# proof of the Cauchy-Riemann equations

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

Suppose that the complex
derivative^{}

$${f}^{\prime}(z)=\underset{\zeta \to 0}{lim}\frac{f(z+\zeta )-f(z)}{\zeta}$$ | (1) |

exists for some $z\in \u2102$. This means that for all $\u03f5>0$, there exists a $\rho >0$, such that for all complex $\zeta $ with $$, we have

$$ |

Henceforth, set

$$f=u+iv,z=x+iy.$$ |

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

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

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

$$\frac{u(x+\xi ,y+\eta )-u(x,y)-(\xi \frac{\partial u}{\partial x}+\eta \frac{\partial u}{\partial y})}{\rho}\to 0,\text{as}\rho \to 0,$$ |

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

$${\rho}^{-1}\parallel \left(\begin{array}{c}\hfill u(x+\xi ,y+\eta )\hfill \\ \hfill v(x+\xi ,y+\eta )\hfill \end{array}\right)-\left(\begin{array}{c}\hfill u(x,y)\hfill \\ \hfill v(x,y)\hfill \end{array}\right)-\left(\begin{array}{cc}\hfill \frac{\partial u}{\partial x}\hfill & \hfill \frac{\partial u}{\partial y}\hfill \\ \hfill \frac{\partial v}{\partial x}\hfill & \hfill \frac{\partial v}{\partial y}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \xi \hfill \\ \hfill \eta \hfill \end{array}\right)\parallel \to 0,\text{as}\rho \to 0.$$ |

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

$f(z)$ | $=$ | $u(x,y)+iv(x,y),$ | ||

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

$\zeta $ | $=$ | $\xi +i\eta $ |

we can therefore rewrite the above limit relation as

$$\left|\frac{f(z+\zeta )-f(z)-{f}^{\prime}(z)\zeta}{\zeta}\right|\to 0,\text{as}\rho \to 0,$$ |

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

Title | proof of the Cauchy-Riemann equations |
---|---|

Canonical name | ProofOfTheCauchyRiemannEquations |

Date of creation | 2013-03-22 12:55:39 |

Last modified on | 2013-03-22 12:55:39 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 6 |

Author | rmilson (146) |

Entry type | Proof |

Classification | msc 30E99 |

Defines | complex derivative |