I want to calculate $\int _{x=0}^{\infty} \frac{\sin(x)}{x}$ with the function $f(z) = \frac{e^{iz}}{z}$.  I thought about using the closed path $\Gamma = \gamma _1 + \gamma _R + \gamma _2 + \gamma _{\epsilon}$, when:  $\gamma_1 (t) = t, t \in [i\epsilon, iR]$  $\gamma_R (t) = Re^{it}, t \in [-\frac{\pi}{2}, \frac{\pi}{2}]$  $\gamma_2 (t) = t, t \in [-iR, -i\epsilon]$  $\gamma_{\epsilon} (t) = \epsilon e^{it}, t \in [-\frac{\pi}{2}, \frac{\pi}{2}]$  I use the fact that $\frac{\sin(x)}{x}$ is an even function and has an anti derivative, so the integral on a closed path is zero.  I managed to show that $\int_{\gamma _{\epsilon}} f = -i\pi$ when $\epsilon \to 0$.  However I am struggling to show that $\int_{\gamma _R} f = 0$ when $R \to \infty$  Help would be appreciated
A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int\limits_0^\infty \frac{\sin x} x \,\mathrm  dx = \frac \pi 2$$ Well, can anyone prove this without using Residue theory?  I actually thought of using the series representation of $\sin x$: $$\int\limits_0^\infty \frac{\sin x} x \, dx = \lim\limits_{n \to \infty} \int\limits_0^n \frac{1}{t} \left( t - \frac{t^3}{3!} + \frac{t^5}{5!} + \cdots \right) \,\mathrm  dt$$ but I don't see how $\pi$ comes here, since we need the answer to be equal to $\dfrac{\pi}{2}$.
I am thinking about buying a new lens specifically to take photos of distant targets (birds, bears, etc). I currently own the Canon 70-300 IS USM, which is a nice lens, but I find that if I want to take a photo of anything further away than about 10 meters, the target does not fill a large amount of the frame (meaning I have to significantly crop in post production).  I am currently looking at the Canon prime 400mm, however I would like to determine at what distance an object will fill a reasonable proportion of the frame.  Is there a formula or rule of thumb I can apply that will help me in this situation? I realise the size of the target I am shooting will play a role here, so if we need to make an assumption about the size of the target please let me know. I want to know is there a way to render Shadows only in Blender's new EEVEE Render Engine. I want my objects in a separate collection and a ground plane on a separate collection(to receive shadows from the objects collection).I want to render the shadows only and not the objects. I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$ I am using Log4Net with the AdoNetAppender to log messages from a simple systray application into a SQL Server 2005 database.  I want to log the machine name along with the log message because this application will be running on multiple machines and I need to know on which one the message originated.  But, I cannot find a way to expose this information via the log4net.Layout.PatternLayout that I am using with the appender.  Is there a way to log the machine name via log4net in this manner? Escaping regex string I want to use input from a user as a regex pattern for a search over some text.  It works, but how I can handle cases where user puts characters that have meaning in regex? For example, the user wants to search for Word (s): regex engine will take the (s) as a group.  I want it to treat it like a string &quot;(s)&quot; .  I can run replace on user input and replace the ( with \( and the ) with \) but the problem is I will need to do replace for every possible regex symbol. Do you know some better way ? Cam I have two GFCIs connected to one double gang box in my laundry room and if so can I do a single line connection so that they both can work independently?  I have a 12/2 going from the panel through conduit (copper wire) to the double gang box. I want to have each GFCI to run independently without using the load. Why, in case one shuts off the other isn't affected. the double gang box is standard. the device is a Leviton 20 amps. I was told that you can run 12/2 from the panel to the first GFCI black and white. Then connect the second GFCI using a jumper and attaching it to he second GFCI. Is this acceptable by code. we use a washer,dryer in the laundry and the garage heavy machinery. so why should this be a problem running two independent? How to run firefox and firefox ESR simultaneously The main reason I want to use firefox ESR is to use a proprietary database tool for work that only allows oracle-java based web logins for linux (or a windows only thin client). I also want to use the full version of the outlook web application which only works in firefox. I don't want to run the NPAPI plugins for normal web browsing and email.   I have the repo version of firefox installed and firefox ESR installed in /opt/firefox and linked to /usr/local/firefox-esr  When I run firefox any attempt to run firefox-esr either brings up a window without the NPAPI plugins.   If I try running with the --new-instance or --no-remote options I receive the following error:  Firefox is already running, but is not responding. To open a new window, you must first close the existing Firefox process, or restart your system.   Regardless which one I start first, I can only open that one, and get the above error if I attempt to sidestep.  What makes this an especially annoying problem is that I often have to respond to emails that I need to access the database to deal with. So either I close and open many windows. Is there any uncountably infinite set that does not generate the reals? Does there exist an uncountably infinite set $X \subseteq \mathbb R$ such that $\mathbb R \neq \left&lt;X\right&gt;$? I can't think of any, but I'm also having trouble trying to prove that no such subset exists.  For example: $\mathbb R$ is uncountable and obviously $\mathbb R = \left&lt;\mathbb R\right&gt;$. The Cantor set $C$ is uncountable, and we know that $C - C = [0, 1]$, so then since $\mathbb R = \left&lt;[0, 1]\right&gt;$ we know that $C$ also generates $\mathbb R$. Also the set of irrationals $\mathbb R \setminus \mathbb Q$ is uncountable, but we can generate all the rational numbers by fixing one irrational number $\alpha$ and then saying the any rational number $x$ shall be $(\alpha + x) - \alpha$, since both $\alpha + x$ and $\alpha$ are irrational.  So the examples that quickly come to mind all generate the reals. Is there a simple counterexample? Number of automorphisms of a direct product of two cyclic $p$-groups Why does the symbol for the multiplication operation change shape? Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen the variable $x$ unless we were already removing $\times$ as the symbol for multiplication. So why do we? I am very curious. It seems like $\times$ is already quite sufficient as a descriptive symbol. Can you determine a formula for this problem? How do I put text over symbols? How to install PHP from source with latest cURL package on Linux? I simply have to use the latest cURL version (7.21.3) with PHP on my webserver. (Has to do with specific bug fixes)  It is not available by default on most current Linux variant repositories. (Natty Narwhal by Ubuntu is the only one and it's still in beta stages) - so I cannot simply run "aptitude install php5-curl" or similar to get it (the very latest version of cURL) installed.  How would one go about installing the very latest PHP + cURL packages?  Is it as simple as changing the to-download file for cURL in a repository somewhere? (ie. changing a text file's entry somewhere from  to  or something..?)  Basically I need the very latest cURL library running with PHP on Linux. How would you personally go about doing that? Can “zealot” have a positive connotation? A zealot is a person who is fanatical and uncompromising in pursuit of their religious, political, or other ideals.  I have never seen this word used with positive connotation, but could it (without causing raised eyebrows)? For example, “Gandhi was a zealot and should be admired by all who believe in non-violence.”  Also, do “zealous” and “zeal” carry the same connotation as “zealot”? How to visualize Reverse Triangle Inequality $||\mathbf{a}| − |\mathbf{b}|| ≤ |\mathbf{a} − \mathbf{b}|$? pp  and ,   illustrates Triangle Inequality. But without referring to Triangle Inequality at all, how to picture  Reverse Triangle Inequality for all $\mathbf{a,b} \in C^n$? I don't want proofs or rigour. Product Notation for Multiplication in Reverse Order Is there a standard notation for multiplication in reverse order? For example consider the problem  $$x_{k+1} = A_k x_k$$  where $x_i \in \mathbb{R}^n$ and $A_i \in M_n(\mathbb{R})$, ($i=0,1,2,\dots$) without any further assumptions on $A_i$. The solution to this problem is  $$x_k = A_{k-1} \dots A_1 A_0 x_0$$  Obviously, writing  $$x_k = \prod_{i=0}^{k-1} A_i x_0$$  is wrong because of the multiplication order. But I'm also unconfortable to use  $$x_k = \prod_{i=k-1}^{0} A_i x_0$$  as the notation suggests that we need to increment $i$, not decrement it. Also, one may need to write $\prod_{i=k}^{m} A_i$. Then, we need to explicitly state $m &lt; k$ to imply ordering.  I can always write the open form but as similar products frequently occur in calculations in my work I want to express them in a more compact way. The question is, is there a standard and nice notation to do so? There are some suggestions in , but I think none of them are "nice enough" to work with.  I came up with the notation  $$x_k = \coprod_{i=k-1}^{0} A_i x_0$$  to imply decrementing $i$ rather than increment it, but I don't know if it is used in somewhere else or there is a standard notation. Arabic Stack Exchange? I have been learning some Arabic mostly for fun. I am looking for the Arabic Stack Exchange site but cannot find it. I am sure I saw it lying around a few months ago or so. Any idea what happened to it? How to find out if a file exists in C# / .NET? I would like to test a string containing a path to a file for existence of that file (something like the -e test in Perl or the os.path.exists() in Python) in C#. Let $\{K_n \}_{n \in \mathbb{N}}$ be a collection of sets, where $K_n = [n, \infty)$. Why is $\bigcap_{n \in \mathbb{N}} K_{n}$ empty? This question is related to Exercise 2.15 in Baby Rudin, in which the reader is instructed to "[s]how that Theorem 2.36 and its Corollary become false (in $R^1$, for example) if the word 'compact' is replaced by 'closed' or by 'bounded.'"  The relevant theorem and corollary are as follows:  Theorem 2.36: If $\{K_{α}\}$ is a collection of compact subsets of a metric space $X$ such that the intersection of every finite subcollection of $\{K_{α}\}$ is nonempty, then   $\{K_{α}\}$ is nonempty.  Corollary: If $\{K_{n}\}$ is a sequence of nonempty compact sets such that $K_{n}$ contains $K_{n+1},  (n = 1, 2, 3, ...)$, then   $K_n$ is not empty.  My Question: I realize this question and similar ones have been asked elsewhere on MSE, but I've yet to see a straightforward (to me, at least) answer to my question specifically. Usually, for the "closed" part of the exercise, the following collection of closed sets, $\{K_n \}_{n \in \mathbb{N}}$ is used:  $$K_n = [n, \infty), n \in \mathbb{N}.$$  For, the intersection of every finite subcollection of ${K_n}$ is clearly nonempty. Indeed,  $$ \bigcap_{i=1}^{m} K_{n_{i}} = [N, \infty) $$  where $N = \max\{n_1, n_2, ..., n_m\}$.  Then, the assertion is made that   $$ \bigcap_{n=1}^{\infty} K_n = \emptyset. $$  However, I don't see how the same argument can't simply be used, i.e., that, for example, $N \in  \bigcap_{n=1}^{\infty} K_n$ (or maybe, say, $n+1$). Intuitively, this certainly makes sense, I'd say. For, given the archimedean property, it seems that as $n$ approaches infinity, we can always find an $l$ in each increasingly smaller set such that $n &lt; l &lt; \infty$; and so $l \in  \bigcap_{n=1}^{\infty} K_n$.  What is it exactly that I'm misunderstanding or overlooking here? I've created a new C++ project in Visual Studio 2008.  No code has been written yet;  Only project settings have been changed.  When I compile the project, I receive the following fatal error:     fatal error LNK1104: cannot open file 'C:\Program.obj'