Example 8. Find the image of the circle $|z|=3$ under the mapping $w=\frac{25}{z}$.
Solution. First method. Let $z=x+i y, w=u+i v$. Then the relation $w=\frac{25}{z}$ can be rewritten as

$$
u+i \dot{v}=\frac{25}{x+i y}=\frac{25 x}{x^{2}+y^{2}}-i \frac{25 y}{x^{2}+y^{2}}
$$

from which

$$
u=\frac{25 x}{x^{2}+y^{2}}, \quad v=-\frac{25 y}{x^{2}+y^{2}}
$$

The equation of the circle $|z|=3$ in Cartesian coordinates is written as

$$
x^{2}+y^{2}=9
$$

By eliminating $x$ and $y$ from (a) and (b), we get

$$
u^{2}+v^{2}=\left(\frac{25}{3}\right)^{2}
$$

i.e., a circle of radius $R=\frac{25}{3}$ centered at the origin in the $w$-plane.

Second method. Write $z$ and $w$ in exponential form

$$
z=\rho e^{i \varphi}, \quad w=r e^{i \theta}
$$

Then under the mapping $w=\frac{25}{z}$ we get

$$
r e^{i \theta}=\frac{25}{\rho e^{i \varphi}}
$$

from which $r=\frac{25}{\rho}, \theta=-\varphi$, where $\rho=3$ and $0 \leqslant \varphi < 2\pi$.

469. Find the images of the following sets under the mapping $w=\frac{1}{z}$:
a) arg $z=\frac{\pi}{3}$
b) $|z|=1, \frac{\pi}{4}<\arg z<\pi$;
c) $2 \leqslant x \leqslant 4, y=0$;
d) $-2<y<-1, x=0$
e) $0<\operatorname{Re} z<1$.

## 3. Fractional-linear function

$$
w=\frac{a z+b}{c z+d}
$$

where $a, b, c, d$ are complex constants and $a d-b c \neq 0$, maps the extended $z$-plane onto the extended $w$-plane in a one-to-one and conformal manner. The transformation performed by a fractional-linear function is called a fractional-linear transformation. Each fractional-linear transformation can be obtained by a sequence of linear transformations and a transformation of the form $w=\frac{1}{z}$.