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### Topics - Victor Ivrii

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31
##### Quiz-7 / Q7 TUT 0301
« on: November 30, 2018, 04:06:33 PM »
(a) Determine all critical points of the given system of equations.

(b) Find the corresponding linear system near each critical point.

(c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?

(d)  Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.
\left\{\begin{aligned} &\frac{dx}{dt} = (1 + x) \sin (y), \\ &\frac{dy}{dt} = 1 - x - \cos (y). \end{aligned}\right.

Bonus: Computer generated picture

32
##### Quiz-7 / Q7 TUT 0201
« on: November 30, 2018, 04:05:34 PM »
(a) Determine all critical points of the given system of equations.

(b) Find the corresponding linear system near each critical point.

(c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?

(d)  Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.
\left\{\begin{aligned} &\frac{dx}{dt} = 1 - xy, \\ &\frac{dy}{dt} = x - y^3. \end{aligned}\right.

Bonus: Computer generated picture

33
##### Quiz-7 / Q7 TUT 0101
« on: November 30, 2018, 04:02:45 PM »
(a) Determine all critical points of the given system of equations.

(b) Find the corresponding linear system near each critical point.

(c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?

(d)  Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.
\left\{\begin{aligned} &\frac{dx}{dt} = x + x^2 + y^2, \\ &\frac{dy}{dt} = y - xy. \end{aligned}\right.

Bonus: Computer generated picture

34
##### Quiz-7 / Q7 TUT 5301
« on: November 30, 2018, 04:00:08 PM »
Using argument principle along line on the picture, calculate the number of zeroes of the following function in the given annulus:
$$z^3- 3z+1 \qquad \text{in } \bigl\{1< |z| < 2\bigr\}.$$

35
##### Quiz-7 / Q7 TUT 5201
« on: November 30, 2018, 03:59:32 PM »
Using argument principle along line on the picture, calculate the number of zeroes of the following function in the first quadrant:
$$f(z)=z^4 - 3z^2 + 3.$$

36
##### Quiz-7 / Q7 TUT 5101
« on: November 30, 2018, 03:58:52 PM »
Using argument principle along line on the picture, calculate the number of zeroes of the following function in the upper half-plane:
$$z^4 + 3iz^2 + z - 2 + i.$$

37
##### Quiz-7 / TUT 0301
« on: November 30, 2018, 03:57:53 PM »
Using argument principle along line on the picture, calculate the number of zeroes of the following function in the first quadrant:
$$f(z)=z^2 + iz + 2 + i.$$

38
##### Quiz-7 / Q7 TUT 0203
« on: November 30, 2018, 03:57:02 PM »
Using argument principle along line on the picture, calculate the number of zeroes of the following function in the given annulus:
$$4z^3- 12z^2 + 2z + 10 \qquad \text{in }\ \bigl\{\frac{1}{2}< |z| < 2\bigr\}.$$

39
##### Quiz-7 / Q7 TUT 0202
« on: November 30, 2018, 03:56:08 PM »
Using argument principle along line on the picture, calculate the number of zeroes of the following function in the first quadrant:
$$f(z)=z^7 + 6z^3 + 7.$$

40
##### Quiz-7 / Q7 TUT 0201
« on: November 30, 2018, 03:53:54 PM »
Using argument principle along line on the picture, calculate the number of zeroes of the following function in the given annulus disk:
$$ze^z-\frac{1}{4} \qquad \text{in }\ \bigl\{0< |z| < 2\bigr\}.$$

41
##### Quiz-7 / Q7 TUT 0102
« on: November 30, 2018, 03:52:38 PM »

Using argument principle along line on the picture, calculate the number of zeroes of the following function in the upper half-plane:
$$2z^4 - 2iz^3 + z^2 + 2iz - 1.$$

42
##### Quiz-7 / Q7 TUT 0101
« on: November 30, 2018, 03:49:48 PM »
Using argument principle along line on the picture, calculate the number of zeroes of the following function in the given annulus:
$$z^4 - 2z - 2 \qquad \text{in }\ \bigl\{\frac{1}{2}< |z| < \frac{3}{2}\bigr\}.$$

43
##### End of Semester Bonus--sample problem for FE / FE Sample--Problem 5A
« on: November 27, 2018, 07:20:10 AM »
Determine the number of zeros of
$$2z^5 + 4z + 1.$$
(a) in the disk $\{z\colon |z|<1\}$;

(b) in the annulus $\{z\colon 1 <|z| < 2\}$.

(c) in the domain $\{z\colon |z|>2\}$.

Show that they are all distinct.

44
##### End of Semester Bonus--sample problem for FE / Readme
« on: November 27, 2018, 04:37:40 AM »
I will post here sample problems for FE. I will post just one version for problems of the type, covered by one of TT, and two versions for problems of the type, not covered by one of TT.

45
##### End of Semester Bonus--sample problem for FE / FE Sample--Problem 6
« on: November 27, 2018, 03:57:44 AM »
Calculate
$$\int_0^\infty \frac{x\sin (x)}{1+x^4}.$$

Hint:
Consider
$$\int _\Gamma f(z)\,dz \qquad \text{with } \ f(z)=\frac{ze^{iz}}{1+z^4}$$
over contour $\Gamma$ on the picture below:

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