Let $\gamma$ be a closed and continuously differentiable path in the upper half plane   $\{z\in C: z= x+iy, x\in R, y\in R, y\gt 0 \}$  not passing through the point $i$. Describe the set of all possible values of the integral $$\frac{1}{2\pi i}\int_{\gamma} \frac{2i}{z^2+1} dz$$  If $f$ doesn't contain $i$ then the value would be zero. If $f$ contains $i$ then it would be $1$.   But is this all??
let $\gamma$ be a closed continuosly differentiable path in the upper half plane  not passing through $i$. What are the possible values of the integral $$\frac{1}{2\pi i}\int_{\gamma}\frac{2i}{z^2+1}dz$$ well the integral can be broken like  $$\frac{1}{2\pi i}\int_{\gamma}\frac{2i}{(z+i)(z-i)}dz=$$  $$\frac{1}{2\pi i}\int_{\gamma}\frac{dz}{z-i}dz-\frac{1}{2\pi i}\int_{\gamma}\frac{dz}{z+i}dz=$$ by Cauchy Integral Formuale $$f(i)-f(-i)$$,so the second integral is $0$ as it is analytic in the upper half plane, but the first integral iss just $n(\gamma,i)$, which iss the windding number of $\gamma$ around $i$ what more I can say? Thank you.
Look at  edit. I didn't really expect that I would able to suggest adding  tag, but I could!  How is it possible? (seems that usual users can add mod-only tags without moderator help!)  Edit  Oded removed  back :-( I installed a nightly or unstable build of a custom ROM and now my phone doesn't work How could I retrieve the exact duplicate value from conditionally formatted list that highlight duplicates? I have imported a webpage table in to Google Sheets. It frequently being updated. A series of calculations has been conducted on the sheet named Shawn Of the link spreadsheet.  I have filtered the my desired results into the Calculation sheet of that linked document... which is in columns A5:A. I then applied =ARRAYFORMULA(if(A5:A200=&quot;&quot;,,A5:A200*{12,13,37,101,111})) in columns B5. The Results of the formula extends from B5:F. This results from a5:F was subsequently conditionally formatted to highlighted duplicates in the aforementioned, i.e. a5:f using the lists of values in K5:K and I5:5. In Columns A5:F, is it possible to generate/place values in the range A5:f with matching name of states and aligned values in J5:K and H5:I? How do I go about porting Ubuntu Touch to different devices? Show that $\mathbb{Z}/n^2\mathbb{Z}$ is not isomorphic to other group I'm currently working on a proof, but didn't get the right idea, a hint would be appreciated:  Show that $\mathbb{Z}/n^2\mathbb{Z}$ is not isomorphic to $\mathbb{Z}/n \times \mathbb{Z}/n$  So, lets assume there is an isomorphism between these groups, my first idea to was to construct a contradiction with the homomorphism property. But I don't know how, there should be an element in  $\mathbb{Z}/n \times \mathbb{Z}/n$ which is not in $\mathbb{Z}/n^2$ to reach this kind of contradiction. Of course there is $n \in \mathbb{Z}/n^2$ which is not in the other group but I don't know if this helps.  Thanks Calculation of subgroups of $(Z_{12}, +)$ How calculate all subgroups of $(Z_{12}, +)$?  I know that the order of subgroups divide the order of the group, but there is such a smart way to calculate the subgroups of order 6? Suppose $X_n$ is the fortune of a gambler after $n$ th game. Then the game is called fair (Breiman 1968) if   $$E[X_{n+1} \mid X_1, \dots, X_n] = X_n \forall n$$  My question is why a fair game is not defined as the following   $$E[X_{n+1}] = E[X_n] \forall n$$ i.e. $$E[X_{n+1}- X_n]=0$$. This should be the proper definition as a fair game is where avg. gain is zero. Nothing conditioning should be there. Drawing Mechanical Systems in LaTeX What is an adjective referring to “someone who puts people down”? I was just thinking about someone I went to school with who had a penchant for putting people down, and I was trying to think of a word that describes this, but couldn’t find it. It would be a synonym of caustic, abrasive, discouraging, and undermining, but these all have slightly different connotations (to wear away", cause discouragement or to make fall). I want an adjective that has the connotation of a hammer hitting you on the head. The word does not necessarily need to describe someone who does this on purpose, out of evil intent. Is there such a word?    The word should give you a mental picture of someone barraging you with negative comments, but it not necessarily affecting you. Think of someone who so often has something negative to “contribute” that many can just either ignore it or roll their eyes at it, and that is the extent of its effect. I keep getting the mental picture of the word buffet as in, how waves buffet the shore. A constant hitting, without the hitting necessarily being devastating in its destruction. “Have you seen...” or “Did you see...?” In the US, when people work together, they may look for a colleague or any person to say something or for any other reason, asking colleagues or other people this type of question:         Have you seen Michael?   (OR)        Did you see Michael?          Which is considered right if the question refers to an activity (in our case "see") during the day and not in the past in general? Using getline(cin, s) after cin Global variables in AngularJS "Access is denied" JavaScript error when trying to access the document object of a programmatically-created  (IE-only) Does the limit $\lim _{x\to 0} \frac 1x \int_0^x \left|\cos \frac 1t \right| dt$ exists? Does the limit $\lim _{x\to 0} \frac 1x \int_0^x \left|\cos \frac 1t \right| dt$ exists ? If it does then what is the value ?   I don't think even L'Hospital's rule can be applied . Please help . Thanks in advance What makes induction (over natural numbers) a valid proof technique? Is $$ \dfrac{ P(0) \quad \forall i \in \mathbb{N}. P(i) \Rightarrow P(i+1) }{ \forall n \in \mathbb{N}.  P(n)} $$ just taken for granted as a proof rule, or can it be derived from more foundational axioms?  Similarly, can the principle of induction over well-founded sets be derived from something more foundational, or does it have to be assumed to be a valid inference rule? Execute a program at login I'm running Arch-Linux ARM on a Raspberry Pi.  My project goal is to have a barcode scanner attached to the Pi that forwards the scanned messages to a server in a LAN.  To harden the system against power outages etc. and minimize required user interaction, I wish to set it up so that a certain user is logged in automatically at startup (done and works) and a specific program is to be run at startup (done and works but not as intended).  I've done this so far by adding  exec mono scannerSoftware.exe 127.0.0.1 1234 randomString   to /home/certainUser/.bash_profile so it gets executed only once even when I'm switching back and forth to root (in contrast to bashrc).    The particular problem I am having: When I launch the system, it works as intended (login as certainUser, starting scannerSoftware.exe); BUT if I want to switch users and hit Ctrl+C, it restarts the login process, logging certainUser in again and running the program again => back to where I came from.  My guess as to why this happens is that the sys detects "login not complete, better start it again".  I could circumvent this by spamming Ctrl+C right around the login so that I practically interrupted it. Of course, this is not good practice. Suffice it to say it is not practical at all.  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