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Messages - Victor Ivrii

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1
MAT334--Lectures & Home Assignments / Re: 2.5 Q19
« on: Today at 05:21:19 PM »
If we don't know that series is converging on the circle, we integrate along, the integration is senseless

2
When considering system
\begin{equation*}
\mathbf{x}'=A\mathbf{x} \qquad \text{with } \
A=\begin{pmatrix} a & b \\ c & d\end{pmatrix}
\end{equation*}
and discovering that it has two adjoint complex (not real) eigenvalues $\lambda_\pm =\mu \pm i\nu$, one should not only tell, if it is an unstable ($\mu>0$) or stable ($\mu <0$) focal point, or a center ($\mu =0$), but also the direction of rotation.

Observe that, $\lambda_\pm$ are roots of the equation $\lambda ^2- (a+d)\lambda + ad -bc=0$ with the discriminant $D:=(a+d)^2-4(ad -bc))=(a-d)^2+4bc$, and we consider the case $D<0\implies bc <0$. Thus $b$ and $c$ are not $0$ and have opposite signs.

Then, if $b<0$ (and $c>0$) rotation is counter-clockwise, and if $b>0$ (and $c<0$) rotation is clockwise.

3
MAT334--Lectures & Home Assignments / Re: 2.5 Q19
« on: Today at 08:17:46 AM »
not clear what do you mean by "let $n=...$". $n$ runs from $-\infty$ to $\infty$ and you must check all values

4
MAT334--Lectures & Home Assignments / Re: 2.5 Q19
« on: Today at 04:43:31 AM »
Well, there was a hint provided:if
$$
f(z)=\sum _{n=-\infty}^\infty a_n (z-z_0)^n =0
$$
as $|z-z_0|=s=$ with $s\in (r,R)$ then
$$
0=\int_\gamma f(z)(z-z_0)^{-m-1}\,dz =\sum _{n=-\infty}^\infty \int_\gamma a_n (z-z_0)^{n -m-1}\,dz
$$
while $\int_\gamma  (z-z_0)^{k}\,dz=0$ for $k\ne -1$ and $2\pi i$ for $k=1$.

5
MAT334--Lectures & Home Assignments / Re: 2.5 Example 5
« on: Today at 04:38:20 AM »
So, we need to calculate residue in each pole of
$$\frac{z+1}{(z^2+4)(z-1)^3}.
$$
Points $\pm 2i$ are easy since they are simple poles, with the residues equal to
$$
\frac{z+1}{(z^2+4)'(z-1)^3}\Bigr|_{z=\pm 2i}=\frac{z+1}{2z (z-1)^3}\Bigr|_{z=\pm 2i}=\frac{\pm 2i+1}{\pm 4i (\pm 2i-1)^3}=...
$$
Point $z=1$ is more tricky since it is a triple pole but the factor $(z-1)^3$ is already separated, so we need to find a coefficient at $(z-1)^2$ in the decomposition of $g(z)=\frac{z+1}{(z^2+4)}$ at $z=1$; it is $\frac{1}{2}g''(z)$ at that point.

But there is a simpler way to find a residue at $z=1$. The function is meromorphic in the extended complex plane, having only isolated points. Then the sum of all residues should be $0$, and we need to include $\infty $ in the tally. But since at infinity the function decays faster than $z^{-1}$, the residue there is $0$. So
$\newcommand{\Res}{\operatorname{Res}}$
$$
\Res (f, 1)= -\Res (f,2i) -\Res (f, -2i).
$$

6
Quiz-5 / Q5 TUT 0501
« on: Today at 04:21:35 AM »
It looks like I missed it

Transform the given system into a single equation of second order and find the solution $(x_1(t),x_2(t))$, satisfying initial conditions
$$\left\{\begin{aligned}
& x'_1= -0.5x_1 + 2x_2, &&x_1(0) = -2,\\
&x'_2= -2x_1 - 0.5x_2, &&x_2(0) = 2
\end{aligned}\right.$$

7
Quiz-5 / Re: Q5 TUT 0601
« on: Today at 04:19:58 AM »
Not sure.... but TUT 0601 actually have this question:
I am really not sure how it could happen.... but it looks as it did. So I will look at this matter and decide how to mitigate this

8
MAT244--Misc / Re: Quiz 5 explanation (was Unfair quiz)
« on: November 17, 2018, 07:04:08 PM »
I checked what I have on my computer... looks like what I posted on forum... Can u reproduce the question -- there could be some mishap (brought the wrong file to printshop?)

9
Quiz-6 / Q6 TUT 5301
« on: November 17, 2018, 04:14:52 PM »
Locate each of the isolated singularities of the given function $f(z)$ and tell whether it is a removable singularity, a pole, or an essential singularity.
If the singularity is removable, give the value of the function at the point; if the singularity is a pole, give the order of the pole:
$$
f(z) =\pi \cot(\pi z).
$$

10
Quiz-6 / Q6 TUT 5201
« on: November 17, 2018, 04:12:56 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.
$$
f(z)=\frac{1}{1-\cos(z)};\qquad z_0=0\quad\text{(four terms of the Laurent series)} .
$$

11
Quiz-6 / Q6 TUT 5101
« on: November 17, 2018, 04:12:01 PM »
Locate each of the isolated singularities of the given function $f(z)$ and tell whether it is a removable singularity, a pole, or an essential singularity.

If the singularity is removable, give the value of the function at the point; if the singularity is a pole, give the order of the pole:
$$
f(z)= \frac{e^z-1}{e^{2z}-1}.
$$

12
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.
$$
f(z)=\frac{\sin(z)}{(z-\pi)^2};\qquad z_0=\pi.
$$

13
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
$$
\Res \bigl(\frac{f'}{f}; z_0\bigr)=-l.
$$

14
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.
$$
f(z)=\frac{1}{e^z-1};\qquad z_0=0\quad \text{(four terms of the Laurent series)} .
$$

15
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.
$$
f(z)=\frac{z^2}{z^2-1};\qquad z_0=1.
$$

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