I have been trying to understand hyperbolic functions for some time now. I have a problem concerning the expressions of inverse hyperbolic functions. The text( G.B. Thomas ) mentions nothing about their expressions.   While plotting the Hyperbolic Sine was easy, plotting its inverse was not. By definition we have, $$\sinh(x) = \frac{e^x - e^{-x}}{2}$$  I plotted its graph below-    The dotted lines are the function's asymptotes. I then looked at the graph of inverse hyperbolic Sine and decided to obtain its expression. I tried by finding the inverses of the asymptotes because I knew that the inverse function would get close to them eventually. I plotted a graph.    I was happy. But not for long. It turned out that I cannot have an algebraic combination of these two asymptotes. This was my second approach, to be honest. A holy approach to finding inverses is, express $x$ in terms of $y$.  I got stuck at the very beginning and could never proceed. Is there a way around this? If there indeed is a way to express $x$ in terms of $y$, could anyone give me a few hints to do it? If there isn't, how do we obtain an expression for the inverse hyperbolic Sine?
$e^{i\theta}=\cos\theta + i\sin \theta$  $e^{i\sin^{-1}x}=\cos(\sin^{-1}x)+i\sin(\sin^{-1}x)$  $i\sin^{-1}x=\ln|\sqrt{1-x^2} + ix|$  $\sin^{-1}x=-i\ln|\sqrt{1-x^2} + ix|$  Now from here I'm kind of lost, since it seems like this should be the definition, but when I look it up, the definition of inverse hyperbolic sine is:  $\sinh^{-1}x=\ln(\sqrt{1+x^2} + x)$  So although they're very similar, I guess I just don't know how to handle the logarithm and anything to the ith power or drop off the absolute value.
Will wolves attack Ocelots? Why does the field of an electric dipole drop off as $1/r^3$? Electric field of a point charge drops off inverse squarely but that of a dipole varies as $1/r^3$. Why the field drops off more quickly for a dipole? Using headset controls like a smart phone Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin...(\sin(x)..)$ Stack Overflow technology makes me write bad answers Today I had some time and felt like writing a relatively good answer on Stack Overflow; explaining both the underlying mechanisms and good practices, unlike usual declarative answers in a "do that" style.  It took me like an hour of time, $50 if I were free lance. It'll bring me nothing but 3-5 occasional views, and - if I get lucky - some ignorant comments.  You know, it's quite discouraging.  Instead of pride for the job that was done, I feel frustrated. I know, this answer may offer help to the OP, if he ever notices it (being already satisfied with other answers).  So the only fate the answer will meet is to sink forever, below thousands other questions, falling like snow.  So, being a programmer, I am thinking of efficiency. It's a waste of time to spend so much effort to answer one question when there is no way to [actually] reuse it. So, I wouldn't do it again but just turn back into short answers mode without explanations.       Seeing other answers, I don't think I am only one with such feeling.  The current policy is driving knowledgeable and educational users away, yet it attracts inexperienced ones in numbers. Ones who will tell you to use [some technology] which they never have laid their hands upon. Ones who write answers just because they have learned some "good practice" from other questions, and this "knowledge" becomes an endless source of rep points.  The idea of Stack Overflow is rotten. It was excellent when trees were high and traffic was low.  It turned bad now; it has become a honeypot for all the "enthusiast programmers" of the world, eager to share their 2 cents faster than you can say the word "close".  You're killing a great resource. Despite all the nice words and proper declarations, it discourages the reuse of the knowledge. And encourages fast on-site answers.  Of course, the rules will tell you contrary. It's all nice and proper (if you don't mind it's strict mutual exclusivity). But mechanically, technologically it works in the way I described.  I tried to write a tag wiki. You know, it was biggest waste of time in my life. As a matter of fact, it is not a wiki at all; you can't even link to a certain section. Not to mention that nobody ever has the idea to read tag wikis.  Reference questions? Don't make me laugh. There is no good mechanism to store and categorize links. Favorites, although intended for the different purpose, are a chunk of crap when you want to use it as a reference. So, you can't compete with 30-seconds answers: a question will get 5 answers before you can even find an appropriate link.  Okay, I can organize links myself but, there are some problems:   I am just a human. I cannot remember everything I am just a human and I cannot answer questions at rate of 1 qpm. One person cannot fight the Tide. It's helpless. When technology makes people writing fast [incomplete] answers -  no good will of proper declarations can do a thing.  Even if a question gets finally closed, it won't hurt anyone - the hydra grew two more already.   You'll call me a brute for insulting honest users and innocent volunteers, but that's fact: There are much more inexperienced users than experienced ones. And inexperienced users have much more spare time to hang over than experts who have their job to do. So, from that fact we can conclude that most of the answers on Stack Overflow have a low quality. Nobody cares. But I do.   There should be a technical way to limit the activity of not-so-knowledgeable-yet-eager-to-answer users. There should be a technical preference for the closing a question over answering it (have a silly answer for the silly question? welcome to comments). There should be a technical solution to accumulate and structure knowledge, a system that lets you choose a good answer instead of writing it by hand.   That's the only way. Otherwise Stack Overflow will remain an automated dispenser of bad practices.  Well, I just felt like letting it out again. Thank you for reading. Capturing console output from a .NET application (C#) How do I invoke a console application from my .NET application and capture all the output generated in the console?   (Remember, I don't want to save the information first in a file and then relist as I would love to receive it as live.) How can I completely remove all traces of Java on my system?   I already know  in case I need it again. How to perform a full system dump/backup with a broken screen? So, yesterday my phone took a serious tumble.  It is now essentially useless, as the screen either fails to turn on, or when it does, there is nothing useful to see (just lines of primary colors, mostly).  I believe (from the fact that it still chirps at me when emails come in, etc), that the phone is still functional, otherwise.  I would like to perform a full system dump, so that I can get off any files from the phone (a TMobile SGS2 (hercules) running slimrom 4.3 build 1).  The sdcard probably isn't important, as I can just pull it, but the phone's own internal storage I should like to get a copy of.  When I plug the phone into a usb port, it 'mounts' the usb via MTP, and I can see all the files.  Unfortunately, when I do a select all | copy | paste operation, the paste op dies before transferring all files.  I've tried a couple other ADB shell operations, in hope of flipping regular USB mounting on, but nothing has worked.  Any ideas how I can get a full dump of my poor phone's contents?  Edit  Currently running a adb pull /sdcard/ ... Its copying files presently, but its too early to tell if its going to get evertying I'm expecting.  Edit #2  looks like its going to work...its taking a rather long time, but it appears to be grabbing everything I was expecting it to. Let $A$ be a commutative ring with $1$. Let $I$ and $J_k$, $k=1,\dots,n$ be ideals of $A$ with $I\subseteq \cup _{k=1}^n J_k$. Then I have obtained the following:  (1) If $J_k$, $k=1,\dots,n$, are prime ideals, then there exists some $j$ such that $I\subseteq J_j$.   (2) If $A$ is a principal ideal ring, then there exists some $j$ such that $I\subseteq J_j$.   I want some counterexamples that for all $k=1,\dots,n$, $I$ is not contained in $J_k$. How to get the counterexamples? Need to schedule a job every hour in jenkins After booting I ran systemd-analyze blame and here are the results:       21.596s systemd-journal-flush.service      18.658s dev-sda8.device      15.099s dev-loop33.device      15.034s dev-loop19.device      15.012s dev-loop34.device      14.989s dev-loop21.device      14.877s dev-loop15.device      14.866s dev-loop26.device      14.773s dev-loop27.device      14.684s dev-loop30.device      14.677s dev-loop32.device      14.649s dev-loop35.device      14.590s dev-loop25.device      14.267s dev-loop23.device      14.192s dev-loop24.device      14.156s dev-loop29.device      14.133s dev-loop16.device      14.065s dev-loop31.device      14.059s dev-loop28.device      13.821s dev-loop20.device      13.531s dev-loop22.device      13.495s dev-loop14.device      13.364s dev-loop18.device   What are these dev-loopxx.device (xx denotes numbers) services and why are they taking so much time? Are they related to the mounting of snaps? Can I reduce the boot time by disabling them? I'm running Ubuntu 18.04 alongside Windows 10. Gollum had his life largely extended because he possessed the One Ring. He appeared to be an old, deformed, and twisted creature.  According to , he had the Ring for 478 years. His loss of the Ring and the events of The Lord of the Rings are separated by around 77 years.  Did the Ring continue to extend his life after he lost it? How do I appropriately examine the dimensionality of binary data? I have 72 binary variables and, at a theoretical level, I am trying to identify groups of variables that seem to vary together. In practice, I am struggling with how to analyze this data properly. I am using R and the psych package. Here is what I have tried and what the outcomes were:  Estimated a tetrachoric correlation matrix (ran without problem) Estimated MSA (relatively strong values, ran without problem) Ran parallel analysis to identify how many factors to extract (ran without problem) EFA using fa() to estimate factors (non-positive definite matrix and strangely high RMSEA) Tried different factor counts, smoothing/unsmoothing, estimation methods, rotations methods, and removing variables (problems persisted) Tried to focus instead on eBIC (always returned as NULL in output) EFA using (inappropriately) Pearson correlations (RMSEA behaved)  From looking at other questions and answers on here, I am now aware that tetrachoric matrices can be problematic like this. However, I don't know what to do with that information. Accept the problems as a limitation and move forward? Conduct a different type of analysis? I considered IRT as is recommended, but am unaware of an exploratory way to conduct IRT analyses. I understand that most of my options are imperfect. My goal is to find an analysis that is at least arguably appropriate for binary variables and that will inform the ways in which I group these variables in my theorizing. Given this, my questions are:  Can I trust EFA on a tetrachoric matrix despite problematic output? Can I somehow recover the lost eBIC value and use that? Is exploratory IRT possible/desirable? Are there other analyses, in R, that can be used to examine the dimensionality of binary data?  I appreciate all feedback! I love learning about data analysis and hearing about how I can improve. As I navigate through physics stackexchange, I noticed .  Electrons radiate light when revolving around nucleus? Why is it so obvious?  Note that I do not know anything about quantum mechanics - and I just want classical stories. Is it possible to remove a terminal command? For example, if there was a command call lookup (not an alias), how would I go about removing it? What's the difference between the atomic and nonatomic attributes? What do atomic and nonatomic mean in property declarations?  @property(nonatomic, retain) UITextField *userName; @property(atomic, retain) UITextField *userName; @property(retain) UITextField *userName;   What is the operational difference between these three? How to resolve "iTunes could not back up the iPhone because the backup was corrupt"? When I try to sync my iPhone with my computer, I get the following message:       iTunes could not back up the iPhone "iPhone Name"   because the backup was corrupt or not compatible   with the iPhone.      Delete the backup for this iPhone, then try again.   Yet, here is no option to delete the backup. Where are backups stored / how can I delete the backup?  This is iTunes version 10.5.3 running under  Mac OS X 10.6.8. how implement the jqueryUI in visualforce For symmetric operators one has: $$A\text{ symmetric}:\quad\mathcal{R}(A\pm\imath)=\mathcal{H}\implies A^*=A$$ How to prove this in an unveiling way? Do muggleborns get their letter on their 11th birthday or the summer before they start Hogwarts? A lot of the fanfiction seems to like to give Hermione her letter in September to coincide with her birthday. That just doesn't seem very logical for the Statute of Secrecy, as you then have 10 out of every 12 10-year-old wizards and witches knowing about magic while finishing the school year (assuming even distribution of birthdays).   It seems to me that this is just asking at least one of them to brag about knowing magic and spill the secret.   Also this would mean that the poor kid born on 30 of August would only have 2 days to adjust to being magical before being shepherded off to a boarding school in the middle of nowhere.  From another point of view, getting such news on your birthday can screw up a lot of celebration plans.  Also, from rereading the memory from Dumbledore where he introduced magic to Tom, there is no mention of snow or even closeness to his birthday from the caretaker.  I believe this view came from the fact that Harry got his letter on his birthday but that piece of fanon seems to forget that letters were being sent up to at least a week or 2 beforehand (leading to the Dursleys hiding on a rock in the sea in the first place).