Example 3. Find the image of the circle

$$
z=R \cos t+i R \sin t \quad(0 \leqslant t<2 \pi)
$$

under the mapping $\boldsymbol{w}=\frac{\boldsymbol{z}}{\bar{z}}$.
Solution. Let $z=x+i y$. The given equation of the circle can be written as

$$
x=R \cos t, \quad y=R \sin t \quad(0 \leqslant t<2 \pi) .
$$

We separate the real and imaginary parts of the function $w=u+i v$. We have

$$
u+i v=\frac{z}{\bar{z}}=\frac{z^{2}}{z \bar{z}}=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}+i \frac{2 x y}{x^{2}+y^{2}} .
$$

Hence,

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

Substituting $x=R \cos t$ and $\boldsymbol{y}=\boldsymbol{R} \sin t$ into $u$ and $\boldsymbol{v}$, we obtain the parametric equations of the image of the circle

$$
\left\{\begin{array}{l}
u=\frac{\cos ^{2} t-\sin ^{2} t}{\cos ^{2} t+\sin ^{2} t}=\cos 2 t \\
v=\frac{2 \cos t \sin t}{\cos ^{2} t+\sin ^{2} t}=\sin 2 t
\end{array}\right.
$$

or $\boldsymbol{u}^{2}+v^{2}=1$.

Thus, the image is the unit circle, traversed twice, which follows from the fact that $0 \leqslant t<2 \pi$, and from the formulas (*).

## Problems for Independent Solution

57. Determine the lines in the $w$-plane to which the following lines in the $z$-plane are mapped by the function $w=\frac{1}{z}$:
a) $|z|=\frac{1}{2}$
b) $\operatorname{Re} z=0$
c) $\arg z=\frac{3 \pi}{4}$
d) $\arg z^{2}=-\frac{\pi}{2}$
e) $\operatorname{Re} z=\operatorname{Im} z$
f) $|z|=z$.
58. Find the images of the coordinate axes $O X$ and $O Y$ under the following mappings:
a) $w=\frac{z+1}{z-1}$
b) $w=1+\frac{1}{z}$

## Basic Elementary Functions of a Complex Variable

1. Rational function

$$
w=\frac{a_{0} z^{n}+a_{1} z^{n-1}+\ldots+a_{n}}{b_{0} z^{m}+b_{1} z^{m-1}+\ldots+b_{m}}
$$

in particular, a rational function is a polynomial

$$
w=a_{0} z^{n}+a_{1} z^{n-1}+\ldots+a_{n}
$$

2. Exponential function $e^{z}$ is defined as the sum of an absolutely convergent power series in the entire complex plane

$$
e^{z}=1+z+\frac{z^{2}}{2!}+\ldots+\frac{z^{n}}{n!}+\ldots
$$

The exponential function $e^{z}$ has the following properties:

a) $e^{z_{1}+z_{2}}=e^{z_{1}} \cdot e^{z_{2}}$, where $z_{1}$ and $z_{2}$ are any complex numbers;

b) $e^{z+2 k \pi i}=e^{z}(k=0, \pm 1, \pm 2, \ldots)$, i.e., $e^{z}$ is a periodic function with period $2 \pi i$,

3. Trigonometric functions $\sin z$ and $\cos z$ are defined by power series:

$$
\begin{aligned}
& \sin z=z-\frac{z^{3}}{3!}+\ldots+(-1)^{n} \frac{z^{2 n+1}}{(2 n+1)!}+\ldots \\
& \cos z=1-\frac{z^{2}}{2!}+\frac{z^{4}}{4!}-\ldots+(-1)^{n} \frac{z^{2 n}}{(2 n)!}+\ldots
\end{aligned}
$$

absolutely convergent for any complex value of $z$. The functions $\sin z$ and $\cos z$ are periodic with real period $2 \pi$ and have only real zeros $z=k \pi$ and $z=\frac{\pi}{2}+k \pi$ respectively, where $k=0, \pm 1, \pm 2, \ldots$

For the functions $e^{z}, \sin z$ and $\cos z$, the Euler formulas hold

$$
e^{i z}=\cos z+i \sin z, e^{-i z}=\cos z-i \sin z
$$

from which

$$
\cos z=\frac{e^{i z}+e^{-i z}}{2}, \quad \sin z=\frac{e^{i z}-e^{-i z}}{2 i}
$$

The functions $\operatorname{tg} z$ and $\operatorname{ctg} z$ are defined by the equations

$$
\operatorname{tg} z=\frac{\sin z}{\cos z} \quad \operatorname{ctg} z=\frac{\cos z}{\sin z}
$$

For trigonometric functions, all trigonometric formulas remain valid.

4. Hyperbolic functions $\operatorname{sh} z$, ch $z$, th $z$, cth $z$ are defined by the equations

$$
\begin{array}{ll}
\operatorname{sh} z=\frac{e^{z}-e^{-z}}{2}, & \operatorname{ch} z=\frac{e^{z}+e^{-z}}{2} \\
\operatorname{th} z=\frac{\operatorname{sh} z}{\operatorname{ch} z}, & \operatorname{cth} z=\frac{\operatorname{ch} z}{\operatorname{sh} z}
\end{array}
$$

5. Trigonometric and hyperbolic functions are related by the following relations:

$$
\begin{array}{rlrl}
\sin z & =-i \operatorname{sh} i z, \quad \operatorname{sh} z & =-i \sin i z \\
\cos z & =\operatorname{ch} i z, & \operatorname{ch} z & =\cos i z \\
\operatorname{tg} z & =-i \operatorname{th} i z, \quad \operatorname{th} z & =-i \operatorname{tg} i z \\
\operatorname{ctg} z & =i \operatorname{cth} i z, \quad \operatorname{cth} z & =i \operatorname{ctg} i z .
\end{array}
$$

6. The logarithmic function $\operatorname{Ln} z$, where $z \neq 0$, is defined as the inverse of the exponential function, and

$$
\operatorname{Ln} z=\ln |z|+i \operatorname{Arg} z=\ln |z|+i \arg z+2 k \pi i \quad(k=0, \pm 1, \pm 2, \ldots)
$$

This function is multivalued. The principal value of $\operatorname{Ln} z$ is the value obtained when $k=0$; it is denoted by $\ln z$:

$$
\ln z=\ln |z|+i \arg z
$$

Obviously,

$$
\operatorname{Ln} z=\ln z+2 k \pi i, \quad k=0, \pm 1, \pm 2, \ldots
$$

The following relations hold:

$$
\operatorname{Ln}\left(z_{1} z_{2}\right)=\operatorname{Ln} z_{1}+\operatorname{Ln} z_{2}, \quad \operatorname{Ln}\left(\frac{z_{1}}{z_{2}}\right)=\operatorname{Ln} z_{1}-\operatorname{Ln} z_{2}
$$

7. Inverse trigonometric functions $\operatorname{Arcsin} z, \operatorname{Arccos} z, \operatorname{Arctg} z$, Arcctg $z$ are defined as the inverses of the functions $\sin w, \cos w, \operatorname{tg} w, \operatorname{ctg} w$ respectively.

For example, if $z=\sin w$, then $w$ is called the arcsine of the number $z$ and is denoted by $w=\operatorname{Arcsin} z$.

All these functions are multivalued and are expressed in terms of logarithmic functions

$$
\begin{aligned}
& \operatorname{Arcsin} z=-i \operatorname{Ln}\left(i z+\sqrt{1-z^{2}}\right) \\
& \operatorname{Arccos} z=-i \operatorname{Ln}\left(z+\sqrt{z^{2}-1}\right) \\
& \operatorname{Arctg} z=-\frac{i}{2} \operatorname{Ln} \frac{1+i z}{1-i z} \\
& \operatorname{Arcctg} z=-\frac{i}{2} \operatorname{Ln} \frac{z+i}{z-i}
\end{aligned}
$$

The principal values of the inverse trigonometric functions $\arcsin z$, $\arccos z, \operatorname{arctg} z, \operatorname{arcctg} z$ are obtained by taking the principal values of the corresponding logarithmic functions.

8. The general power function $w=z^{a}$, where $a=\alpha+i \beta$ is any complex number, is defined by the equation

$$
z^{a}=e^{a \operatorname{