3.7. Using the function $w=z^{2}$ to map the lines $x=3$ and $y=2$ onto the Ouv plane.
Solution. The function $w=z^{2}$ maps circles centered at $z=0$ to circles centered at $w=0$; lines passing through the point $z=0$ - to lines passing through the point $w=0$, indeed:

$$
w=z^{2}=\rho^{2} e^{2 i \varphi} \Rightarrow\left[\begin{array}{l}
\rho=\rho_{0}=\text { const }, w=\rho_{0}^{2} e^{2 i(\rho},|w|=\text { const } \\
\varphi=\varphi_{0}=\text { const, } w=\rho^{2} e^{2\left(\varphi_{0}\right)}, \arg w=\text { const. }
\end{array}\right.
$$

Let's find out what the images of the lines $x=x_{0}$ and $y=y_{0}$ look like under the mapping $w$. Based on the rule for raising a complex number to a natural power, we conclude that:

$$
w^{2}=(x+i y)^{2}=x^{2}-y^{2}+i 2 x y=u+i v
$$

Let's write the parametric equations of the images of the lines $x=x_{0}$ and $y=y_{0}$, produced by the mapping $w$ on the Ouv plane:

$$
\begin{aligned}
& x=x_{0} \Leftrightarrow\left[\begin{array} { l l } 
{ x = x _ { 0 } } \\
{ y = t }
\end{array} \quad \stackrel { w } { \rightarrow } \left[\begin{array}{l}
u=x_{0}^{2}-t^{2} \\
v=2 x_{0} t
\end{array}\right.\right. \\
& y=y_{0} \Leftrightarrow\left[\begin{array} { l } 
{ x = t } \\
{ y = y _ { 0 } }
\end{array} \quad \stackrel { w } { u } \left[\begin{array}{l}
u=t^{2}-y_{0}^{2} \\
v=2 y_{0} t
\end{array}\right.\right.
\end{aligned}
$$

If we switch to an explicit representation, we get:

$$
\begin{aligned}
& x=x_{0} \xrightarrow{w} u=x_{0}^{2}-\frac{v^{2}}{4 x_{0}^{2}} ; \\
& y=y_{0} \xrightarrow{w} u=\frac{v^{2}}{4 y_{0}^{2}}-y_{0}^{2} .
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_05b7516b505cf6654012g-185.jpg?height=346&width=370&top_left_y=1355&top_left_x=161)

Fig. 3.6

Noting that $x_{0}=3, y_{0}=2$, we finally have:

$$
\begin{aligned}
& x=3 \xrightarrow{w} u=9-\frac{v^{2}}{36} \\
& y=2 \stackrel{w}{\rightarrow} u=\frac{v^{2}}{16}-4
\end{aligned}
$$

The images of the lines $x=3, y=2$ are shown in Fig. 3.6.

## 3.6 .2 . LIMIT OF A FUNCTION OF A COMPLEX VARIABLE

A number $A=\alpha+i \beta$ is called the limit of the function $f(z)$ at the point $z_{0}$ if for any number $\varepsilon>0$ there exists a $\delta(\varepsilon)>0$ such that for all values of $z$ in the domain of definition of $f(z)$, satisfying the condition $\left|z-z_{0}\right|<\delta(\varepsilon)$, the inequality $|f(z)-A|<\varepsilon$ holds. This definition of the limit of a function of a complex variable is analogous to the definition of the limit of a function of a real variable by Cauchy. An equivalent definition of the limit of a function of a complex variable $f(z)$, analogous to the definition of the limit of a function of one variable by Heine, can also be given: a number $A=\alpha+i \beta$ is called the limit of the function $f(z)$ at the point $z_{0}$ if for any sequence of points $\left\{z_{n}\right\}\left(z_{n} \neq z_{0}\right)$ converging to $z_{0}$, the corresponding sequence of function values $f\left(z_{n}\right)$ has a unique limit equal to $A$. This is written as follows:

$$
\lim _{z \rightarrow z_{0}} f(z)=A
$$

The limit of a function of a complex variable has the same properties as the limit of a function of one or several variables: if at the point $z_{0}$ the limits

$$
\lim _{z \rightarrow z_{0}} f(z)=A \text { and } \lim _{z \rightarrow z_{0}} g(z)=B
$$

exist, then at the point $z_{0}$ the limits of the functions $f(z) \pm g(z)$, $f(z) \cdot g(z), f(z) / g(z)(B \neq 0)$ also exist, and:

$$
\lim _{z \rightarrow z_{0}}(f(z) \pm g(z))=\lim _{z \rightarrow z_{0}} f(z) \pm \lim _{z \rightarrow z_{0}} g(z),
$$

$$
\begin{gathered}
\lim _{z \rightarrow z_{0}} f(z) g(z)=\lim _{z \rightarrow z_{0}} f(z) \cdot \lim _{z \rightarrow z_{0}} g(z), \\
\lim _{z \rightarrow z_{0}} \frac{f(z)}{g(z)}=\frac{\lim _{z \rightarrow z_{0}} f(z)}{\lim _{z \rightarrow z_{0}} g(z)} .
\end{gathered}
$$

In particular, since the limit of a constant $C$ is the constant $C$ itself, i.e., $\lim _{z \rightarrow z_{0}} C=C$, it follows from (3.15) that for $g(z)$, $\lim _{z \rightarrow z_{0}} C f(z)=C \cdot \lim _{z \rightarrow z_{0}} f(z)$.

For the function $f(z)=u(x, y)+i v(x, y)$ to have a limit equal to $A=\alpha+i \beta$ at the point $z_{0}=x_{0}+i y_{0}$, it is necessary and sufficient that the limits exist at this point:

$$
\begin{aligned}
& \lim _{\substack{x \rightarrow x_{0} \\
y \rightarrow y_{0}}} u(x, y)=\alpha, \\
& \lim _{\substack{x \rightarrow x_{0} \\
y \rightarrow y_{0}}} v(x, y)=\beta .
\end{aligned}
$$

Let's recall that, as in the case of functions of several variables, the limit of a function of a complex variable exists only if the result of the limiting transition in (3.13), and thus in (3.17), (3.18), does not depend on the path (method) of the limiting transition to the point $z_{0}$, and for (3.17), (3.18) to the point $M_{0}\left(x_{0}, y_{0}\right)$.

## Examples.