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Problem: Find the image of domain \[ D=\{|z|<1\} \cup\{\operatorname{Im} z>0\} \backslash\left\{z=i t, 0

10.5.38 Conformal mappings

6.35 $

Problem: Find the image of domain \[ D=\left\{|\operatorname{Re} z|<\frac{\pi}{2}\right\} \backslash\left\{z=t, \quad t \in\left[0 ; \frac{\pi}{2}\right)\right\} \] for the mapping \( f(z)=e^{i z} \).

10.5.39 Conformal mappings

5.08 $

Problem: Find the image of domain \[ D=\left\{|\operatorname{Re} z|<\frac{\pi}{2}\right\} \cup\{\operatorname{Im} z>0\} \backslash\{z=i t, t \in(0 ; 1]\} \] for the mapping \( f(z)=\sin z \).

10.5.40 Conformal mappings

6.35 $

Problem: Find the image of domain \( D=\{0<\operatorname{Im} z<\pi\} \) for the mapping \( f(z)=\operatorname{cth} z \).

10.5.41 Conformal mappings

6.35 $

Problem: Find the image of domain \[ D=\{0<\operatorname{Re} z<\pi\} \backslash\left\{z=t, t \in\left(0 ; \frac{\pi}{2}\right)\right\} \] for the mapping \( f(z)=\operatorname{ch} i z \).

10.5.42 Conformal mappings

6.35 $

Problem: Find the image of domain \( D \) for the linear-fractional mapping \( w=f(z) \), where \[ D=\{z \in \mathbb{C}:|| z+1-i \mid>\sqrt{2}\}, \quad f(z)=\frac{z+1}{z-i} . \]

10.5.43 Conformal mappings

6.35 $

Problem: Find the image of domain \( D \) when mapped by the Joukowsky transform: \[ D:\left\{\begin{array}{c} \operatorname{Re} z>0, \quad \operatorname{Im} z>0 \\ |z|>1 \end{array}, \quad f(z)=\frac{1}{2}\left(z+\frac{1}{z}\right)\right. \text {. } \]

10.5.44 Conformal mappings

4.32 $

Problem: Find the image of line \( L \) for the mapping \( w=f(z) . L \) is the segment, connecting the points \( 1, i \), where \( w=(1+i) z+1-i \).

10.5.45 Conformal mappings

0 $

Problem: Construct a linear-fractional function that maps the points \( z_{1}, z_{2}, z_{3} \) corresponding to the points \( w_{1}, w_{2}, w_{3} \). \[ z_{1}=3, z_{2}=2, z_{3}=3 i, w_{1}=1+i, w_{2}=0, w_{3}=\infty . \]

10.5.46 Conformal mappings

2.54 $

Problem: Prove that the number \( \sqrt{3}-\sqrt{2} \) is irrational.

12.2.43 Number theory

0 $

Problem: A complex number is given: \( z=\frac{-4 \sqrt{3}-4 i}{1+i \sqrt{3}} \). 1) Write the complex number in algebraic, trigonometric and exponential forms. 2) Write the algebraic, trigonometric and exponential forms of the number \( u=z^{n} \), where \( n(-1)^{N}(N+4), N=25 \). 3) Write the exponential and trigonometric forms of the roots \( W_{k}=\sqrt[3]{z}, k=0,1,2 \). 4) Draw \( z \) and \( W_{0}, W_{1}, W_{2} \) on the same complex plane.

10.3.10 Operations with complex numbers

3.81 $

Problem: Find the gradient of the scalar field \( U(x, y, z)= \) \( \arctan \left(\frac{z}{y+z}\right) \) at an arbitrary point and at the point \( M_{0}=(1,1,0) \). For the resulting vector field \( \bar{a}= \) \( \nabla U(x, y, z) \) find \( \operatorname{div} \bar{a} \) and \( \operatorname{rot} \bar{a} \) at the point \( M_{0} \).

1.12.2 Vector analysis

3.81 $

Problem: Find the general solution of the equation, bringing it to canonical form. \[ u_{x x}+2 u_{x y}+u_{y y}-7 u_{x}-3 u_{y}=0 \]

11.5.4.3 With constant coefficients

5.08 $

Problem: Find all values of the parameter \( a \), for which the minimum value of the function \( f(x)=4 x^{2}-4 a x+a^{2}-2 a++2 \) on the segment \( x \in[0 ; 2] \) is equal to 3 .

17.20 USE problems

0 $

Problem: Find all the values of the parameter for which the roots of the equation \( (1+k) x^{2}-3 k x+4 k=0 \) belong to the segment \( [0 ; 5] \) ?

17.3 USE problems

3.3 $

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