Conformal Mappings :
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Theorem: If $f$ is analytic at the point $z_{0}$ and if MATH then $f$ is conformal at $z_{0}$.

Corollary: If $f$ is both analytic and one-to-one on a domain $D$, then $f$ is conformal at all points of $D$.

Example 1: Since $f(z) = e^{z}$ is analytic in $\QTR{Bbb}{C}$ and MATH for all $z \in \QTR{Bbb}{C}$, the map $w = e^{z}$ is conformal at all points in $\QTR{Bbb}{C}$.

Example 2: The map $w = z^{2}$ is conformal at all non-zero complex points. Whereas, the map $w = z^{2}$ is not conformal at $z = 0$. To see this, the angle between the two rays $R_{1}:\theta = 0$ and MATH in the $z$-plane is $\pi /2$. Under the map $w = z^{2}$, the image of $R_{1}$ is given by MATH and the image of $R_{2}$ is given by MATH. The angle between $R_{1}^{\prime}$ and $R_{2}^{\prime}$ is $\pi$ and is not equal to the angle between $R_{1}$ and $R_{2}$ which is $\pi / 2$. Therefore, the map $w = z^{2}$ is not conformal at $z = 0$.

Example 3: The map $w = \overline{z}$ is not conformal at any point in $\QTR{Bbb}{C}$. The map $w = \overline{z}$ preserves the magnitude of the angle between the two curves, but does not preserve the sense in which the angle is measured. A map that preserves the magnitude, but not the sense of the angle between two curves is called an isogonal mapping.


 
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