|
|
|
|
holomorphic mapping of curve and tangent
|
(Topic)
|
|
|
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 $f'$ has in $z_0$ a nonzero value
 |
(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 $$\Delta z \,=\, ke^{i\alpha}, \quad \Delta w \,=\, k_we^{i\alpha_w},$$ 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 $$\lim_{\Delta z \to 0}\frac{\Delta w}{\Delta z} \;=\; f'(z_0).$$ This implies, by (1), that
 |
(2) |
From this we infer, because $\varrho \neq 0$ that, up to a multiple of $2\pi$ ,
 |
(3) |
But the limit of $\alpha$ is the slope angle $\varphi$ of the tangent of $\gamma$ at $z_0$ . Hence (3) implies that
 |
(4) |
Accordingly, we have the
Theorem 1. If a curve $\gamma$ has a tangent line in a point $z_0$ where the derivative $f'$ 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 $$\omega \;=\; \arg f'(z_0).$$
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:
 |
(5) |
Conformality
If we have besides $\gamma$ another curve $\gamma'$ 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): $$\varphi_w \;=\; \varphi+\omega, \quad \varphi_w' \;=\; \varphi'+\omega$$ By subtracting we obtain
 |
(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 $f'(z) \neq 0$ . 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.
|
"holomorphic mapping of curve and tangent" is owned by pahio.
|
|
(view preamble | get metadata)
| Also defines: |
directly conformal |
| Keywords: |
conformal mapping |
This object's parent.
|
|
Cross-references: antiholomorphic, inversely conformal, orientation, angle between two curves, preserves, mapping, arc lengths, equation, smooth, angle, vanish, theorem, tangent, limit, multiple, implies, difference quotient, secant lines, slope angles, derivative, tangent line, image, curve, continuous, point, holomorphic, function, complex plane, domain
This is version 6 of holomorphic mapping of curve and tangent, born on 2009-01-06, modified 2009-01-07.
Object id is 11469, canonical name is HolomorphicMappingOfCurveAndTangent.
Accessed 365 times total.
Classification:
| AMS MSC: | 30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions) | | | 53A30 (Differential geometry :: Classical differential geometry :: Conformal differential geometry) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|