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# holomorphic mapping of curve and tangent

Let $D$ be a domain of the complex plane and the function $f\!:\,D\to\mathbb{C}$ be holomorphic. Then for each point $z$ of $D$ there is a corresponding point $w=f(z)\,\in\mathbb{C}$; 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 $z_{0}\!+\!\Delta z$ be two points on $\gamma$ and $w_{0}$ and $w_{0}\!+\!\Delta w$ 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

$\displaystyle f^{{\prime}}(z_{0})\,=\,\varrho e^{{i\omega}}.$ | (1) |

If the slope angles of the secant lines $(z_{0},\,z_{0}\!+\!\Delta z)$ and $(w_{0},\,w_{0}\!+\!\Delta w)$ are $\alpha$ and $\alpha_{w}$, then we have

$\Delta z\,=\,ke^{{i\alpha}},\quad\Delta w\,=\,k_{w}e^{{i\alpha_{w}}},$ |

and the difference quotient of $f$ has the form

$\frac{\Delta w}{\Delta z}\;=\;\frac{f(z_{0}\!+\!\Delta z)-f(z_{0})}{\Delta z}% \,=\,\frac{k_{w}}{k}e^{{i(\alpha_{w}-\alpha)}}.$ |

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

$\lim_{{\Delta z\to 0}}\frac{\Delta w}{\Delta z}\;=\;f^{{\prime}}(z_{0}).$ |

This implies, by (1), that

$\displaystyle\lim_{{\Delta z\to 0}}\frac{k_{w}}{k}\;=\;\varrho.$ | (2) |

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

$\displaystyle\lim_{{\Delta z\to 0}}(\alpha_{w}-\alpha)\;=\;\omega.$ | (3) |

But the limit of $\alpha$ is the slope angle $\varphi$ of the tangent of $\gamma$ at $z_{0}$. Hence (3) implies that

$\displaystyle\varphi_{w}\;=\;\lim_{{\Delta z\to 0}}\alpha_{w}\;=\;\varphi+\omega.$ | (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

$\omega\;=\;\arg f^{{\prime}}(z_{0}).$ |

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:

$\displaystyle\lim_{{\Delta z\to 0}}\frac{s_{w}}{s}\;=\;|f^{{\prime}}(z_{0})|.$ | (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):

$\varphi_{w}\;=\;\varphi+\omega,\quad\varphi_{w}^{{\prime}}\;=\;\varphi^{{% \prime}}+\omega$ |

By subtracting we obtain

$\displaystyle\varphi_{w}^{{\prime}}-\varphi_{w}\;=\;\varphi^{{\prime}}-\varphi,$ | (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 $f^{{\prime}}(z)\neq 0$. 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.

## Mathematics Subject Classification

53A30*no label found*30E20

*no label found*

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