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**Home Assignment 2 / Re: problem4 (20)**

« **on:**January 19, 2019, 07:03:34 AM »

Please learn how to post math properly Also, asking for help, copy the problem.

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Please learn how to post math properly Also, asking for help, copy the problem.

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Since integral curves are rays (straight half-lines) from $(0,0)$ the solution in the plane wit the punched out origin is any function, constant along these rays, in particular $u=f(y/x)$.

But if we want solution in the whole plane, $u$ must be continuous at $(0,0)$ and since all rays intersect there $u$ is just a constant.

But if we want solution in the whole plane, $u$ must be continuous at $(0,0)$ and since all rays intersect there $u$ is just a constant.

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Read Section 2.1, Subsection "semilinear equations"

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Junjing, you are right but I am not sure if anyone but me would be able to read your solution

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You draw just a bit more than one period and say "$\pi$-periodic"

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Physical interpretation is useful. However Quizzes, Tests and Exam neither test nor require any knowledge of Physics. Like in ODE or Calculus I, II classes.

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The main purpose of this forum is not the communication with instructors but peer-to-peer. I noticed that some of you answer the questions of your peers and I really appreciate, but I would like to see more -- much more -- of this!

It runs as aid to the class website. More permanent and systematic information is placed on website, while forum is for discussion.

- You can select any
**username**(login) you want (take a rather short one). Username is what only you (and Admin) can see. - However change your (screen)
**Name**(the one which everybody sees) to one easily identifiable (Admin should be able to identify you with a student enrolled to this course--and matching BlackBoard name; otherwise your posting privileges could be reduced or revoked): Open Profile > Account Settings and look for "Name". - It would be preferable to use your University of Toronto email address.
- BTW you can have a cool avatar too. Please do not display any confidential info.
- You cannot change the loginname (used for login). But there is no need. Nobody except you knows your password - even Admins - but Admins don't need to know it.

- Please use different boards according to their description (I will create new ones if needed).
- Do not
**hijack topics**: if a topic is devoted to some question, do not post anything which is not related. Start a new topic. - On the other hand, do not start new topic answering to the existing post; use the same topic.
- Do not put in the same post several not related things - make several posts instead (in different topics).
- Do not post solution which coincides with the solution already posted by someone else; exception: typed solution beats the scanned one.
- Do not
**spam**(post irrelevant messages, especially to promote something),**flood**(post many messages with very little or no sense, like "me too", "I like this post", ...) or**flame**(messages, containing personal attacks and heated arguments). - Do not thank instructors.
- Do not post snapshots of paper documents, scan them (with the scanner or smartphone).
- Do not use red color (it belongs to Admins and Moderators), large fonts
- Read and search before postings.
- Use common sense.

- This semester
**karma does not translates into Bonus mark points**(there are plenty reasons for this) - On the other hand,
**your posts and active participation in the class will be a decisive factor if you ask me for a recommendation letter for a Graduate School**. To write such letter I need to know something about you beyond your marks (and your final mark is in the transcript anyway). Participation in the forum automatically leaves a trace.

- If you have a question which you think is of general interest--ask it on the Forum rather than via email.

- Both this forum and website use MathJax to display mathematics.

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For those who had < 4 for this Quiz Quiz mark was based on the best 4.5 quizzes (which means that the 5-th best had a half weight).

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Karma * 0.4. I extracted karma from database but because I did it in rush to submit emarks before Dean's office closes , I could miss several students, especially those whose Screen name on forum differed from their Quercus name. Please email me, if it is the case: I am planning to amend marks Dec 24. My changes will not affect ACORN marks until Jan 8 for sure, and may be a bit later

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😈 👿 👹 👺 💀 👻 👽 🤖 💩 😺 😸 😹 😻 😼 😽 🙀 😿 😾

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for Quizzes and Bonuses Fri

Final -- I hope Fri (actually, I am required to do it in 7 business days) but it depends on circumstances beyond my control. First, other graders. I graded the most difficult to grade P5 in 4 days, then I graded MAT334, and some of graders still have not finished. If they finish tonight, I will do my overhead overnight and submit Friday morning.

Second, marks need to be approved. If they are not approved by our undergraduate chair and then Dean's office before 5 pm, they are delayed until the first day of the next semester, and I am not allowed to release Crowdmark links until their approval.

Final -- I hope Fri (actually, I am required to do it in 7 business days) but it depends on circumstances beyond my control. First, other graders. I graded the most difficult to grade P5 in 4 days, then I graded MAT334, and some of graders still have not finished. If they finish tonight, I will do my overhead overnight and submit Friday morning.

Second, marks need to be approved. If they are not approved by our undergraduate chair and then Dean's office before 5 pm, they are delayed until the first day of the next semester, and I am not allowed to release Crowdmark links until their approval.

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\begin{gather*}

g(w)=\lambda \frac{w-a}{1- w\bar{a}}\\

\text{with $|\lambda|=1$ and $a=\frac{1}{2}$:}\\

g(w)=\lambda \frac{2w-1}{2- w}.

\end{gather*}

Since $g(-1)=1$ we have $1=\lambda \frac{-2-1}{2+1}\implies \lambda=-1 $ and therefore

$$

g(w)= - \frac{2w-1}{2- w}=z \implies (2w-1)=-z(2-w)\implies w= \frac{2z-1}{z-2}=: f(z). $$

$$

\frac{2z-1}{z-2}=z\implies 2z-1=z^2-2z\implies z^2-4z +1=0\implies z=2\pm \sqrt{3}.$$

These are "black" points on the picture.

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$$

|P(z)-Q(z)|=|2iz -3-i|\le 4+|3+i|=4+\sqrt{10}<8;$$

therefore $P$ has as many roots in $\{z\colon |z|<2\}$ as $Q(z)$ has, which is $3$ (we count orders).

$$

|P(w)-Q(w)|=|w^3+3w^2 -i| < 5.

$$

Indeed, $|P(w)-Q(w)|\le |w+3|+1< 5$ except $w=1$, and for $w=1$ we have $|w^3+3w^2 -i|=|4-i|=\sqrt{17}<5$. Therefore $P(w+1)$ has as many roots in $\{w\colon |w|<1\}$ as $Q(w)$ has, which is $1$ (we count orders).

Finally, in the domain $\{z\colon |z-1|>1, |z|<2\}$, there are $3-1=2$ roots.

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$\newcommand{\Res}{\operatorname{Res}}$

**(a)** Singular points are of $g(z)=\tan(z)$ and $h(z)=z\cot ^2(z)$, that is $z_n= (n+\frac{1}{2})\pi$ and $w_n = \pi n$.

**(b)** $z_n$ are simple poles and $\Res (f,z_n)= \Res (\tan(z), z_n)= \frac{\sin(z)}{(\cos (z))'}\bigr|_{z=z_n} =-1$.

$w_0$ is a simple pole

$$\Res (f,w_0)= \Res (z\cot^2(z), w_0)= \Res (z\cot (z) \times \cot(z), 0)= \Res (\cot(z),0)= 1$$

because $\lim _{z\to 0} z\cot(z)=1$.

$w_n$ with $n\ne 0$ are double poles and

\begin{align*}

\Res (f,w_n)=&\Res (z\cot^2 (z), w_n)= \Res ((\pi n +w)\cot^2 (w) , 0) =\\

&\pi n\Res (\cot^2 (w) , 0) +\Res (w\cot^2(w),0) =1

\end{align*}

because $\Res (\cot^2 (w) , 0)=0$ (since $\cot^2(w)$ is an even function) and $\Res (w\cot^2(w),0) =1$ we already calculated.

$\infty$ is a not isolated singularity and therefore residue here is not defined.

$w_0$ is a simple pole

$$\Res (f,w_0)= \Res (z\cot^2(z), w_0)= \Res (z\cot (z) \times \cot(z), 0)= \Res (\cot(z),0)= 1$$

because $\lim _{z\to 0} z\cot(z)=1$.

$w_n$ with $n\ne 0$ are double poles and

\begin{align*}

\Res (f,w_n)=&\Res (z\cot^2 (z), w_n)= \Res ((\pi n +w)\cot^2 (w) , 0) =\\

&\pi n\Res (\cot^2 (w) , 0) +\Res (w\cot^2(w),0) =1

\end{align*}

because $\Res (\cot^2 (w) , 0)=0$ (since $\cot^2(w)$ is an even function) and $\Res (w\cot^2(w),0) =1$ we already calculated.

$\infty$ is a not isolated singularity and therefore residue here is not defined.

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\begin{align*}

w= \frac{1}{2}\bigl(re^{it} +r^{-1}e^{-it}\bigr)&= \cosh(s)\cos(t)+i\sinh(s)\sin (t)

\end{align*}

with $s=\ln (r)$ (and $r=e^s$).

Then $w=u+iv$ with $\frac{u^2}{a^2}+\frac{v^2}{b^2}=1$, $a=\cos(s)$, $b=\sinh(s)$ and also

$\frac{u^2}{A^2}-\frac{v^2}{B^2}=1$,

with $A=\cos (\theta)$, $B=\sin(\theta)$, $A^2+B^2=1$.

$w =\frac{1}{2}(e^{it}+e^{-it})=\cos(t)$ and it runs a line segment $[-1,1]\subset\mathbb{R}$. So $f$ maps the unit disk onto compliment of it $\mathbb{C}^*\setminus [-1,1]$ (and $0$ is mapped to $\infty$).

$z_1z_2=1$ and $z_1+z_2=2w$; exactly one of them for $w\notin [-1,1]$ is in unit disk, and the second one is in $\{z\colon |z|>1\}$.