Example 7. Find the linear function that maps the triangle with vertices at points $0,1, i$ in the $z$-plane to a similar triangle with vertices $1+i, 0,2$ in the $w$-plane.
Solution. First method. From Fig. 16, we see that $\triangle ABC$ transforms into a similar $\triangle A_{1} B_{1} C_{1}$ through the following operations:

1) rotation around the origin by an angle $\frac{5}{4} \pi$, which corresponds to the transformation

$$
w_{1}=e^{\frac{5}{4} \pi} z
$$

2) similarity transformation with the center at the origin and coefficient $r=\sqrt{2}$ (since $\frac{A_{1} B_{1}}{A B}=\sqrt{2}$):

$$
\boldsymbol{w}_{2}=\sqrt{2} \boldsymbol{w}_{1} ;
$$

3) parallel translation, shifting point $C(0,0)$ to point $C_{1}(1,1)$ (we get $b=1+i)$:

$$
\boldsymbol{w}=\boldsymbol{w}_{2}+1+\boldsymbol{i}
$$

Considering that $e^{i 5 \pi / 4}=-\frac{\sqrt{2}}{2}-i \frac{\sqrt{2}}{2}$, we finally obtain

$$
w=\sqrt{2}\left(-\frac{\sqrt{2}}{2}-i \frac{\sqrt{2}}{2}\right) z+1+i=(1-z)(1+i)
$$

Second method. Let the desired function be $w=a z+b$, where $a$ and $b$ are as yet undetermined constants. According to the problem, points $z_{1}=0$ and $z_{2}=1$ should transform into points $w_{1}=1+i$ and $w_{2}=0$, respectively. We obtain the system of equations to determine $a$ and $b$:

$$
\left\{\begin{array}{l}
1+i=b \\
0=a+b
\end{array}\right.
$$

From this, $a=-1-i, b=1+i$, and thus,

$$
w=(1+i)(1-z)
$$

## Problems for Independent Solution

464. Indicate the geometric meaning (translation, scaling, rotation) of the following transformations:
a) $w=z+3 i$
b) $w=z+5$
c) $w=i z$
d) $\boldsymbol{w}=e^{i \pi / 6} z$
e) $w=3 z$
f) $w=\frac{1-i}{\sqrt{2}} z$
465. Find the general form of linear functions that perform the following transformations:

a) the upper half-plane onto itself;

b) the upper half-plane onto the lower half-plane;

c) the upper half-plane onto the right half-plane.

466. Find the linear mapping $w=a z+b$ that leaves the point $z_{0}$ fixed and maps the point $z_{1}$ to the point $w_{1}$:
a) $z_{0}=1-i, \quad z_{1}=2+i, \quad w_{1}=4-3 i$;

b) $z_{0}=-i, \quad z_{1}=1-2 i, \quad w_{1}=2-3 i$

c) $z_{0}=-1-i, z_{1}=3-2 i, \quad w_{1}=3 i$.

467. Find the linear function $w=f(z)$ that maps the strip between the lines $x=\dot{a}, x=a+h$ to the strip $0<u<1$ in the $w$-plane.

## 2. Function

$$
w=\frac{1}{z}
$$

Points $M$ and $M^{\prime}$ are called symmetric with respect to the circle $\Gamma$ if

1) they lie on the same ray emanating from the center of the circle;
2) the product of their distances from the center of the circle is equal to the square of the radius of the circle: $O M \cdot O M^{\prime}=R^{2}$ (Fig. 17).

![](https://cdn.mathpix.com/cropped/2024_05_22_f7d63c3a5e94c3f2f1bbg-126.jpg?height=366&width=447&top_left_y=327&top_left_x=170)

Fig. 17

![](https://cdn.mathpix.com/cropped/2024_05_22_f7d63c3a5e94c3f2f1bbg-126.jpg?height=551&width=479&top_left_y=142&top_left_x=674)

Fig. 18

Remark. Points on the circle $\Gamma$ are symmetric to themselves with respect to this circle.

For the center $O$ of the circle $\Gamma$, the symmetric point with respect to $\Gamma$ is the point at infinity.

If the center of the circle $\Gamma$ is at the origin and one of the points symmetric with respect to $\Gamma$ represents the complex number $z$, then the other corresponds to the complex number $\frac{\boldsymbol{R}^{2}}{\bar{z}}$.

The transformation $w=\frac{1}{z}$ consists of two symmetric reflections: with respect to the unit circle and with respect to the real axis (Fig. 18) and is called inversion.

The transformation $w=\frac{1}{z}$ is conformal in the entire extended plane, with the point $z=0$ corresponding to the point $w=\infty$, and the point $z=\infty$ corresponding to the point $w=0$. (It is assumed that the angle between lines at the infinitely distant point in one of the planes ($z$ or $w$) is equal to the angle between the images of these lines at the origin in the other plane). Circles (as well as lines) under the mapping $w=\frac{1}{z}$ transform into circles or lines. The fixed points are $z_{1}=+1$ and $z_{2}=-1$.