# holomorphic mapping of curve and tangent

Let $D$ be a domain of the complex plane and the function $f:D\to \u2102$ be holomorphic. Then for each point $z$ of $D$ there is a corresponding point $w=f(z)\in \u2102$; we think that $z$ and $w$ both lie in their own complex planes, $z$-plane and $w$-plane.

Since $f$ is continuous^{} in $D$, if $z$ draws a continuous curve $\gamma $ in $D$ then its image point $w$ also draws a continuous curve ${\gamma}_{w}$. Let ${z}_{0}$ and ${z}_{0}+\mathrm{\Delta}z$ be two points on $\gamma $ and ${w}_{0}$ and
${w}_{0}+\mathrm{\Delta}w$ their image points on ${\gamma}_{w}$.

We suppose still that the curve $\gamma $ has a tangent line^{} at the point ${z}_{0}$ and that the value of the derivative^{} ${f}^{\prime}$ has in ${z}_{0}$ a nonzero value

${f}^{\prime}({z}_{0})=\varrho {e}^{i\omega}.$ | (1) |

If the slope angles of the secant lines^{} $({z}_{0},{z}_{0}+\mathrm{\Delta}z)$ and $({w}_{0},{w}_{0}+\mathrm{\Delta}w)$ are $\alpha $ and ${\alpha}_{w}$, then we have

$$\mathrm{\Delta}z=k{e}^{i\alpha},\mathrm{\Delta}w={k}_{w}{e}^{i{\alpha}_{w}},$$ |

and the difference quotient of $f$ has the form

$$\frac{\mathrm{\Delta}w}{\mathrm{\Delta}z}=\frac{f({z}_{0}+\mathrm{\Delta}z)-f({z}_{0})}{\mathrm{\Delta}z}=\frac{{k}_{w}}{k}{e}^{i({\alpha}_{w}-\alpha )}.$$ |

Let now $\mathrm{\Delta}z\to 0$. Then the point ${z}_{0}+\mathrm{\Delta}z$ tends on the curve $\gamma $ to ${z}_{0}$ and

$$\underset{\mathrm{\Delta}z\to 0}{lim}\frac{\mathrm{\Delta}w}{\mathrm{\Delta}z}={f}^{\prime}({z}_{0}).$$ |

This implies, by (1), that

$\underset{\mathrm{\Delta}z\to 0}{lim}{\displaystyle \frac{{k}_{w}}{k}}=\varrho .$ | (2) |

From this we infer, because $\varrho \ne 0$ that, up to a multiple of $2\pi $,

$\underset{\mathrm{\Delta}z\to 0}{lim}({\alpha}_{w}-\alpha )=\omega .$ | (3) |

But the limit of $\alpha $ is the slope angle $\phi $ of the tangent^{} of $\gamma $ at ${z}_{0}$. Hence (3) implies that

${\phi}_{w}=\underset{\mathrm{\Delta}z\to 0}{lim}{\alpha}_{w}=\phi +\omega .$ | (4) |

Accordingly, we have the

Theorem 1. If a curve $\gamma $ has a tangent line in a point ${z}_{0}$ where the derivative ${f}^{\prime}$ does not vanish, then the image curve $f(\gamma )$ also has in the corresponding point ${w}_{0}$ a certain tangent line with a direction obtained by rotating the tangent of $\gamma $ by the angle

$$\omega =\mathrm{arg}{f}^{\prime}({z}_{0}).$$ |

If the curve $\gamma $ is smooth, then also ${\gamma}_{w}$ is smooth, and it follows easily from (2) the corresponding limit equation between the arc lengths^{}:

$\underset{\mathrm{\Delta}z\to 0}{lim}{\displaystyle \frac{{s}_{w}}{s}}=|{f}^{\prime}({z}_{0})|.$ | (5) |

Conformality

If we have besides $\gamma $ another curve ${\gamma}^{\prime}$ emanating from ${z}_{0}$ with its tangent, the mapping $f$ from $D$ in $z$-plane to $w$-plane gives two curves and their tangents emanating from ${w}_{0}$. Thus we have two equations (4):

$${\phi}_{w}=\phi +\omega ,{\phi}_{w}^{\prime}={\phi}^{\prime}+\omega $$ |

By subtracting we obtain

${\phi}_{w}^{\prime}-{\phi}_{w}={\phi}^{\prime}-\phi ,$ | (6) |

whence we have the

Theorem 2. The mapping created by the holomorphic function $f$ preserves the magnitude of the angle between two curves in any point $z$ where ${f}^{\prime}(z)\ne 0$. The equation (6) tells also that the orientation of the angle is preserved.

The facts in Theorem 2 are expressed so that the mapping is directly conformal. If the orientation were reversed the mapping were called inversely conformal; in this case $f$ were not holomorphic but antiholomorphic.

Title | holomorphic mapping of curve and tangent |
---|---|

Canonical name | HolomorphicMappingOfCurveAndTangent |

Date of creation | 2013-03-22 18:42:19 |

Last modified on | 2013-03-22 18:42:19 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Topic |

Classification | msc 53A30 |

Classification | msc 30E20 |

Defines | directly conformal |