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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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