holomorphic mapping of curve and tangent

Let $D$ be a domain of the complex plane and the function  $f:D\to ℂ$  be holomorphic.  Then for each point $z$ of$D$ there is a corresponding point  $w=f(z)\in ℂ$;  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 $z_0+\mathrm{\Delta }z$ 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

$$\omega =\arg f^{\prime }(z_0).$$

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.