Prove that $\int_{0}^{\infty}\frac{\sin(x)}{x}dx=\frac{\pi}{2}$ using contour integral
Computing $\int_{-\infty}^\infty \frac{\sin x}{x} \mathrm{d}x$ with residue calculus
How do I get query string value from script path? Why is the Principle of Superposition true in EM? Does it hold more generally? In the theory of electromagnetism (EM), why is the principle of superposition true? Can we read it off from Maxwell's equations directly?  Does it have any limit of applicability or is it a fundamental property of nature? We have set up GA on several websites to gather information about traffic to the site. We know about the organic keywords that are displayed under the Acquisition section of the traffic report. But we have been asked to find a way to display "not provided" keywords typed in by the end user in our Google Analytics report.  Is it possible to accomplish this? How is the derivative of $\textrm{Trace}\left\{ X^T A X B\right\}$ with respect to $X$ equal to $AXB + A^TXB^T$? How do you count the lines of code in a Visual Studio solution? Each  has a compiled cost that contributes towards the  compiled character formula limit.  What are the current compiled costs associated with each formula function? How to reinstall Windows 10 on new hard disk Fixing "Page is not mobile friendly" report from Google test In search results google reports "this page is not mobile-friendly". I want to fix this.   when I test my website in the mobile-friendly test, it reports:   Content wider than screen. Clickable elements too close together.   How can I find the items/content that are creating these problems?  Any help would be appreciated. How do you add tags to blocks so they can only be placed on 1 type of block? Your visa application has been concluded - what does this mean? I got two mails from the UK embassy in Abu, saying your visa application has been concluded. I'm really confused, did I get the visa or not? What could the possible results be? I need a cell that contains an image to also contain a hyperlink.  Following formula is the only one I found but doesn't work (error analyzing formula)  =HYPERLINK("URL", image("URL"))   Following formula I tried shows a blank cell but the link works  =hyperlink("URL"; Image("URL")) $(a,b) \mathbin\# (c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse? Let $r\in \mathbb{R}$ and let $0\neq s \in \mathbb{R}$. Define operations $\#$ and $\&amp;$ on $\mathbb{R}$ x $\mathbb{R}$ by $(a,b) \mathbin\#(c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&amp;(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$ as addition and multiplication, respectively. Do these operations form a field?  I have shown that every axiom of the field is satisfied except for the "existence of multiplicative inverses" axiom. This is where I need help. We either need to show that this axiom fails and so we don't have a field or that the axiom $aa^{-1}=1$ is satisfied for every $a\neq 0 \in \mathbb{R}$  The multiplicative identity in this problem is $(1,0)$.  Any solutions or hints are greatly appreciated. I have the following reference:  @MISC{EX2012,   author = {{S}ingle {A}uthor {O}rganization},   year = {2012},   title = {Example} }   A citation for this is breaking after "Author", but extending beyond the margin, as shown in the following example layout in the PDF:  Demo text demo text demo to margin Demo text which cites it (Single Author Organization, 2012) and flows over the margin.   How can I solve this? I am using natbib and citing with \citep{EX2012}. Is there any difference between "color" and "colour"? What is the difference between color and colour? Return a blank when cell reference is blank I am using the following formula in a Google Sheets to calculate the number of days between two dates (dates are in columns G and AL):  =ARRAYFORMULA(if(row(A:A)=1, "Readmission Days", (G:G)-(AL:AL)))   It has to be an array because I am working with a Google Forms and entries are automatically populated into new rows.  The problem is that some of the cells in col AL are blank, so the formula is reading those dates as something else and returning a very large number. Also, when both cells are blank, the formula returns a 0, messing up my data.  How can I include something to ignore blank cells or just return a blank when one or both cell references is blank? When I visit a website it sometimes puts cookies on my browser. Usually cookies keep track of browsing info from that particular site. If I don't delete those cookies and browse other websites, will the cookies be able to know which site I am browsing? Also, can any website read browsing history which is saved by browsers? Can any website know about my browsing history by help of cookies, browsing histories saved by browsers, or by any scripting language e.g. javascript? I'm currently working through the book Heisenberg's Quantum Mechanics (Razavy, 2010), and am reading the chapter on classical mechanics. I'm interested in part of their derivative of a generalized Lorentz force via a velocity-dependent potential.   I understand the generalized force   $$F_i = -\frac{\partial V}{\partial x_i} + \frac{d}{dt}\left(\frac{\partial V}{\partial v_i}\right)$$  that they derive from a Lagrangian of the form $L = \frac{1}{2}m|\vec v|^2 - V(\vec r,\vec v,t)$.  However, in the next (critical) step of the derivation, the author cites a theorem from Helmholtz saying      ...according to Helmholtz, for the existence of the Lagrangian, such a generalized force can be at most a linear function of acceleration, and it must satisfy the Helmholtz identities.   The three Helmholtz identities are then listed as:  $$\frac{\partial F_i}{\partial \dot{v_j}} = \frac{\partial F_j}{\partial \dot{v_i}},$$  $$\frac{\partial F_i}{\partial v_j} +  \frac{\partial F_j}{\partial v_i} = \frac{d}{dt}\left(\frac{\partial F_i}{\partial \dot{v_j}} +  \frac{\partial F_j}{\partial \dot{v_i}}\right),$$  $$\frac{\partial F_i}{\partial x_j} -  \frac{\partial F_j}{\partial x_i} = \frac{1}{2}\frac{d}{dt}\left(\frac{\partial F_i}{\partial v_j} - \frac{\partial F_j}{\partial v_i}\right).$$  I'm trying to understand where this theorem comes from. Razavy cited a 1887 paper by Helmholtz. I was able to find , but it is in German, so I could not verify whether or not it proved the theorem. Additionally, I could not find it in any recent literature. I searched online and in Goldstein's Classical Mechanics.   The only similar concept that I can find is in the  where we have three equations known as Helmholtz conditions. Are these two concepts one in the same? If so, how should I interpret the function $\Phi$ and the matrix $g_{ij}$ that appear in the Helmholtz conditions I found online?   If the cited theorem from Razavy does not relate from the inverse Lagrangian problem, could I have some help finding the right direction? Fill the area under minimum of two functions using tikz and pgfplots If gravity isn't a force, then why do we learn in school that it is? I have studied some of Einstein's Theory of General Relativity, and I understand that it states that gravity isn't a force but rather the effects of objects curving space-time. If this is true, then why are we instructed in middle school that it is a force? A Hollow Black Hole I was just reading a . It was a trivial question, obviously there is no force felt. But then it got me thinking.  Suppose you had a hollow sphere that was massive enough that outside of it, it became a black hole. It's not hard to imagine a situation where the magnitude of the force of gravity at the surface (while strong) is equal to the magnitude of the force on the particles of the sphere due to some phenomenon involving one of the other 3 forces (just so I can allow this hollow sphere to continue being hollow).  Obviously, I don't have to do the integral to know that any object inside the hollow would experience no net gravitational force due to the shell. However, being a few months short of General Relativity, I am curious to know whether there would be any effects from Gravity observed.  Would an observer inside experience any time dilation? There may not be any net force due to gravity, but one cannot deny that there are still some fairly powerful gravitational fields present that happen to superimpose to zero. Furthermore, (assuming a relatively massless observer) would they have access to any of the information that was "lost" in the black hole? If so, then if they were long-lived enough for the black hole to be able to evaporate around them, does this mean that not only would the black hole radiate that information, but it would also store a copy of it in the hollow?  I want to apologize in advance. I know this question seems a bit more like a discussion than a Q&amp;A thing.  To be clear, I am asking: In my scenario, would there be any effects of gravity on things in the hollow? And would information lost in the black hole be accessible to the hollow (which is essentially its own isolated bubble of the universe)? A good answer can address these two questions. If you want to add in your own opinions/contribute to a discussion about it, that'd be a bonus.