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(3) Problem: Find the total charge on the surface \( S \), if the surface charge density is equal to \( f(x, y, z) \) : \[ S: z=\sqrt{4-x^{2}-y^{2}}, \quad f(x, y, z)=x^{2} . \]

9.8.11 Surface integrals

3.25 $

Problem: Is the vector field \( \vec{a}=x z^{2} \vec{\imath}+(y-1) \vec{\jmath}+x^{2} z \vec{k} \) : a) potential. If yes, find its potential, b) solenoidal?

9.9.4 Potential and solenoidal fields

3.75 $

3) Problem: Calculate the area of a figure bounded by the lemniscate \( \rho^{2}=2 \cos 2 \varphi \) and the circle \( \rho=1 \), and located outside the circle.

9.7.13 The area of a region

2.5 $

(3) Problem: Find the area of the surface formed by the rotation of the cardioid \[ x=a(2 \cos t-\cos 2 t), \quad y=a(2 \sin t-\sin 2 t) \] around the axis \( O X(0 \leq t \leq \pi) \).

9.7.14 The area of a region

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3) Problem: Find the area of the figure, formed by the graphs of the functions \( y=x^{2}, y=8-x^{2} \).

9.7.15 The area of a region

0 $

as Problem: Find the area of the region bounded by the graphs of the functions: \[ y=e^{x}-1, \quad y=e^{2 x}-3, \quad x=0 . \]

9.7.16 The area of a region

1.25 $

Problem: Find the singular points of the function, indicate their type and find the residues in them: \[ f(z)=\frac{1}{z^{4}+1} e^{\frac{1}{z+1}}+\frac{\tan ^{2} z}{z} . \]

10.2.5 Singular points and residues

3 $

Problem: Find the singular points of the function and indicate their type: \[ f(z)=\frac{1}{z^{2}-1} \sin \frac{1}{2 z+1}+\frac{1}{e^{z}+i} . \]

10.2.6 Singular points and residues

3 $

Problem: Solve the equation: \[ f(t)=\int_{0}^{t} \frac{\operatorname{ch} \tau-1}{\tau} d \tau . \]

10.4.10 Laplace transform

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Problem: Find the Laplace transform \( F(p) \) by the given signal: \[ f(t)=\int_{0}^{t} \frac{\cos \beta \tau-\cos \alpha \tau}{\tau} d \tau . \]

10.4.11 Laplace transform

2 $

Problem: Let function \( f_{1}(z), f_{2}(z) \) are analytic in a domain \( D \subset \mathbb{C} \) of the complex plane, and they coincide at \( E \subset \) \( D\left(f_{1}(z) \equiv f_{2}(z), \forall z \in E\right) \) and let \( E \) have at least one limit point in \( D \), then \( f_{1}(z) \equiv f_{2}(z), \forall z \in D \).

10.6.16 Analytic functions

4.99 $

Problem: a) Point \( z_{0} \in \mathbb{C} \) is a zero of multiplicity \( m \) of the analytic function in \( z_{0} \) of function \( f(z) \) only if \( f(z) \) is represented as \( f(z)=\left(z-z_{0}\right)^{m} \cdot \varphi(z) \), where \( \varphi(z) \), is analytic in \( z_{0} \), and \( \varphi\left(z_{0}\right) \neq 0 \). b) Let the zeros of function \( f(z) \) be \( z_{1}, z_{2}, \ldots, z_{n} \in \mathbb{C} \), with orders \( m_{1}, m_{2}, \ldots, m_{n} \in \mathbb{N} \), respectively, and \( f(z) \) is analytic in points \( z_{1}, z_{2}, \ldots, z_{n}\left(z_{i} \neq z_{j}, i \neq j\right) \). Then \( f(z) \) is represented in the form of \( f(z)=\left(z-z_{1}\right)^{m_{1}}(z- \) \( \left.-z_{2}\right)^{m_{2}} \cdot \ldots \cdot\left(z-z_{n}\right)^{m_{n}} \cdot \varphi(z) \), where \( \varphi\left(z_{i}\right) \neq 0, i=\overline{1, n} \) and \( \varphi(z) \) is analytic in points \( z_{1}, \ldots, z_{n} \).

10.6.17 Analytic functions

7.49 $

Problem: Calculate the integral using residues: \[ \int_{-\infty}^{\infty} \frac{x^{2}}{\left(x^{2}+4\right)\left(x^{2}+9\right)} d x \text {. } \]

10.7.19 Calculating integrals of a real variable using residues

2.5 $

Problem: Calculate the integral using residues: \[ \int_{-\infty}^{\infty} \frac{1}{x^{2}+4 x+29} d x \text {. } \]

10.7.20 Calculating integrals of a real variable using residues

2 $

Problem: Find the values of \( \tan \left(\frac{\pi}{6}+\frac{i}{2} \ln 3\right) \).

10.3.7 Operations with complex numbers

1.25 $

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