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proof of the Cauchy-Riemann equations
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(Proof)
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Suppose that the complex derivative \begin{equation} \label{eq:cder} f'(z) = \lim_{\zeta\rightarrow 0} \frac{f(z+\zeta)-f(z)}{\zeta} \end{equation}exists for some $z\in \cnums$ . This means that for all $\epsilon>0$ , there exists a $\rho>0$ , such that for all complex $\zeta$ with $\vert \zeta\vert<\rho$ , we have $$\left\vert f'(z) - \frac{f(z+\zeta)-f(z)}{\zeta} \right \vert<\epsilon.$$
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. $$f'(z) = \frac{\partial f}{\partial x} = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x},$$ Taking the limit with an imaginary $\zeta$ we deduce that $$f'(z) = -i\frac{\partial f}{\partial y} = -i \frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}.$$ 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.
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 $(x,y)\in\reals^2$ , 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 $\xi,\eta\in\reals$ 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 $$ \rho^{-1} \left\Vert \begin{pmatrix} u(x+\xi,y+\eta) \\ v(x+\xi,y+\eta) \end{pmatrix} - \begin{pmatrix} u(x,y) \\ v(x,y) \end{pmatrix} - \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x} & \frac{\partial
v}{\partial y}\\ \end{pmatrix} \begin{pmatrix} \xi \\ \eta \end{pmatrix} \right\Vert \rightarrow 0,\;\mbox{as } \rho\rightarrow 0.$$ 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 \begin{eqnarray*} f(z) &=& u(x,y)+i v(x,y),\\ f'(z) &=& \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x}\\ \zeta &=& \xi+i\eta \end{eqnarray*}we can therefore rewrite the above limit relation as $$ \left\vert
\frac{ f(z+\zeta)-f(z) - f'(z)\zeta}{ \zeta}\right\vert \rightarrow 0,\;\mbox{as } \rho\rightarrow 0,$$ which is the complex limit definition of $f'(z)$ shown in (![[*]](http://images.planetmath.org:8080/cache/objects/3282/js//usr/share/latex2html/icons/crossref.png "[*]") ).
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"proof of the Cauchy-Riemann equations" is owned by rmilson.
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complex derivative |
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Cross-references: complex numbers, equivalent, product, vector, scalar, similar, directional derivatives, implies, continuous at, fixed, equations, Cauchy-Riemann equations, imaginary parts, relation, imaginary, partial derivative, limit, real, complex, derivative
There are 8 references to this entry.
This is version 3 of proof of the Cauchy-Riemann equations, born on 2002-08-10, modified 2005-03-11.
Object id is 3282, canonical name is ProofOfTheCauchyRiemannEquations.
Accessed 15255 times total.
Classification:
| AMS MSC: | 30E99 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Miscellaneous) |
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Pending Errata and Addenda
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