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Problem: Make sure that the multi-valued analytic functions, under the integral sign, allow the selection of a given domain \( \Omega \) of single-valued branches that satisfy the given conditions, and calculate the integral of this branch. \[ \int_{\partial \Omega} \frac{d z}{(2+\sqrt{z-1}) \sin z}, \] where \( \Omega=\left\{|z|<\frac{1}{2}\right\} \) and \( \left.\sqrt{z-1}\right|_{z=0}=i \).

10.1.19 Integral of a complex variable

3 $

Problem: Make sure that the multi-valued analytical functions, standing under the sign of the integral, allow selection of the given domain \( \Omega \) of singlevalued branches that satisfy the given conditions, and calculate the integral of this branch. \[ \int_{\partial \Omega} \frac{d z}{\ln z-3 \pi i}, \] where \( \Omega=\left\{|z+2|<\frac{3}{2}\right\} \) and \( \left.\ln z\right|_{z=-e}=1-\pi i \).

10.1.20 Integral of a complex variable

1.5 $

Problem: Make sure that the multi-valued analytical functions standing under the sign of the integral, allow selection of the given domain \( \Omega \) of single-valued branches that satisfy the given conditions, and calculate the integral of this branch. \[ \int_{\partial \Omega} \frac{z+2}{2 \pi i-\ln (1+z)}, \] where \( \Omega=\left\{|z-2|<\frac{5}{2}\right\} \) and \( \left.\ln (1+z)\right|_{z=e-1}=1-2 \pi i \).

10.1.21 Integral of a complex variable

2.5 $

Problem: Make sure that the multi-valued analytical functions standing under the sign of the integral, allow selection of the given domain \( \Omega \) of singevalued branches, that satisfy the given conditions, and calculate the integral of this branch. \[ \int_{\partial \Omega} \frac{z+2}{z+\ln (1-z)}, \] where \( \Omega=\left\{|z+2|<\frac{5}{2}\right\} \) and \( \left.\ln (1+z)\right|_{z=e-1}=1 \).

10.1.22 Integral of a complex variable

4.25 $

Problem: Appropriately applying the Cauchy residue theorem, calculate the following integrals along the boundaries of infinite domains: \[ \int_{\partial \Omega} \frac{z e^{-z}}{z^{2}-1} d z, \quad \text { where } \Omega=\{\operatorname{Re} z>0\} . \]

10.1.23 Integral of a complex variable

4.25 $

Problem: Appropriately applying the Cauchy residue theorem, calculate the following integrals along the boundaries of infinite domains: \[ \int_{\partial \Omega} \frac{e^{z}}{\sinh 2 z} d z \text {, where } \Omega=\left\{-\frac{\pi}{4}<\operatorname{Im} z<\frac{\pi}{4}\right\} \text {. } \]

10.1.24 Integral of a complex variable

4.25 $

Problem: Appropriately applying the Cauchy residue theorem, calculate the following integrals along the boundaries of infinite domains: \[ \int_{\partial \Omega} \frac{z^{3}}{(z-1)^{2}} e^{-z^{2}} d z \text {, where } \Omega=\left\{-\frac{\pi}{6}<\arg z<\frac{\pi}{6}\right\} \text {. } \]

10.1.25 Integral of a complex variable

4.99 $

Problem: Appropriately applying the Cauchy residue theorem, calculate the following integrals along the boundaries of infinite domains: \[ \int_{\partial \Omega} \frac{d z}{\left(z^{4}+1\right) \sqrt{z^{2}+1}} d z \] where \( \Omega=\left\{\operatorname{Re} z>(\operatorname{Im} z)^{2}\right\} \), and \( \left.\sqrt{z^{2}+1}\right|_{z=0}=1 \).

10.1.26 Integral of a complex variable

7.49 $

Problem: Calculate the integral of the complex variable: \[ \oint_{|z-3|=1} \frac{1-\cos z}{z^{2}-\pi^{2}} d z . \]

10.1.27 Integral of a complex variable

0 $

Problem: Calculate the integral of the complex variable: \[ \oint_{|z|=2} \frac{4 z^{4} \alpha z}{(z+1)^{4}} d z . \]

10.1.28 Integral of a complex variable

1.25 $

Problem: Calculate the integral of the complex variable: \[ \oint_{|z|=\frac{1}{2}} \frac{\operatorname{ch} 3 z-\cos 3 z}{z^{2} \operatorname{sh} 4 z} d z . \]

10.1.29 Integral of a complex variable

3 $

Problem: Calculate the integral along the curve \( \mathcal{L} \). \[ \begin{array}{l} \int_{\mathcal{L}} z \operatorname{Re} z^{2} d z, \\ \mathcal{L}=\widetilde{A B}=\left\{z:|z|=2, \quad \operatorname{Im} z \geq 0, \quad z_{A}=2, \quad z_{B}=-2 .\right. \end{array} \]

10.1.30 Integral of a complex variable

1.25 $

Problem: Write the canonical and parametric equations of the given straight line, as well as the equation of the plane in "segments" perpendicular to the given line and passing through the point \( M_{0}\left(x_{0} ; y_{0} ; z_{0}\right) \). Construct a plane. \( L:\left\{\begin{array}{l}6 x-5 y-4 z+8=0 \\ 6 x+5 y+3 z+4=0\end{array} \quad M_{0}(7 ; 2 ;-5)\right. \).

3.3.8 Line and plane in space

2 $

Problem: Prove the statement using mathematical induction: (8 \( { }^{n}-1 \) ) is a multiple of 7 for all natural \( n \geq 1 \).

6.6.20 Combinatorics

0 $

Problem: A company of 8 people went fishing. In order to organize dinner and overnight stay, they need to prepare firewood, make a fire, prepare food, and put up tents. To complete all these tasks, it is necessary for them to split into groups "firemakers", “cooks", "house builders". a) How many different ways are there for such a division if no group should include less than 2 people? b) How many different ways are there to sleep in three completely identical tents, without being arranged one by one?

6.6.21 Combinatorics

3 $

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