The question:  Let $\gamma$ be a contour such that $0 \in I(\gamma),$ where $I$ is the interior of the contour. Show that   $$\int_\gamma z^n \, \text{d}z = \begin{cases} 2\pi i &amp; \text{if }  n = -1 \\ 0 &amp; \text{otherwise} \end{cases}$$  By taking $\gamma$ as the ellipse  $$\{ (x,y) : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \},$$ show that $$ \int_0^{2 \pi } \frac{dt}{a^2 \cos^2 t + b^2 \sin^2 t} = \frac{2 \pi}{ab}.$$  The question's answer uses the Deformation theorem and the fact that the first integral has the given value if the contour is a unit circle. However, the two contours must not overlap, so it seems like this should only be true for a contour that either always has magnitude less than one or greater than one. In Mathematica, the final integral was true for the case that $a = 1.5$ and $b=0.4$. What am I missing?  Edit: The radius of the circle cancels in the integral if $n = -1$, so then it does hold irrespective of the radius.
I want to prove:  $$\int_{0}^{2\pi}\frac{1}{a^{2}\cos^2t+b^{2}\sin^{2}t}dt=\frac{2\pi}{ab}$$  by the deformation theorem of complex variable.  Then I consider a parameterization $\gamma:[0,2\pi]\rightarrow A$, traveled in the opposite direction of the clock hand, of the ellipse (in $\mathbb{C}$):  $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$  I thought of searching for a function $f:A\rightarrow \mathbb{C}$ (analytic in $A$) and a curve $\lambda:[0,2\pi]\rightarrow A$ (homotopic to $\gamma$) and use the deformation theorem:  $$\int_{\gamma}f = \int_{\lambda}f$$  But I can not find a function $f(z)$ that meets what I need, can you help me to find the f?   P.D: Necessarily I have to use the theorem of deformation
"Is there" versus "Are there" Are there any questions I should be asking?   Is there any articles available on the subject?   My instinct is that in the two questions above, it should be 'are' as the subjects of the sentences (questions/articles) are plural.  However I also recall being told that the singular 'is' is valid here too, and of course see it very often in usage online, and this has left me a bit unsure.  Update: I have stumbled upon the reason for the confusion.     Is there any water?   Are there any questions?   The sentence structure is the same but of course water is not plural, although it is not singular either. Meta sites in "Your Communities" display different reputation While using the top bar to navigate across communities, I noticed that the reputation displayed on meta sites on that list does not match the one of their corresponding main sites:    I have noticed on past situations that reputation gained takes a bit to "update" on the display on Meta sites (that is, if you gain rep on SO, for example, it will take some time for you to see your updated rep on MSO).  However, I have never noticed this behavior here, on the Sites List of the top bar. Furthermore, the "delay" here seems to be more severe, as the "outdated" rep showed is of several days ago (for example, I turned 3k on SO almost exactly a week ago, but the rep shown on MSO is still &lt;3k).  Any idea what could be causing this behavior?    Update: 2 months later this persists: my main site rep updates correctly but their corresponding meta rep remains the same as in the image above.     Update: As per Catija's  I removed the site (MSO) and put it back in, but it didn't update the rep.   However, after removing one site (TWP Meta), waiting at least a day, and then adding it back I see that although the reputations still don't match, the number displayed did changed a bit (previously 29,924 but now 33,112): Get variable from PHP to JavaScript I want to use a PHP variable in JavaScript. How is it possible? Collection of clopen sets form a base for a countable completely regular space I have $X$ as a countable, Tychonoff space, and I want to show that the collection of clopen subset of $X$ form a base for the topology on $X$.  Can I first just define a base $\mathscr{B}$ of X, let $x\in X$ and $E \subset X$ is closed such that $x\notin E$. So $x\in X-E$, which is open.   I also noticed that the interval $[0,1]$ is uncountable. Can I define a function $f: X\to [0,1]$, then f is onto. Do I need to show that $f$ continuous next? Story where radically different cultures live together and are defined by rulesets, some choose not to see each other, a house is used as a spaceship I think this novel came out in the 2000s.  There are many cultures that live close together but are very different, for example some pretend to be pre-industrial.  One of the main characters is a youth who grew up in a pretend pre-industrial society and is now exploring the rest of the solar system? ring-world?   Each culture experiences their own reality that is defined by a ruleset.  One of the rulesets becomes aggressively self-propagating and this is the problem that drives the plot.  At some point a tumbling house, just a normal house, is used to travel across space. Cross-thread operation not valid: Control accessed from a thread other than the thread it was created on I often read outputs in the terminal that get wrapped as they are too long. Since there is no way that I can modify the printed output, I need to let the Terminal allow for horizontal scrolling.  Is there a modification I can make to the default OS X Terminal to disable line wrapping and allow for horizontal scrolling?  Or do you know another alternative I can use? Show that:  If $U$ is a principal ultrafilter, then the canonical inmersion $j$ is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$ How to mount electronics on a vehicle chassis safely I want to implement a control system I designed for a light electric motorbike. I would like to mount it as professionally as possible on the chassis. It consists of a power system pcb and a SAMC21 development board which I'll merge later once this all works properly. I have tried screwing it directly on the aluminium chassis but have found that shocks and vibrations from the horn disturbed the system or even destroyed it upon large shocks like a few hammer hits. Is there a specific way car and motorcycle manufacturers mount their electronics to make their system as robust as possible? Thank you for your help! Why is $a^n - b^n$ divisible by $a-b$? What does this triangle/diamond schematic symbol represent? Could someone please tell me what the symbol ringed in red represents?     I know it has ON Switch pointing to it, but the symbol has other lables in different parts of the document. Why isn't there a truly unified package manager for Linux? Why isn't there a unified package manager that acts as an interface between the end-user and the underlying low-level package manager (apt, yast, pacman, etc.)?  Is it hard to do and therefore not practical, or is there a genuine obstacle making it impossible to do? Can you create a strong blockcipher with small blocksize, given a strong blockcipher of conventional blocksize? Suppose I want a strong 20-bit blockcipher.  In other words, I want a function that takes a key (suppose the key is 128 bits), and implements a permutation from 20 bits to 20 bits.  The set of permutations should be close to a randomly-chosen subset of size $2^{128}$ of all $2^{20}!$ permutations on 20 bits.  I don't want you to build a new blockcipher from scratch.  Instead, assume you have a "strong" blockcipher like AES (128-bit blocksize, 128-bit keysize) and use this to build your 20-bit cipher.   Of course, your construction should be practical (i.e., you should be able to run it in both directions with reasonable amounts of time and memory). Diffeomorphism group of the unit circle I am given to understand that the group of diffeomorphisms of the unit circle, $\operatorname{Diff}(\mathbb{S}^1)$, has two connected components, $\operatorname{Diff}^+(\mathbb{S}^1)$ and $\operatorname{Diff}^-(\mathbb{S}^1)$, the diffeomorphisms that preserve or reverse the canonical (counterclockwise) orientation, respectively.  Question 1: How does one prove that?  Question 2: Given $\Phi, \Psi \in \operatorname{Diff}^+(\mathbb{S^1})$, can one construct an explicit path joining them?  I'd be satisfied already with a proof that we can join $\Psi, \Phi \in \operatorname{Diff}^+ (\mathbb{S}^1)$, without giving the path explicitly. I tried the obvious path, $t \mapsto \dfrac{t\Phi + (1-t)\Psi}{|t\Phi + (1-t)\Psi|}$, but that doesn't seem to work.  Thanks. Accumulation Points for $S = \{(-1)^n + \frac1n \mid n \in \mathbb{N}\}$ I was recently asked to find the accumulation points of the set $$S = \{(-1)^n + \frac{1}{n} \mid n \in \mathbb{N}\}$$  I answered that the accumulation points are $\{-1,1\}$, because despite the fact that $\frac{1}{n}$ diverges, we can still use $\frac{1}{n}$ to find an element arbitrarily close to either $1$ or $-1$, i.e. to show that there is an element in an arbitrarily small neighborhood of $1$ or $-1$. (No proof was required for this question --- this is my intuition.)  Am I right at all? Identify geographic location to points I am quite new in QGIS and I really need your help.  I have a map of Kenya divided into 68 districts and I have coordinates of 30.000 schools. The points have one variable besides the latitude and longitude data called "distance to nearest school".   First I imported the map. Then I imported the spreedsheet and transformed it to a SHP file like the map. So now I have two SHP files.    The points have three variables: "Latitude", "Longitude" and "Distance to nearest school". I want to have a fourth variable called "District" and take the mean of the "Distance to nearest school" for each district. What is the difference between single-quoted and double-quoted strings in PHP? For f continuous on $[0,1]$, show that there exist points $\alpha_k$ such that $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $ How would you know if your (high dimensional) data exhibits enough clustering so that results from kmeans or other clustering algorithm is actually meaningful?  For k-means algorithm in particular, how much of a reduction in within-cluster variance should there be for the actual clustering results to be meaningful (and not spurious)?  Should clustering be apparent when a dimensionally-reduced form of the data is plotted, and are the results from kmeans (or other methods) meaningless if the clustering cannot be visualized? Finding $\lim_{n\to\infty}\binom{2n}{n}$ I found a limit problem which is $\lim_{n\to\infty}{2n \choose n}$ as we know that $\sum_{i=0}^{n}{n \choose i}^2={2n \choose n}$ now we could transform the given problem as $\lim_{n\to\infty} \sum_{i=0}^{n}{n \choose i}^2$ and as $\sum_{i=0}^{n}{n \choose i}^2$ is a polynomial in $n$, therefore, $\lim_{n\to\infty}\sum_{i=0}^{n}{n \choose i}^2=\infty$ Please tell me if my solution is correct or not,and if it is incorrect then please provide me a solution.  Thanks