| Version 4 |
Version 3 |
| 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 |
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. |
$D$ 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. |
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| 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 |
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 $\gamma_w$. |
$w_0\!+\!\Delta w$ their image points on $\gamma_w$. |
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\rput(-2.75,1.96){$z_0\!+\!\Delta z$} |
| \rput(-3.76,0.6){$\Delta s$} |
\rput(-3.76,0.6){$\Delta s$} |
| \rput(-4.7,1.1){$k$} |
\rput(-4.7,1.1){$k$} |
| \rput(-2.2,-1.5){$(\varphi)$} |
\rput(-2.2,-1.5){$(\varphi)$} |
| \rput(-2.75,3.6){$\gamma$} |
\rput(-2.75,3.6){$\gamma$} |
| \rput(-4.5,-2.5){$z$-plane} |
\rput(-4.5,-2.5){$z$-plane} |
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| \rput(0.5,1.3){$k_w$} |
\rput(0.5,1.3){$k_w$} |
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| 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 |
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 |
| \begin{align} |
\begin{align} |
| f'(z_0) \,=\, \varrho e^{i\omega}. |
f'(z_0) \,=\, \varrho e^{i\omega}. |
| \end{align} |
\end{align} |
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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
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If the arguments 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
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| $$\Delta z \,=\, ke^{i\alpha}, \quad \Delta w \,=\, k_we^{i\alpha_w},$$ |
$$\Delta z \,=\, ke^{i\alpha}, \quad \Delta w \,=\, k_we^{i\alpha_w},$$ |
| and the difference quotient of $f$ has the form |
and the difference quotient of $f$ has the form |
| $$\frac{\Delta w}{\Delta z} \;=\; |
$$\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)}.$$ |
\frac{f(z_0\!+\!\Delta z)-f(z_0)}{\Delta z} \,=\, \frac{k_w}{k}e^{i(\alpha_w-\alpha)}.$$ |
| Let now\, $\Delta z \to 0$.\, Then the point $z_0\!+\!\Delta z$ tends on the curve $\gamma$ to $z_0$ and |
Let now\, $\Delta z \to 0$.\, Then the point $z_0\!+\!\Delta z$ tends on the curve $\gamma$ to $z_0$ and |
| $$\lim_{\Delta z \to 0}\frac{\Delta w}{\Delta z} \;=\; f'(z_0).$$ |
$$\lim_{\Delta z \to 0}\frac{\Delta w}{\Delta z} \;=\; f'(z_0).$$ |
| This implies, by (1), that |
This implies, by (1), that |
| \begin{align} |
\begin{align} |
| \lim_{\Delta z \to 0}\frac{k_w}{k} \;=\; \varrho. |
\lim_{\Delta z \to 0}\frac{k_w}{k} \;=\; \varrho. |
| \end{align} |
\end{align} |
| From this we infer, because\, $\varrho \neq 0$\, that, up to a multiple of $2\pi$, |
From this we infer, because\, $\varrho \neq 0$\, that, up to a multiple of $2\pi$, |
| \begin{align} |
\begin{align} |
| \lim_{\Delta z \to 0}(\alpha_w-\alpha) \;=\; \omega. |
\lim_{\Delta z \to 0}(\alpha_w-\alpha) \;=\; \omega. |
| \end{align} |
\end{align} |
| But the limit of $\alpha$ is the slope angle $\varphi$ of the tangent of $\gamma$ at $z_0$.\, Hence (3) implies that |
But the limit of $\alpha$ is the slope angle $\varphi$ of the tangent of $\gamma$ at $z_0$.\, Hence (3) implies that |
| \begin{align} |
\begin{align} |
| \varphi_w \;=\; \lim_{\Delta z \to 0}\alpha_w \;=\; \varphi+\omega. |
\varphi_w \;=\; \lim_{\Delta z \to 0}\alpha_w \;=\; \varphi+\omega. |
| \end{align} |
\end{align} |
| Accordingly, we have the |
Accordingly, we have the |
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| \textbf{Theorem.}\, 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 |
\textbf{Theorem.}\, 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 |
| $$\omega \;=\; \arg f'(z_0).$$ |
$$\omega \;=\; \arg f'(z_0).$$ |
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| [Not ready...] |
[Not ready...] |
