Math Problems Vault

Problem 3.1

Prove that the sum and product of functions continuous on a set $E$ are also functions continuous on this set. The quotient of two functions continuous on a set $E$ is also a function continuous on this set, provided the denominator does not vanish at any point of the set $E$ .

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Problem 3.2

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

Determine whether the following functions are uniformly continuous in the domain $0 < \left|z\right| < 1$ :

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(1)

$f = e^{-1 /\left|z\right|}$ ;

(2)

$f = \frac{\mathrm{Re}\left(z\right)}{\left|z\right|}$ ;

(3)

$f = \frac{\left(\mathrm{Re}\left(z^{2}\right)\right)^{2}}{z^{2}}$ ;

(4)

$f = e^{-1 / z^{2}}$ .

Problem 3.3

(1)(2)

Let the function $f(z)$ be defined and continuous on a closed bounded set $E$ . Prove that:

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(1)

the function $\left|f(z)\right|$ is bounded on the set $E$ and attains its greatest and least values;

(2)

the function $f(z)$ is uniformly continuous on the set $E$ .

Problem 3.4

The functions $f(z)$ and $g(z)$ are uniformly continuous on a set $E$ . Can one assert that the function $f(z) g(z)$ is uniformly continuous on the set $E$ ?

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Problem 3.5

Let the function $f(z)$ be defined and uniformly continuous in a bounded domain $D$ . Prove that at each boundary point of the domain $D$ the function $f(z)$ has a limit, and that the function $f(z)$ , extended to the boundary of the domain by these limiting values, is continuous in the closed domain $\bar{D}$ .

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Problem 3.6

Let the domain $D$ be bounded by a simple piecewise-smooth curve. Prove that a function continuous in the domain $D$ up to its boundary is uniformly continuous in this domain.

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Problem 3.7

Let the domain $D$ be capable of being divided into a finite number of domains $D_{1}, D_{2}, \ldots , D_{n}$ , each of which is bounded by a simple piecewise-smooth curve. Prove that for the continuity of the function $f(z)$ up to the boundary of the domain $D$ it is necessary and sufficient that the function $f(z)$ be uniformly continuous in each of the domains $D_{1}, \ldots , D_{n}$ .

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Problem 3.8

Compute the values of the function $e^{z}$ at the points:

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(1)

$z = 2 \pi i$ ;

(2)

$z = \pi i$ ;

(3)

$z = \pi i / 2$ ;

(4)

$z = -\pi i / 2$ ;

(5)

$z = \pi i / 4$ .

Problem 3.9

Prove the formulas:

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(1)

$\cos (-z) = \cos z$ ;

(2)

$\sin (-z) = -\sin z$ ;

(3)

$\sin \left(z+\frac{\pi }{2}\right) = \cos z$ ;

(4)

$\sin (\pi +z) = -\sin z$ ;

(5)

$\sin (z+2 \pi ) = \sin z$ ;

(6)

$\cos \left(z+\frac{\pi }{2}\right) = -\sin z$ ;

(7)

$\sin \left(z_{1}+z_{2}\right) = \sin z_{1} \cos z_{2}+\cos z_{1} \sin z_{2}$ ;

(8)

$\cos \left(z_{1}+z_{2}\right) = \cos z_{1} \cos z_{2}-\sin z_{1} \sin z_{2}$ .

Problem 3.10

Prove the formulas:

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(1)

$\operatorname {ch}(-z) = \operatorname {ch} z$ ;

(2)

$\operatorname {sh}(-z) = -\operatorname {sh} z$ ;

(3)

$\operatorname {ch}^{2} z-\operatorname {sh}^{2} z = 1$ ;

(4)

$\operatorname {ch}\left(z_{1}+z_{2}\right) = \operatorname {ch} z_{1} \operatorname {ch} z_{2}+\operatorname {sh} z_{1} \operatorname {sh} z_{2}$ ;

(5)

$\operatorname {sh}\left(z+\frac{\pi i}{2}\right) = i \operatorname {ch} z$ ;

(6)

$\operatorname {ch}\left(z+\frac{\pi i}{2}\right) = i \operatorname {sh} z$ ;

(7)

$\operatorname {sh}(z+\pi i) = -\operatorname {sh} z$ ;

(8)

$\operatorname {ch}(z+\pi i) = -\operatorname {ch} z$ ;

(9)

$\operatorname {th}(z+\pi i) = \operatorname {th} z$ ;

(10)

$\operatorname {ch}(z+2 \pi i) = \operatorname {ch} z$ .

Problem 3.11

Prove the formulas:

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(1)

$\operatorname {sh} z = -i \sin (i z)$ ;

(2)

$\operatorname {sh}(i z) = i \sin z$ ;

(3)

$\cos (i z) = \operatorname {ch} z$ ;

(4)

$\operatorname {ch}(i z) = \cos z$ ;

(5)

$\operatorname {tg}(i z) = i \operatorname {th} z$ ;

(6)

$\operatorname {th}(i z) = i \operatorname {tg} z$ ;

(7)

$\operatorname {ctg}(i z) = -i \operatorname {cth} z$ ;

(8)

$\operatorname {cth}(i z) = -i \operatorname {ctg} z$ .

Problem 3.12

Let $x = \mathrm{Re}\left(z\right), y = \mathrm{Im}\left(z\right)$ . Prove that:

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(1)

$\mathrm{Re}\left(\sin z\right) = \sin x \cdot \operatorname {ch} y, \quad \mathrm{Im}\left(\sin z\right) = \cos x \cdot \operatorname {sh} y ;$

(2)

$\mathrm{Re}\left(\cos z\right) = \cos x \cdot \operatorname {ch} y, \quad \mathrm{Im}\left(\cos z\right) = -\sin x \cdot \operatorname {sh} y$ ;

(3)

$\mathrm{Re}\left(\operatorname {sh} z\right) = \operatorname {sh} x \cdot \cos y, \quad \mathrm{Im}\left(\operatorname {sh} z\right) = \operatorname {ch} x \cdot \sin y$ ;

(4)

$\mathrm{Re}\left(\operatorname {ch} z\right) = \operatorname {ch} x \cdot \cos y, \quad \mathrm{Im}\left(\operatorname {ch} z\right) = \operatorname {sh} x \cdot \sin y$ ;

(5)

$\mathrm{Re}\left(\operatorname {tg} z\right) = \frac{\sin 2 x}{\cos 2 x+\operatorname {ch} 2 y}, \quad \mathrm{Im}\left(\operatorname {tg} z\right) = \frac{\operatorname {sh} 2 y}{\cos 2 x+\operatorname {ch} 2 y}$ ;

(6)

$\mathrm{Re}\left(\operatorname {ctg} z\right) = \frac{\sin 2 x}{\operatorname {ch} 2 y-\cos 2 x}, \quad \mathrm{Im}\left(\operatorname {ctg} z\right) = \frac{-\operatorname {sh} 2 y}{\operatorname {ch} 2 y-\cos 2 x}$ .

Problem 3.13

Let $x = \mathrm{Re}\left(z\right), y = \mathrm{Im}\left(z\right)$ . Prove that:

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(1)

$\left|\sin z\right| = \sqrt{\operatorname {ch}^{2} y-\cos^{2} x}$ ;

(2)

$\left|\cos z\right| = \sqrt{\operatorname {ch}^{2} y-\sin^{2} x}$ ;

(3)

$\left|\operatorname {sh} z\right| = \sqrt{\operatorname {ch}^{2} x-\cos^{2} y}$ ;

(4)

$\left|\operatorname {ch} z\right| = \sqrt{\operatorname {ch}^{2} x-\sin^{2} y}$ ;

(5)

$\left|\operatorname {tg} z\right| = \sqrt{\frac{\operatorname {ch} 2 y-\cos 2 x}{\operatorname {ch} 2 y+\cos 2 x}}$ ;

(6)

$\left|\operatorname {cth} z\right| = \sqrt{\frac{\operatorname {ch} 2 x+\cos 2 y}{\operatorname {ch} 2 x-\cos 2 y}}$ .

Problem 3.14

Describe the points $z$ at which the following functions take real values:

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(1)

$\cos z$ ;

(2)

$\operatorname {ch} z$ ;

(3)

$\sin z$ ;

(4)

$\operatorname {tg} z$ ;

(5)

$\operatorname {cth} z$ .

Problem 3.15

Describe the points $z$ at which the following functions take purely imaginary values:

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(1)

$\sin z$ ;

(2)

$\operatorname {sh} z$ ;

(3)

$\cos z$ ;

(4)

$\operatorname {ctg} z$ ;

(5)

$\operatorname {th} z$ .

Problem 3.16

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

Find all points at which the following functions vanish:

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(1)

$\sin z$ ;

(2)

$\cos z$ ;

(3)

$\operatorname {sh} z$ ;

(4)

$\operatorname {ch} z$ .

Problem 3.17

Find all solutions of the following equations:

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(1)

$\sin z = \frac{4 i}{3}$ ;

(2)

$\sin z = \frac{5}{3}$ ;

(3)

$\cos z = \frac{3 i}{4}$ ;

(4)

$\cos z = \frac{3+i}{4}$ ;

(5)

$\operatorname {tg} z = \frac{5 i}{3}$ ;

(6)

$\operatorname {ctg} z = -\frac{3 i}{5}$ ;

(7)

$\operatorname {sh} z = \frac{i}{2}$ ;

(8)

$\operatorname {ch} z = \frac{1}{2}$ .

Problem 3.18

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

Prove that for $\left|z\right| \leqslant R$ the following inequalities hold:

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(1)

$\left|\operatorname {ch} z\right| \leqslant \operatorname {ch} R$ ;

(2)

$\left|\operatorname {sh} z\right| \leqslant \operatorname {sh} R$ ;

(3)

$\left|\cos z\right| \leqslant \operatorname {ch} R$ ;

(4)

$\left|\sin z\right| \leqslant \operatorname {sh} R$ .

Problem 3.19

(1)(2)(3)

Let $x = \mathrm{Re}\left(z\right), \quad y = \mathrm{Im}\left(z\right)$ . Prove that:

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(1)

The function $e^{z}$ tends to infinity as $x$ tends to $+\infty$ , and this tendency is uniform in $y$ .

(2)

The function $e^{z}$ tends to zero as $x \rightarrow -\infty$ , and this tendency is uniform in $y$ .

(3)

The functions $\sin z$ and $\cos z$ tend to infinity as $y \rightarrow \pm \infty$ , and this tendency is uniform in $x$ .

Problem 3.20

(1)(2)

Prove that:

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(1)

The function $e^{z^{2}}$ tends to infinity as $z \rightarrow \infty$ in any angle of the form

\left|\arg z-\pi \right| \leqslant \alpha

and in any angle of the form $\left|\arg z\right| \leqslant \alpha$ , provided only that $\alpha < \frac{\pi }{4}$ .

(2)

The function $e^{z^{2}}$ tends to zero as $z \rightarrow \infty$ in any angle of the form

\left|\arg z \pm \frac{\pi }{2}\right| \leqslant \alpha < \frac{\pi }{4}

Problem 3.21

Let $P(z) = z^{n}+a_{1} z^{n-1}+\ldots +a_{n}$ . Prove that the function $e^{P(z)}$ tends to infinity as $z \rightarrow \infty$ in the angles

\left|\arg z-\frac{2 k \pi }{n}\right| \leqslant \alpha < \frac{\pi }{2 n} \quad (k = 0,1, \ldots , n-1),

and tends to zero as $z \rightarrow \infty$ in the angles

\left|\arg z-\frac{(2 k+1) \pi }{n}\right| \leqslant \alpha < \frac{\pi }{2 n} \quad (k = 0,1, \ldots , n-1).

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Problem 3.22

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

Let $x = \mathrm{Re}\left(z\right), y = \mathrm{Im}\left(z\right)$ . Prove the inequalities:

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(1)

$\frac{2 e^{-2 y}}{1+e^{-2 y}} \leqslant \left|\operatorname {tg} z-i\right| \leqslant \frac{2 e^{-2 y}}{1-e^{-2 y}} \quad (y > 0)$ ;

(2)

$\frac{2 e^{2 y}}{1+e^{2 y}} \leqslant \left|\operatorname {tg} z+i\right| \leqslant \frac{2 e^{2 y}}{1-e^{-2 y}} \quad (y < 0)$ ;

(3)

$\frac{2 e^{-2 y}}{1+e^{-2 y}} \leqslant \left|\operatorname {ctg} z+i\right| \leqslant \frac{2 e^{-2 y}}{1-e^{-2 y}} \quad (y > 0)$ ;

(4)

$\frac{2 e^{2 y}}{1+e^{2 y}} \leqslant \left|\operatorname {ctg} z-i\right| \leqslant \frac{2 e^{2 y}}{1-e^{2 y}} \quad (y < 0)$ .

Problem 3.23

(1)(2)

Compute the integral $\int_{C}\left|z\right| d z$ in the cases when the curve $C$ is:

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(1)

the straight-line segment going from the point $z = -i$ to the point $z = i$ ;

(2)

the semicircle

\left|z\right| = 1, \quad \mathrm{Re}\left(z\right) \geqslant 0

going from the point $z = -i$ to the point $z = i$ .

Problem 3.24

Compute the integral $\int_{C} z \sin z \; d z$ , where $C$ is the straight-line segment going from the point $z = 0$ to the point $z = i$ .

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Problem 3.25

Let the function $f(z)$ be continuous on the whole extended plane. Denote by $C_{a}$ the straight-line segment going from the point $a$ to the point $a+1$ . Prove that

\lim _{a \rightarrow \infty } \int _{C_{a}} f(z) d z = f(\infty )

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Problem 3.26

Let the function $f(z)$ be continuous in the half-plane $\mathrm{Im}\left(z\right) \geqslant 0$ and satisfy the inequality

\left|f(z)\right| \leqslant M\left|z\right|^{m}

Denote by $C_{R}$ the semicircle $\left|z\right| = R, \mathrm{Im}\left(z\right) \geqslant 0$ , going from the point $z = R$ to the point $z = -R$ . Prove the inequality

\left|\int _{C_{R}} f(z) e^{i z} d z\right| \leqslant \pi M R^{m}

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Problem 3.27

Let the function $f(z)$ be continuous in the angle

-\alpha \leqslant \arg z \leqslant \alpha \quad (0 < \alpha < \pi ),

and let $z f(z) \rightarrow A$ as $z \rightarrow \infty , \quad \left|\arg z\right| \leqslant \alpha$ . Denote by $C_{R}$ the arc of the circle $\left|z\right| = R, \quad \left|\arg z\right| \leqslant \alpha$ , going from the point $z = R e^{-i \alpha }$ to the point $z = R e^{i \alpha }$ . Prove that

\lim _{R \rightarrow \infty } \int _{C_{R}} f(z) d z = 2 i \alpha A

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