Compute the following sum in terms of an expression having only one binomial coefficient
Truncated alternating binomial sum
When playing with the complete elliptic integral of the first kind and its Fourier-Legendre expansion, I discovered that a consequence of $\sum_{n\geq 0}\binom{2n}{n}^2\frac{1}{16^n(4n+1)}=\frac{1}{16\pi^2}\,\Gamma\left(\frac{1}{4}\right)^4 $ is:     $$\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \tfrac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2\tag{A}$$   which might be regarded as a sort of Ahmed's integral under steroids.  I already have a proof of this statement (through Fourier-Legendre expansions), but I would be happy to see a more direct and elementary proof of it, also because it might have some consequences about the moments of $K(x)$ of the form $\int_{0}^{1}K(x)\,x^{m\pm 1/4}\,dx$, which are associated with peculiar hypergeometric functions. Have you used any of the C++ interpreters (not compilers)? I am curious if anyone have used UnderC, Cint, Cling, Ch, or any other C++ interpreter and could share their experience. Why use apparently meaningless do-while and if-else statements in macros? Signature/date line with fixed width I'm trying to create a signature and date line in a document.  I need the date line to always be in the same spot on the page no matter what name is under the signature line, such as:  ----------------------   ---------------------- Representative           Date  ----------------------   ---------------------- Program Manager          Date   Here is the code I currently have:  \makebox[2.5in]{\hrulefill} \hspace {1.0in}\makebox[2.5in]{\hrulefill} \\ User Representative \makebox[2.5in][r]{Date} \\  \vspace{.2in} \makebox[2.5in]{\hrulefill} \hspace {1.0in}\makebox[2.5in]{\hrulefill} \\ Program Manager \makebox[2.5in][r]{Date} \   I want to be able to change the names, such as change Program Manager to Joe Foo and have the word 'Date' stay in the exact place.  Any help would be greatly appreciated.  I'm still a newbie at TeX so please forgive my ignorance. Can I have multiple primary keys in a single table? Can I have multiple primary keys in a single table? What is the 'rev' dropdown supposed to do? I only see this on sites where I don't have edit privileges, and only where there have been multiple revisions:        What exactly is this supposed to do?  Intuitively, I would expect it to change the text in the box to what it was at that revision, but it doesn't. And, if that's it, why only where I need to suggest edits? There are several things that I don't understand. I know that zero is a real number, but I'm confused on the how and why aspects. What defines a real number, compared to a number that's considered non-real? For what reason is zero real, and how is it that a value such as infinity isn't when zero is?  Also, what purpose does zero even have? It's obvious that it's used as a place holder, but what value does it have alone? What makes it any different from other numbers? I realize that it is typically interchanged with nothing, but that would technically mean that zero is nothing—and that's not true. People have told me already so many times that zero does not necessarily mean nothing, and I figured it out a long while ago anyway. So, what I want to know is the actual value that can be associated with zero.  However, my question brings up another topic that has been argued and disputed about and disagreed on for years. Zero divided by zero, or 0/0. There are a lot of varying answers about this, but what's been constant throughout everything is that people claim 0/0 is not a real number. So, if I may ask, why is it that zero is considered a real number, and not 0/0? And why is it considered non-real in the first place? True, there is no defined value, but the expression is anything but meaningless; if it was, then there wouldn't be so many people trying to understand it. I guess what I'm trying to figure out is the reason behind the expression's classification as non-real—especially when the sole number of zero is real, despite not having an evident value.  This covers a few different things, so I was uncertain about what to make the title of these questions. I apologize for that. If anyone can answer, though, I'd be extremely grateful! I have a sentence that contains a number of long dataset names. I would like to allow LaTeX to break these names at certain positions that I would like to specify somehow just like I can use \- to allow a word break with hyphenation. But: I would like LaTeX not to insert a dash (which could be mistaken as belonging to the dataset name). How can I do this?   (Something similar is realized in the url package so I guess I could just read that code :) I would not like to use the url package here to be able to make a syntactical (and typographic) difference between urls and datset names.) Login to Stack Exchange account doesn't use HTTPS/SSL? How to round a number to n decimal places in Java There is an exercise in Humphreys's An Introduction to Lie Algebras and Representation Theory: "Any nilpotent Lie algebra contains a codimension 1 ideal".  The proof I am thinking of is the following.  Suppose the Lie algebra $L$ satisfies $L\neq[L,L]$. Then $\pi\colon L\to L/[L,L]$ is a projection onto an abelian algebra. Choose a codimension 1 subspace $I\subset L/[L,L]$. Then $I$ is an ideal and its preimage $\pi^{-1}(I)$ is necessarily an ideal in $L$, because any subspace containing the derived algebra is an ideal. Moreover, $\pi^{-1}(I)$ has codimension 1. $\square$  The only condition on $L$ is that $L\neq[L,L]$, which holds for solvable Lie algebras as well. Is this proof correct? How come Humphreys didn't set the exercise with "nilpotent" replaced by "solvable"? Short connection at IAD (Dulles international airport) I am flying from New Orleans to Dulles (IAD) on United Express, and from IAD to Frankfurt on United. I have a short connection (45 minutes) in IAD.   I'm trying to figure out how can I make it:  Will I need to go throughout security and border check since I'm leaving the US? How big is IAD airport? and are there different terminals for domestic and international flights? If so, how long would it take to walk between them? How compute $\lim_{p\rightarrow 0} \|f\|_p$ in a probability space? Two different solutions to integral Given the very simple integral  \begin{equation} \int -\frac{1}{2x} dx \end{equation}  The obvious solution is  \begin{equation} \int -\frac{1}{2x} dx = -\frac{1}{2} \int \frac{1}{x} dx = -\frac{1}{2} \ln{|x|} + C \end{equation}  However, by the following integration rule \begin{equation} \int \frac{1}{ax + b} dx = \frac{1}{a} \ln{|ax + b|} + C \end{equation}  the following solution is obtained \begin{equation} \int -\frac{1}{2x} dx = -\frac{1}{2}\ln{|-2x|} + C \end{equation}  Why are these solutions different? Which is correct?   The second solution can be simplified \begin{equation} -\frac{1}{2}\ln{|-2x|} + C = -\frac{1}{2}\ln{|-2|} -\frac{1}{2}\ln{|x|} + C= -\ln{\frac{1}{\sqrt{2}}} - \frac{1}{2}\ln{|x|} + C \end{equation} but they still differ. How can I make an X-Y scatter plot with histograms next to the X-Y axes? Can anyone help me with a solution? Proof of Steinitz Theorem I want a source containing the proof of Steinitz Isomorphism Theorem stating:     For any Dedekind domain $R$ and any two nonzero ideals $I$ and $J$ of $R$ we have $I⊕J≅R⊕IJ$.   Thanks! Are there systems in classical mechanics that are provably non-deterministic? How to detect Safari, Chrome, IE, Firefox and Opera browser? Is there a way to take a screenshot using Java and save it to some sort of image? Simple as the title states: Can you use only Java commands to take a screenshot and save it?  Or, do I need to use an OS specific program to take the screenshot and then grab it off the clipboard?