The Geometric Meaning of the Derivative

Let a ray $\left\{ z: \arg \left(z-z_{0}\right) = \varphi \right\}$ emanating from the point $z_{0}$ be given. Find the coefficient of linear stretching $R(\varphi )$ at the point $z_{0}$ and the angle of rotation $\alpha (\varphi )$ at the point $z_{0}$ for this ray under the following mappings:

Find the sets of all points $z_{0}$ at which the coefficient of linear stretching equals one for the following mappings:

Find the set of all points $z_{0}$ at which the angle of rotation is zero for the following mappings:

Let the function $w(z)$ be regular at the point $z_{0}$, and let the smooth curves $\Gamma_{1}$ and $\Gamma_{2}$, passing through the point $z_{0}$, have the property that

Prove that if $w^{\prime }\left(z_{0}\right) \neq 0$, then the curves $\Gamma_{1}$ and $\Gamma_{2}$ intersect at a right angle.

Let the function $w(z)$ be regular at the point $z_{0}$, and let the smooth curves $\Gamma_{1}$ and $\Gamma_{2}$, passing through the point $z_{0}$, have the property that

Prove that if $w^{\prime }\left(z_{0}\right) \neq 0$, then the curves $\Gamma_{1}$ and $\Gamma_{2}$, upon intersecting at the point $z_{0}$, form angles $\pm \arg w\left(z_{0}\right)+k \pi$.

Find the lengths of the images of the following curves Г under the indicated mappings:

Find the areas of the images of the domains $D$ under the indicated mappings:

Find the region onto which the function $w = e^{z}$ maps the rectangle

Compute the area of the region $w(D)$ using formula (4) and explain why this formula gives an incorrect answer.

Formula (4): if a function $f$ is regular and univalent in a region $D$ and maps it onto a region $G = f(D)$, then the area of $G$ equals $S(G) = \iint_{D}\left|f^{\prime }(z)\right|^{2} \, dx \, dy$.

Let

Denote by $L(r)$ the length of the image of the circle $\left\{ z:\left|z\right| = r\right\}$ under the mapping $w = P(z)$, and by $S(r)$ the area of the image of the disk $\left\{ z:\left|z\right| < r\right\}$ under the same mapping. Prove that:

Let $Q(z) = z+\sum_{k = 1}^{n} c_{k} z^{-k}$. Find the area of the image of the annulus $\left\{ z: r < \left|z\right| < R\right\}$ under the mapping $w = Q(z)$, counting the area of each elementary patch centered at the point $w_{0}$ as many times as the function $Q(z)$ takes the value $w_{0}$ in the annulus $\left\{ z: r < \left|z\right| < R\right\}$.

Let the function $w(z)$ be regular in the domain $D$, and let its mapping $w = w(z)$ preserve the Euclidean distance between points, i.e., for any pair of points $z_{1} \in D$ and $z_{2} \in D$ the equality holds

Prove that $w(z) = e^{i \varphi } z+a$, where $\varphi$ is a real and $a$ a complex constant.