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Messages - Victor Ivrii

Pages: 1 [2] 3 4 ... 148
16
Quiz-6 / Q6 TUT 0301
« on: November 17, 2018, 04:11:06 PM »
Find the Laurent series for the given function $f(z)$ about the indicated point. Also, give the residue of the function at the point.
$$
f(z)=\frac{\sin(z)}{(z-\pi)^2};\qquad z_0=\pi.
$$

17
Quiz-6 / Q6 TUT 0203
« on: November 17, 2018, 04:10:26 PM »
$\newcommand{\Res}{\operatorname{Res}}$
If $f$ is analytic in $\{z\colon 0< |z - z_0| < R\}$ and has a pole of order $l$ at $z_0$ , show that
$$
\Res \bigl(\frac{f'}{f}; z_0\bigr)=-l.
$$

18
Quiz-6 / Q6 TUT 0202
« on: November 17, 2018, 04:09:19 PM »
Find the Laurent series for the given function $f(z)$ about the indicated point. Also, give the residue of the function at the point.
$$
f(z)=\frac{1}{e^z-1};\qquad z_0=0\quad \text{(four terms of the Laurent series)} .
$$

19
Quiz-6 / Q6 TUT 0201
« on: November 17, 2018, 04:08:42 PM »
Find the Laurent series for the given function $f(z)$ about the indicated point. Also, give the residue of the function at the point.
$$
f(z)=\frac{z^2}{z^2-1};\qquad z_0=1.
$$

20
Quiz-6 / Q6 TUT 0102
« on: November 17, 2018, 04:08:01 PM »
Find the Laurent series for the given function $f(z)$ about the indicated point. Also, give the residue of the function at the point.
$$
f(z)=\frac{z}{\sin^2(z)};\qquad z_0=0\quad \text{(four terms of the Laurent series)} .
$$

21
Quiz-6 / Q6 TUT 0101
« on: November 17, 2018, 04:07:17 PM »
$\newcommand{\Res}{\operatorname{Res}}$
If $f$ is analytic in $\{z\colon |z - z_0| < R\}$ and has a zero of order $m$ at $z_0$ , show that
$$
\Res \bigl(\frac{f'}{f}; z_0\bigr)=m.
$$

22
Quiz-6 / Q6 TUT 5102
« on: November 17, 2018, 04:01:01 PM »
The coefficient matrix contains a parameter $\alpha$.

(a) Determine the eigenvalues in terms of $\alpha$.
(b)  Find the critical value or values of  $\alpha$  where the qualitative nature of the phase portrait for the system changes.
(c) Draw a phase portrait for a value of  $\alpha$ slightly below, and for another value slightly above, each critical value.
$$\mathbf{x}' =\begin{pmatrix}
4 &\alpha\\
8 &-6
\end{pmatrix}\mathbf{x}.$$

23
Quiz-6 / Q6 TUT 5101
« on: November 17, 2018, 03:59:47 PM »
The coefficient matrix contains a parameter $\alpha$ . In each of these problems:

(a) Determine the eigenvalues in terms of $\alpha$.
(b)  Find the critical value or values of  $\alpha$  where the qualitative nature of the phase portrait for the system changes.
(c) Draw a phase portrait for a value of  $\alpha$ slightly below, and for another value slightly above, each critical value.
$$\mathbf{x}' =\begin{pmatrix}
2 &-5\\
\alpha & -2
\end{pmatrix}\mathbf{x}.$$

24
Quiz-6 / Q6 TUT 0801
« on: November 17, 2018, 03:58:12 PM »
Find the general solution of the given system of equations:
$$\mathbf{x}'=
\begin{pmatrix}
1 &1 &1\\
2 &1 &-1\\
-8 &-5 &-3
\end{pmatrix}\mathbf{x}.$$

25
Quiz-6 / Q6 TUT 0701
« on: November 17, 2018, 03:57:36 PM »
Find the general solution of the given system of equations:
$$\mathbf{x}'=
\begin{pmatrix}
3 &2 &4\\
2 &0 &2\\
4 &2 &3
\end{pmatrix}\mathbf{x}.$$

26
Quiz-6 / Q6 TUT 0601
« on: November 17, 2018, 03:56:07 PM »
The coefficient matrix contains a parameter $\alpha$.

(a) Determine the eigenvalues in terms of $\alpha$.
(b)  Find the critical value or values of  $\alpha$  where the qualitative nature of the phase portrait for
the system changes.
(c) Draw a phase portrait for a value of  $\alpha$ slightly below, and for another value slightly above,
each critical value.
$$\mathbf{x}' =\begin{pmatrix}
0 &-5\\
1 &\alpha
\end{pmatrix}\mathbf{x}.$$

27
Quiz-6 / Q6 TUT 0501
« on: November 17, 2018, 03:54:44 PM »
Express the general solution of the given system of equations in terms of real-valued functions:
$$\mathbf{x}' = \begin{pmatrix}
-3 &0 &2\\
1 &-1 &0\\
-2 &-1 &0
\end{pmatrix}\mathbf{x}.$$

28
Quiz-6 / Q6 TUT 0401
« on: November 17, 2018, 03:54:02 PM »
The coefficient matrix contains a parameter $\alpha$.

(a) Determine the eigenvalues in terms of $\alpha$.
(b) Find the critical value or values of  $\alpha$  where the qualitative nature of the phase portrait for
the system changes.
(c) Draw a phase portrait for a value of  $\alpha$ slightly below, and for another value slightly above,
each critical value.

$$\mathbf{x}' =\begin{pmatrix}
\alpha  &1\\
-1 &\alpha
\end{pmatrix}\mathbf{x}.$$

29
Quiz-6 / Q6 TOT 0301
« on: November 17, 2018, 03:52:31 PM »
Find the general solution of the given system of equations:
$$\mathbf{x}'=
\begin{pmatrix}
1 &1 &2\\
1 &2 &1\\
2 &1 &1
\end{pmatrix}\mathbf{x}.$$

30
Quiz-6 / Q6 TUT 0201
« on: November 17, 2018, 03:52:00 PM »
 Express the general solution of the given system of equations in terms of real-valued functions:
$$\mathbf{x}' = \begin{pmatrix}
1 &0 &0\\
2 &1 &-2\\
3 &2 &1
\end{pmatrix}\mathbf{x}.$$

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