Math Problems Vault

Problem 27.1

(1)(2)(3)

Let $x = \mathrm{Re}\left(z\right), y = \mathrm{Im}\left(z\right)$ , and let $C$ be a positive constant. Find the images of each line of the indicated families under the mapping $w = \frac{1}{z}$ :

?

(1)

the family of circles $\left\{ (x, y): x^{2}+y^{2} = C x\right\}$ ;

(2)

the family of lines $\left\{ (x, y): y = x+C\right\}$ ;

(3)

the family of lines $\left\{ (x, y): y = C x\right\}$ .

Problem 27.2

Let $x = \mathrm{Re}\left(z\right), \quad y = \mathrm{Im}\left(z\right)$ . Find the images of each domain of the indicated families under the mapping $w = \frac{1}{z}$ :

?

(1)

the family of disks $\left\{ (x, y): x^{2}+y^{2} < C x\right\}$ (here $C$ is a positive constant);

(2)

the family of disks $\left\{ (x, y): x^{2}+y^{2} < C x\right\}$ (here $C$ is a negative constant);

(3)

the family of disks $\left\{ (x, y): x^{2}+y^{2} < C y\right\}$ ( $C$ is a positive constant);

(4)

the family of half-planes $\left\{ (x, y): y > C x\right\}$ ( $C$ is a positive constant);

(5)

the family of disks $\left\{ z:\left|z-a\right| < R\right\}$ , where $a$ is a fixed point, and the positive constant $R$ satisfies the condition $R < \left|a\right|$ ;

(6)

the family of disks $\left\{ z:\left|z-a\right| < R\right\}$ , where $a$ is a fixed point, and the constant $R$ satisfies the condition $R > \left|a\right|$ .

Problem 27.3

(1)(2)(3)(4)

Find the image of the disk $\left\{ z:\left|z-1\right| < 2\right\}$ under the following mappings:

?

(1)

$w = 1-2 i z$ ;

(2)

$w = \frac{2 i z}{z+3}$ ;

(3)

$w = \frac{z+1}{z-2}$ ;

(4)

$w = \frac{z-1}{2 z-6}$ .

Problem 27.4

Find the image of the half-plane $\left\{ z: \mathrm{Re}\left(z\right) < 1\right\}$ under the following mappings:

?

(1)

$w = (1+i) z+1$ ;

(2)

$w = \frac{z}{z-1+i}$ ;

(3)

$w = \frac{z}{z-2}$ ;

(4)

$w = \frac{4 z}{z+1}$ ;

(5)

$w = \frac{z-3+i}{z+1+i}$ .

Problem 27.5

(1)(2)(3)(4)

Find the images of the indicated domains $D$ under the indicated mappings:

?

(1)

$D = \left\{ z:\left|z\right| < 1, \mathrm{Im}\left(z\right) > 0\right\} , w = \frac{1-z}{1+z}$ ;

(2)

$D = \left\{ z: z \notin [-2,1]\right\} , w = \frac{z+2}{1-z}$ ;

(3)

$D = \left\{ z:\left|z-i\right| > 1, \mathrm{Im}\left(z\right) > 0\right\} , w = \frac{1}{z}$ ;

(4)

$D = \left\{ z: 1 < \left|z\right| < 2\right\} , w = \frac{2}{z-1}$ .

Problem 27.6

(1)(2)(3)

Find linear-fractional functions $w(z)$ satisfying the conditions:

?

(1)

$w(0) = 4, w(1+i) = 2+2 i, w(2 i) = 0$ ;

(2)

$w(0) = 0, w(1+i) = 2+2 i, w(2 i) = 4$ ;

(3)

$w(0) = 0, w(1+i) = \infty , w(2 i) = 2 i$ .

Find the image of the disk $\left\{ z:\left|z-i\right| < 1\right\}$ under the mappings given by these functions.

Problem 27.7

(1)(2)(3)

Find linear-fractional functions $w(z)$ satisfying the conditions:

?

(1)

$w(i) = 2, w(\infty ) = 1+i, w(-i) = 0$ ;

(2)

$w(i) = 0, w(\infty ) = 1, w(-i) = \infty$ ;

(3)

$w(i) = -2, w(\infty ) = 2 i, w(-i) = 2$ .

Find the image of the half-plane $\left\{ z: \mathrm{Re}\left(z\right) > 0\right\}$ under the mappings given by these functions.

Problem 27.8

Find a function $w(z)$ that conformally maps the domain $D$ onto the domain $D_{1}$ and satisfies the indicated conditions:

?

(1)

$D = \left\{ z:\left|z\right| < 1\right\} , \quad D_{1} = \left\{ w:\left|w\right| < 1\right\} , \quad w\left(z_{0}\right) = 0, \quad \arg w^{\prime }\left(z_{0}\right) = \alpha \left(\left|z_{0}\right| < 1\right)$ ;

(2)

$D = \left\{ z:\left|z\right| < 1\right\} , D_{1} = \left\{ w:\left|w\right| < 1\right\} , w\left(z_{0}\right) = w_{0}, \arg w^{\prime }\left(z_{0}\right) = \alpha \left(\left|z_{0}\right| < 1,\left|w_{0}\right| < 1\right)$ ;

(3)

$D = \left\{ z:\left|\mathrm{Im}\left(z\right)\right| > 0\right\} , D_{1} = \left\{ w:\left|w\right| < 1\right\} , w\left(z_{0}\right) = 0, \arg w^{\prime }\left(z_{0}\right) = \alpha \left(\mathrm{Im}\left(z_{0}\right) > 0\right)$ ;

(4)

$D = \left\{ z:\left|\mathrm{Im}\left(z\right)\right| > 0\right\} , D_{1} = \left\{ w:\left|w\right| < 1\right\} , w\left(z_{0}\right) = w_{0}, \arg w^{\prime }\left(z_{0}\right) = \alpha \left(\mathrm{Im}\left(z_{0}\right) > 0,\left|w_{0}\right| < 1\right)$ ;

(5)

$D = \left\{ z:\left|\mathrm{Im}\left(z\right)\right| > 0\right\} , D_{1} = \left\{ w: \mathrm{Im}\left(w\right) > 0\right\} , w\left(z_{0}\right) = w_{0}$ , $\arg w^{\prime }\left(z_{0}\right) = \alpha \left(\mathrm{Im}\left(z_{0}\right) > 0, \mathrm{Im}\left(w_{0}\right) > 0\right) ;$

(6)

$D = \left\{ z:\left|z\right| < 1\right\} , D_{1} = \left\{ w:\left|w\right| < 1\right\} , w(1) = 1, w(i) = \frac{3 i-4}{5}$ , $w(-1) = -1$ ;

(7)

$D = \left\{ z:\left|z\right| < 1\right\} , D_{1} = \left\{ w:\left|w\right| < 1\right\} , w(i) = i, w\left(\frac{i}{2}\right) = \frac{4 i}{5}$ ;

(8)

$D = \left\{ z:\left|z\right| < 1\right\} , D_{1} = \left\{ w:\left|w\right| < 1\right\} , w\left(\frac{1}{2}\right) = \frac{1}{2}$ , $\arg w^{\prime }\left(\frac{1}{2}\right) = \frac{\pi }{2}$ ;

(9)

$D = \left\{ z: \mathrm{Im}\left(z\right) > 0\right\} , D_{1} = \left\{ w:\left|w\right| < 1\right\} , w(0) = i, w(-1) = 1$ , $w(\infty ) = -1$ ;

(10)

$D = \left\{ z: \mathrm{Im}\left(z\right) > 0\right\} , D_{1} = \left\{ w:\left|w\right| < 1\right\} , w(0) = -i, w(2 i) = \frac{i}{3}$ ;

(11)

$D = \left\{ z: \mathrm{Im}\left(z\right) > 0\right\} , \quad D_{1} = \left\{ w:\left|w\right| < 1\right\} , \quad w(1+i) = 0$ , $\arg w^{\prime }(1+i) = \pi$ ;

(12)

$D = \left\{ z: \mathrm{Im}\left(z\right) > 0\right\} , D_{1} = \left\{ w: \mathrm{Im}\left(w\right) > 0\right\} , w(-1) = 0, w(0) = 2$ , $w(1) = \infty$ ;

(13)

$D = \left\{ z: \mathrm{Im}\left(z\right) > 0\right\} , D_{1} = \left\{ w: \mathrm{Im}\left(w\right) > 0\right\} , w(-1) = -2$ , $w(-2+i) = 1+3 i ;$

(14)

$D = \left\{ z: \mathrm{Im}\left(z\right) > 0\right\} , \quad D_{1} = \left\{ w: \mathrm{Im}\left(w\right) > 0\right\} , \quad w(1+i) = i$ , $\arg w^{\prime }(1+i) = \frac{\pi }{2}$ ;

(15)

$D = \left\{ z:\left|z-1-i\right| < 2\right\} , D_{1} = \left\{ w:\left|w\right| < 1\right\} , w(i) = 0, \arg w^{\prime }(i) = \frac{\pi }{2}$ ;

(16)

$D = \left\{ z: \mathrm{Re}\left(z\right) > -1\right\} , D_{1} = \left\{ w:\left|w\right| < 1\right\} , w(0) = 0, \arg w^{\prime }(0) = \pi$ .

Problem 27.9

(1)(2)

Find the general form of the conformal mapping of the following domains onto the annulus $\left\{ w: 1 < \left|w\right| < R\right\}$ :

?

(1)

$\left\{ z:\left|z-3\right| > 9,\left|z-8\right| < 16\right\}$ ;

(2)

$\left\{ z:\left|z-5\right| > 4\text{, Re }z > 0\right\}$ .

Problem 27.10

Let $w(z)$ be an arbitrary linear-fractional function, and let $z_{1}, z_{2}, z_{3}, z_{4}$ be four pairwise distinct points of the extended complex plane. Denote

w_{k} = w\left(z_{k}\right), \quad k = 1,2,3,4.

Prove that

\frac{w_{4}-w_{1}}{w_{4}-w_{2}}: \frac{w_{3}-w_{1}}{w_{3}-w_{2}} = \frac{z_{4}-z_{1}}{z_{4}-z_{2}}: \frac{z_{3}-z_{1}}{z_{3}-z_{2}}.

?

Problem 27.11

Prove that the level lines of the modulus of a linear-fractional function are circles or straight lines.

?

Problem 27.12

Prove that the level lines of the real part of a linear-fractional function are circles or straight lines.

?

Problem 27.13

Find the condition that the points $z_{1}, z_{1}^{*}, z_{2}, z_{2}^{*}$ must satisfy for there to exist a circle (or line) with respect to which the points $z_{k}$ are symmetric to the points $z_{k}^{*}(k = 1,2)$ .

?

Problem 27.14

(1)(2)

Let the function $w(z)$ be meromorphic in the domain $D$ . Prove the following statements:

?

(1)

if under the mapping $w = w(z)$ the image of any line segment (lying in the domain $D$ ) is a line segment, then $w(z)$ is a linear function;

(2)

if under the mapping $w = w(z)$ the image of any line segment (lying in the domain $D$ ) is an arc of a circle or a line segment, then $w = w(z)$ is a linear-fractional function.

Remark.

?

A point $a \in \overline{\mathbb {C}}$ is called a fixed point of the linear-fractional transformation $f$ if $f(a) = a$ . This term is used in problems 27.15-27.17 below.

Problem 27.15

(1)(2)

Prove the statements:

?

(1)

every linear-fractional transformation has at least one fixed point (finite or infinite);

(2)

every linear-fractional transformation other than the identity has at most two fixed points (finite and infinite).

Problem 27.16

Prove that a linear-fractional transformation $f$ with a unique fixed point $a \in \overline{\mathbb {C}}$ satisfies the equation

\frac{1}{f(z)-a} = \frac{1}{z-a}+A, \quad A \in \mathbb {C}

for $a \neq \infty$ , while for $a = \infty$ it has the form

f(z) = z+A, \quad A \in \mathbb {C}.

?

Problem 27.17

Prove that a linear-fractional transformation $f$ with two distinct finite fixed points $a$ and $b$ satisfies the equation

\frac{f(z)-a}{f(z)-b} = A \cdot \frac{z-a}{z-b}, \quad A \in \mathbb {C}

?