Example 7. Solve the equation $\sin z=3$.
Solution. The problem reduces to finding the value of

$$
z=\operatorname{Arcsin} 3
$$

We will use formula (7):

$$
\operatorname{Arcsin} t=-i \operatorname{Ln}\left(t+\sqrt{1-t^{2}}\right)
$$

We will have

$$
z=\operatorname{Arcsin} 3=-i \operatorname{Ln}(3 i+\sqrt{-8})
$$

or, considering that

$$
\sqrt{-8}= \pm \sqrt{8} i
$$

we get

$$
z=-i \operatorname{Ln}[(3+\sqrt{8}) i], \quad z=-i \operatorname{Ln}[(3-\sqrt{8}) i]
$$

Since

$$
\begin{gathered}
\arg [(3+\sqrt{8}) i]=\arg [(3-\sqrt{8}) i]=\frac{\pi}{2} \\
|(3+\sqrt{8}) i|=3+\sqrt{8}, \quad|(3-\sqrt{8}) i|=3-\sqrt{8}
\end{gathered}
$$

then

$$
\operatorname{Ln}[(3 \pm \sqrt{8}) i]=\ln (3 \pm \sqrt{8})+\frac{\pi}{2} i+2 k \pi i
$$

where $k=0, \pm 1, \pm 2, \ldots$ Therefore,

$$
z=\frac{\pi}{2}+2 k \pi-i \ln (3 \pm \sqrt{8}) \quad(k=0, \pm 1, \pm 2, \ldots)
$$

## Problems for Independent Solution

Solve the following equations:

73. $e^{-z}+1=0$.
74. $e^{z}+i=0$.
75. $4 \cos z+5=0$.
76. $\operatorname{sh} i z=-i, \quad$ 77. $\sin z=\pi i$.
77. $e^{i x}=\cos \pi x(x-$ real).
78. $e^{2 z}+2 e^{z}-3=0$.
79. $\operatorname{ch} z=i$.
80. a) $\ln (z+i)=0$;

b) $\ln (i-z)=1$.

## § 3. Limit of a Sequence of Complex Numbers. Limit and Continuity of a Function of a Complex Variable

Let a sequence $\left\{z_{n}\right\}$ of complex numbers be given

$$
z_{1}, z_{2}, \ldots, z_{n}, \ldots
$$

Definition 1. A complex number $a$ is called the limit of the sequence $\left\{z_{n}\right\}$ if for any positive number $\varepsilon$ there exists a number $N=N(\varepsilon)$ such that starting from this number all elements $z_{n}$ of the sequence satisfy the inequality

$$
\left|z_{n}-a\right|<\varepsilon \text { for } n \geqslant N(\varepsilon)
$$

A sequence $\left\{z_{n}\right\}$ that has a limit $a$ is called convergent to the number $a$, which is written as $\lim _{n \rightarrow \infty} z_{n}=a$.

To each sequence of complex numbers $\left\{z_{n}\right\}$ there correspond two sequences of real numbers $\left\{x_{n}\right\}$ and $\left\{y_{n}\right\}$, where $z_{n}=x_{n}+i y_{n}, n=1,2, \ldots$.

Theorem 1. A sequence $\left\{z_{n}=x_{n}+i y_{n}\right\}$ converges to the number $a=\alpha+i \beta$ if and only if

$$
\lim _{n \rightarrow \infty} x_{n}=\alpha, \quad \lim _{n \rightarrow \infty} y_{n}=\beta
$$

Definition 2. A sequence $\left\{z_{n}\right\}$ is called bounded if there exists a positive number $M$ such that for all elements $z_{n}$ of the sequence the inequality $\left|z_{n}\right| \leqslant M$ holds.

Theorem 2. Every convergent sequence is bounded.

Properties of convergent sequences of complex numbers

If $\lim _{n \rightarrow \infty} z_{n}=a$ and $\lim _{n \rightarrow \infty} \tau_{n}=b$, then

1. $\lim \left(z_{n} \pm \tau_{n}\right)=a \pm b$

$n \rightarrow \infty$
2. $\lim _{n \rightarrow \infty}\left(z_{n} \tau_{n}\right)={ }^{\prime} a b ;$
3. $\lim _{n \rightarrow \infty} \frac{z_{n}}{\tau_{n}}=\frac{a}{b} \quad\left(\tau_{n} \neq 0, b \neq 0\right)$.