| Version 2 |
Version 1 |
| \paragraph{Existence of complex derivative implies the Cauchy-Riemann |
\paragraph{Existence of complex derivative implies the Cauchy-Riemann |
| equations.} |
equations.} |
|
|
| Suppose that the complex |
Suppose that the complex |
| derivative |
derivative |
| \begin{equation} |
\begin{equation} |
| \label{eq:cder} |
\label{eq:cder} |
| f'(z) = \lim_{\zeta\rightarrow 0} \frac{f(z+\zeta)-f(z)}{\zeta} |
f'(z) = \lim_{\zeta\rightarrow 0} \frac{f(z+\zeta)-f(z)}{\zeta} |
| \end{equation} |
\end{equation} |
| exists for some $z\in \cnums$. |
exists for some $z\in \cnums$. |
| This means that for all $\epsilon>0$, there exists a $\rho>0$, such that |
This means that for all $\epsilon>0$, there exists a $\rho>0$, such that |
| for all complex $\zeta$ with |
for all complex $\zeta$ with |
| $\vert \zeta\vert<\rho$, we have |
$\vert \zeta\vert<\rho$, we have |
| $$\left\Vert f'(z) - \frac{f(z+\zeta)-f(z)}{\zeta} \right \Vert<\epsilon.$$ |
$$\left\Vert f'(z) - \frac{f(z+\zeta)-f(z)}{\zeta} \right \Vert<\epsilon.$$ |
|
|
| Henceforth, set |
Henceforth, set |
| $$f=u+iv,\quad z=x+iy.$$ |
$$f=u+iv,\quad z=x+iy.$$ |
| If $\zeta$ is |
If $\zeta$ is |
| real, then the above limit reduces to a partial derivative in $x$, i.e. |
real, then the above limit reduces to a partial derivative in $x$, i.e. |
| $$f'(z) = \frac{\partial f}{\partial x} = \frac{\partial u}{\partial |
$$f'(z) = \frac{\partial f}{\partial x} = \frac{\partial u}{\partial |
| x} + i \frac{\partial v}{\partial x},$$ |
x} + i \frac{\partial v}{\partial x},$$ |
| Taking the limit with an |
Taking the limit with an |
| imaginary $\zeta$ we deduce that |
imaginary $\zeta$ we deduce that |
| $$f'(z) = -i\frac{\partial f}{\partial y} = -i \frac{\partial u}{\partial |
$$f'(z) = -i\frac{\partial f}{\partial y} = -i \frac{\partial u}{\partial |
| y} + \frac{\partial v}{\partial y}.$$ |
y} + \frac{\partial v}{\partial y}.$$ |
| Therefore |
Therefore |
| $$\frac{\partial f}{\partial x} = -i\frac{\partial f}{\partial y},$$ |
$$\frac{\partial f}{\partial x} = -i\frac{\partial f}{\partial y},$$ |
| and breaking this relation up into its real and imaginary parts gives |
and breaking this relation up into its real and imaginary parts gives |
| the Cauchy-Riemann equations. |
the Cauchy-Riemann equations. |
|
|
| \paragraph{The Cauchy-Riemann |
\paragraph{The Cauchy-Riemann |
| equations imply the existence of a complex derivative.} |
equations imply the existence of a complex derivative.} |
| Suppose that the Cauchy-Riemann |
Suppose that the Cauchy-Riemann |
| equations |
equations |
| $$ |
|
| \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\quad |
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\quad |
| \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}, |
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}, |
| $$ |
|
| hold for a fixed $(x,y)\in\reals^2$, |
hold for a fixed $(x,y)\in\reals^2$, |
| and that all the |
and that all the |
| partial derivatives are continuous at $(x,y)$ as well. The continuity |
partial derivatives are continuous at $(x,y)$ as well. The continuity |
| implies that all directional derivatives exist as well. In |
implies that all directional derivatives exist as well. In |
| other words, for $\xi,\eta\in\reals$ and $\rho=\sqrt{\xi^2+\eta^2}$ |
other words, for $\xi,\eta\in\reals$ and $\rho=\sqrt{\xi^2+\eta^2}$ |
| we have |
we have |
| $$ |
|
| \frac{ |
\frac{ |
| u(x+\xi,y+\eta) - u(x,y) - (\xi \frac{\partial u}{\partial x } + \eta |
u(x+\xi,y+\eta) - u(x,y) - (\xi \frac{\partial u}{\partial x } + \eta |
| \frac{\partial u}{\partial y})}{\rho} \rightarrow |
\frac{\partial u}{\partial y})}{\rho} \rightarrow |
| 0,\;\mbox{as } \rho\rightarrow 0,$$ |
0,\;\mbox{as } \rho\rightarrow 0,$$ |
| with a similar relation holding for $v(x,y)$. Combining the two scalar |
with a similar relation holding for $v(x,y)$. Combining the two scalar |
| relations into a vector relation we obtain |
relations into a vector relation we obtain |
| $$ |
|
| \rho^{-1} \left\Vert |
\rho^{-1} \left\Vert |
| \begin{pmatrix} |
\begin{pmatrix} |
| u(x+\xi,y+\eta) \\ v(x+\xi,y+\eta) |
u(x+\xi,y+\eta) \\ v(x+\xi,y+\eta) |
| \end{pmatrix} |
\end{pmatrix} |
| - |
|
| \begin{pmatrix} |
\begin{pmatrix} |
| u(x,y) \\ v(x,y) |
u(x,y) \\ v(x,y) |
| \end{pmatrix} |
\end{pmatrix} |
| - |
|
| \begin{pmatrix} |
\begin{pmatrix} |
| \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\\ |
\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\\ |
| \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\\ |
\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\\ |
| \end{pmatrix} |
\end{pmatrix} |
| \begin{pmatrix} |
\begin{pmatrix} |
| \xi \\ \eta |
\xi \\ \eta |
| \end{pmatrix} |
\end{pmatrix} |
| \right\Vert \rightarrow 0,\;\mbox{as } \rho\rightarrow 0.$$ |
\right\Vert \rightarrow 0,\;\mbox{as } \rho\rightarrow 0.$$ |
| Note that |
Note that |
| the Cauchy-Riemann equations imply that the matrix-vector product |
the Cauchy-Riemann equations imply that the matrix-vector product |
| above is equivalent to the product of two complex numbers, namely |
above is equivalent to the product of two complex numbers, namely |
| $$\left(\frac{\partial u}{\partial x} +i\frac{\partial v}{\partial |
$$\left(\frac{\partial u}{\partial x} +i\frac{\partial v}{\partial |
| x}\right)(\xi+i\eta).$$ |
x}\right)(\xi+i\eta).$$ |
| Setting |
Setting |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| f(z) &=& u(x,y)+i v(x,y),\\ |
f(z) &=& u(x,y)+i v(x,y),\\ |
| f'(z) &=& \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial |
f'(z) &=& \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial |
| x}\\ |
x}\\ |
| \zeta &=& \xi+i\eta |
\zeta &=& \xi+i\eta |
| \end{eqnarray*} |
\end{eqnarray*} |
| we can therefore rewrite the above limit relation as |
we can therefore rewrite the above limit relation as |
| $$ |
|
| \left\Vert \frac{ f(z+\zeta)-f(z) - f'(z)\zeta}{ \zeta}\right\Vert \rightarrow |
\left\Vert \frac{ f(z+\zeta)-f(z) - f'(z)\zeta}{ \zeta}\right\Vert \rightarrow |
| 0,\;\mbox{as } \rho\rightarrow 0,$$ |
0,\;\mbox{as } \rho\rightarrow 0,$$ |
| which is the complex limit definition of $f'(z)$ shown |
which is the complex limit definition of $f'(z)$ shown |
| in \eqref{eq:cder}. |
in \eqref{eq:cder}. |
