Example 7. Find the coefficient of stretching and the angle of rotation for the mapping $w=z^{2}$ at the point $z_{0}=\sqrt{2}+i \sqrt{2}$.
Solution. We have $w^{\prime}(z)=2 z$, so $\left.w^{\prime}\right|_{z=\sqrt{2}+i \sqrt{2}}=2 \sqrt{2}+i 2 \sqrt{2}$.

Transitioning from the algebraic form of the complex number $2 \sqrt{2}+i 2 \sqrt{2}$ to the trigonometric form, we get

$$
2 \sqrt{2}+i 2 \sqrt{2}=4\left(\frac{\sqrt{2}}{2}+i \frac{\sqrt{2}}{2}\right)=4\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)
$$

Thus,

$$
\left|f^{\prime}(z)\right|_{z=\sqrt{2}+i \sqrt{2}}=4,\left.\quad \arg f^{\prime}(z)\right|_{z=\sqrt{2}+i \sqrt{2}}=\frac{\pi}{4}
$$

i.e., the coefficient of dilation $r=4$, and the angle of rotation $\varphi=\frac{\pi}{4}$.

## Problems for Independent Solution

Find the coefficient of dilation $r$ and the angle of rotation $\varphi$ for the given mappings $\boldsymbol{w}=f(z)$ at the specified points:

132. a) $w=e^{z}$ at points $z_{1}=\ln 2+i \frac{\pi}{4}$ and $z_{2}=-1-i \frac{\pi}{2} ;$ b) $w=\sin z$ at points $z_{1}=0$ and $z_{2}=1+i ;$ c) $w=z^{3}$ at points $z_{1}=2-i$ and $z_{2}=1+i \frac{\pi}{2}$.
133. Determine which part of the complex plane is stretched and which is compressed for the following mappings:
a) $w=e^{z}$
b) $w=\ln z$;
c) $w=\frac{1}{z}$
d) $w=z^{3}$

If the function $w=f(z)$ is analytic in some domain $D$, and maps this domain bijectively onto the domain $\widetilde{D}$, then a curve $L$ lying in the domain $D$ will be mapped to some curve $\widetilde{L}$ in the $w$-plane, with the length of the curve given by

$$
l_{w}=\int_{L}\left|f^{\prime}(z)\right| \cdot|d z|
$$

The domain $D$ in the $z$-plane, under the mapping $w=f(z)$, transitions to the domain $\widetilde{D}$ in the $w$-plane, with the area of the domain $\widetilde{D}$ expressed by the formula

$$
S_{\tilde{D}}=\iint_{D}\left|f^{\prime}(z)\right|^{2} d x d y
$$

Thus, $\left|f^{\prime}(z)\right|^{2}$ is the coefficient of area distortion under the mapping $w=f(z)$.