Topics

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.