Example 10. Find the number of roots of the equation

$$
\lambda-\boldsymbol{z}-e^{-z}=0, \quad \lambda>1
$$

in the right half-plane $\operatorname{Re} z>0$.
Solution. Consider the contour composed of the segment $[-i R, i R]$ and the right semicircle $|z|=R$. Let $f(z)=z-\lambda$ and $\varphi(z)=e^{-z}$. On the segment $[-i R, i R]$, where $z=i y$, we have

$$
\begin{aligned}
& |f(z)|=|i y-\lambda|=\sqrt{\lambda^{2}+y^{2}} \geqslant \sqrt{\lambda^{2}}=\lambda>1 \\
& |\varphi(z)|=\left|e^{-z}\right|=\left|e^{-i y}\right|=1
\end{aligned}
$$

and, consequently, $|f(z)|>|\varphi(z)|$.

On the semicircle $|z|=R$, where $\operatorname{Re} z=x>0$ for sufficiently large $R$ $(R>\lambda+1)$, we have $|f(z)|>|\varphi(z)|$, since

$$
\begin{aligned}
& |f(z)|=|z-\lambda| \geqslant|z|-\lambda=R-\lambda>1 \\
& |\varphi(z)|=\left|e^{-z}\right|=\left|e^{-x-i y}\right|=\left|e^{-x} e^{-i y}\right|=e^{-x}\left|e^{-i y}\right|=e^{-x} \leqslant 1 \quad(x>0)
\end{aligned}
$$

By Rouché's theorem, inside the specified contour for any sufficiently large $R$, the given equation has as many roots as the equation $f(z)=z-\lambda=0$, i.e., one root. Therefore, in the entire right half-plane, the given equation has a unique root.

## Problems for Independent Solution

457. Show that the equation $z e^{\lambda-z}=1$, where $\lambda>1$, has a unique real and positive root in the unit disk $|z| \leqslant 1$.
458. Show that the equation $1+z+\alpha z^{n}=0$, where $n$ is a natural number greater than one, has at least one root in the disk $|z| \leqslant 2$ for any $\alpha$.
459. Let $f(z)$ and $\varphi(z)$ be functions that are analytic in some neighborhood of the point $a$, and $C$ be a circle centered at the point $a$ such that along the circumference of this circle we have

$$
|\alpha f(z)|+|\beta \varphi(z)|<r .
$$

Show that the equation $F(z)=z-a-\alpha f(z)-\beta f(z)=0$ has one and only one root inside the circle $C$.

## CHAPTER

## Conformal Mappings

## § 13. Conformal Mappings

## $1^{\circ}$. Concept of Conformal Mapping.

Definition. A mapping of a neighborhood of a point $z_{0}$ onto a neighborhood of a point $w_{0}$, realized by the function $w=f(z)$, is called conformal if at the point $z_{0}$ it has the property of preserving angles between curves and the constancy of dilations (Fig. 14).

This means that:

1) if under the mapping $w=f(z)$ the curves $\gamma_{1}$ and $\gamma_{2}$ are transformed into the curves $\Gamma_{1}$ and $\Gamma_{2}$, respectively, then the angle $\varphi$ between the tangents $k_{1}$ and $k_{2}$ to the curves $\gamma_{1}$ and $\gamma_{2}$ at the point $z_{0}$ will be equal to the angle $\Phi$ between the corresponding tangents $K_{1}$ and $K_{2}$ to the curves $\Gamma_{1}$ and $\Gamma_{2}$ at the point $w_{0}$, i.e., $\Phi=\varphi$.
2) if in the complex plane $z$ we take an infinitesimally small circle centered at the point $z_{0}$, then in the plane $w$ it will correspond to an infinitesimally small circle centered at the point $w_{0}$.

Therefore, it is said that a conformal mapping has the property of angle conservation and similarity in the small.

If under the mapping $w=f(z)$ the angles between corresponding directions are equal not only in magnitude but also in the direction of measurement, then such a mapping is called a conformal mapping of the first kind.

A conformal mapping in which angles are preserved only in absolute magnitude but the direction of their measurement is reversed is called a conformal mapping of the second kind.

A simple example of a conformal mapping of the first kind is the mapping $w=z$, and of the second kind is the mapping $w=\bar{z}$.

In the following, we will consider only conformal mappings of the first kind.

A mapping $w=f(z)$ is called conformal in a domain $D$ if it is conformal at every point of this domain.

Criterion for Conformality. For the mapping $w=f(z)$ to be conformal in a domain $D$, it is necessary and sufficient that in this domain the function $w=f(z)$ is univalent and analytic, and $f^{\prime}(z) \neq 0$ for all $z \in D$.

If we do not assume the univalence of $f(z)$, then the mapping realized by this function will not be one-to-one, and thus will not be conformal. For example, the function $w=z^{4}$, defined in the half-ring $1 \leqslant|z| \leqslant 2,0 \leqslant \arg z \leqslant \pi$, is analytic in it, and, moreover, the condition $w^{\prime}=4 z^{3} \neq 0$ is satisfied everywhere in the half-ring. However, the function $w=z^{4}$ maps the given half-ring onto the region $1 \leqslant|w| \leqslant 16,0 \leqslant \arg w \leqslant 4 \pi$, i.e., a region that twice covers the corresponding ring on the $w$-plane, which violates the one-to-one correspondence.

![](https://cdn.mathpix.com/cropped/2024_05_22_f7d63c3a5e94c3f2f1bbg-119.jpg?height=466&width=946&top_left_y=159&top_left_x=216)

Fig. 14