofOfTheCauchyRiemannEquations 2013-03-22 12:55:39 2013-03-22 12:55:39 rmilson (146) rmilson (146) 6 rmilson (146) Proof msc 30E99 complex derivative