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### Topics - Victor Ivrii

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61
##### 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:
$$z^3- 3z+1 \qquad \text{in } \bigl\{1< |z| < 2\bigr\}.$$

62
##### 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:
$$f(z)=z^4 - 3z^2 + 3.$$

63
##### 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:
$$z^4 + 3iz^2 + z - 2 + i.$$

64
##### 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:
$$f(z)=z^2 + iz + 2 + i.$$

65
##### 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:
$$4z^3- 12z^2 + 2z + 10 \qquad \text{in }\ \bigl\{\frac{1}{2}< |z| < 2\bigr\}.$$

66
##### 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:
$$f(z)=z^7 + 6z^3 + 7.$$

67
##### 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:
$$ze^z-\frac{1}{4} \qquad \text{in }\ \bigl\{0< |z| < 2\bigr\}.$$

68
##### Quiz-7 / Q7 TUT 0102
« on: November 30, 2018, 03:52:38 PM »
$\renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}}$

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

69
##### 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:
$$z^4 - 2z - 2 \qquad \text{in }\ \bigl\{\frac{1}{2}< |z| < \frac{3}{2}\bigr\}.$$

70
##### 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.

71
##### 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.

72
##### 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:

73
##### End of Semester Bonus--sample problem for FE / FE Sample--Problem 5
« on: November 27, 2018, 03:57:31 AM »
Show that the equation
$$e^{z}=e^2z$$
has a real root in the unit disk $\{z\colon |z|<1\}$.

Are there non-real roots?

74
##### End of Semester Bonus--sample problem for FE / FE Sample--Problem 4
« on: November 27, 2018, 03:57:15 AM »
(a) Find the Mobius's transformation $f(z)$ mapping the unit disk $\{z\colon |z|<1\}$ onto exterior $\{w\colon |w|>1\}$ of the unit disk, such that $f(0)=5$ and $f(-1)=-1$.

(b) Find the fixed points of $f$ (points s.t. $f(z)=z$).

(c) Find the stretch ($|f'(z)|$) and the rotation angle ($\arg(f'(z))$) of $f$ at $z$.

75
##### End of Semester Bonus--sample problem for FE / FE Sample--Problem 3
« on: November 27, 2018, 03:56:33 AM »
Find all singular points, classify them, and find residues at these points of
$$f(z)= \frac{\cos(z/6)}{\sin^2(z)} + \frac{z}{\sin(z)}.$$
infinity included.

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