NOTE: I see it suggested that this is a duplicate question. Certainly there is substantial overlap between the two questions. This one explicitly mentions the Beta function and explicitly invites arguments for doing it the other way. One of the more able regular posters here stated today in a comment that it ought to be done the other way; I would be curious to see that case made. end of NOTE  One sometimes sees it opined that instead of $$ \Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x} \,dx $$ one should follow a convention that puts simply $\alpha$ where $\alpha-1$ appears. Similarly with both $\alpha$ and $\beta$ in $$ \operatorname{B}(\alpha,\beta) = \int_0^1 x^{\alpha-1} (1-x)^{\beta-1} \, dx. $$  What facts can be adduced on which to base arguments for one or the other convention?  (I will post a brief answer of my own, but possibly some very different answers can be readily written.)
As a physics student, I have occasionally run across the gamma function  $$\Gamma(n) \equiv \int_0^{\infty}t^{n-1}e^{-t} \textrm{d}t = (n-1)!$$  when we want to generalize the concept of a factorial.  Why not define the gamma function so that   $$\Gamma(n) = n!$$  instead?  I realize either definition is equally good, but if someone were going to ask me to choose one, I would choose the second option.  Are there some areas of mathematics where the accepted definition looks more natural?  Are there some formulas that work out more cleanly with the accepted definition?
How does Java calculate negative numbers? How to represent kWh usage by year against average temperature? I guess nobody really knows the true nature of black holes, however, based on everything I know about black holes, there is a "singularity" at their center, which has finite mass but is infinitely small and thus infinitely dense.  So a black hole is really just a point mass that dramatically warps the space-time around it so that an event horizon is achieved.  What doesn't make sense to me is how a true mathematical "point" can rotate, or at least how it is even possible to determine if a point is rotating (or if it should even make a difference about whether or not it is rotating, for the same reason it would be impossible to tell if a perfect, featureless sphere were rotating (I know it would have higher rotational energy but lets assume you can't measure that because a true "point" can't attain rotational energy since it has no radius)).  The only conceptual solution I have to describe a rotating black hole would be to compare it to a rotating whirlpool, where the surface of the water is the fabric of space-time... but this is different because there is no "singularity" in a whirlpool that causes the water around it to rotate, whirlpools form from flow of water around a point (put into motion perhaps by some kid swirling water in a 2-liter bottle... it's due to an external cause), but the only thing that could cause rotation around a black hole would be the singularity itself, which doesn't make sense because, once again, how can we know if a true point is actually rotating?  Aside from everything I've said, here is what I am actually asking:  Why/how do some black holes rotate... what is the mechanism of their rotation (is analogous to a spinning point or a 3-d whirlpool, or something different)? Is there any way of doing something like,  &lt;script runat=server&gt;     Platform.Load("core","1");     //initialize DE     var _Sent = DataExtension.Init("_Sent");     //retrieve data     var _Sent_data = _Sent.Rows.Retrieve();     Write("Length: " + _Sent_data.length); //returns 0 &lt;/script&gt;   What would the external key for a system dataview be? Why is $\langle \textbf R \rangle =Tr \rho R$ here $\rho$ is the state density matrix We have a org wide email addressed which has been "Verified" into our sandbox. I understant org wide email addressed is unsupported metadata, so this was achieved as pre deployment step.   Into our repo, we have a workflow that send email alert to that org wide email address. "  Workflow   ... &lt;senderAddress&gt;noreply@mycompany.com.au&lt;/senderAddress&gt; &lt;senderType&gt;OrgWideEmailAddress&lt;/senderType&gt;    The problem we are facing is that we are still getting the following error even if this org wide email address is verified:      You must specify a verified Org Wide Email Address for this sender   type.   Is that a bug in the metadata api or expected behaviour ? Ideally we would like to not have to do that as a post deployment step.   PS: We are running DX under api version 43.0 Is there a regular expression to detect a valid regular expression? Is it possible to detect a valid regular expression with another regular expression?  If so please give example code below. has been solved, in XeTeX, by this code:  \def\kright#1{\leavevmode #1\kern-\XeTeXglyphbounds3 \the\XeTeXcharglyph`#1 } \def\kleft#1{\leavevmode \kern-\XeTeXglyphbounds1 \the\XeTeXcharglyph`#1 #1}   which allows me to align text more precisely to the ink by accessing the side-bearing measurements (\XeTeXglyphbounds1 and 3).  This XeTeX dependency is currently the only thing stopping me from moving to LuaTeX (mainly for microtype). Is an equivalent function available in LuaTeX, or planned?    After reading a bit more on microtype, could the equivalent effect be achieved by an appropriately-tailored set of character protrusions? That way, my move to LuaTeX might be a lot simpler. How to run a complete GUI as root? I don't care if I mess up the system (which is likely if I do things like this), or the potential security issues that this could imply. Note, that I don't just want to "", but run my GUI from bottom-up as root. I don't want to type my password for anything but just for login and have total control over it.  Of course, I want any application running as root, like Firefox, LibreOffice, etc. without having to mess up with the terminal. So, how do I do this?     This question tries to address users that do not like the current permission system with root being separated from normal users. I do not endorse such behaviors, but if a user wants to, for whatever reason, this question should fit their corncerns.   Not advisable for new users What's the difference between absolute path &amp; relative path when using any web server or Tomcat? Is the integral $\int_1^\infty\frac{x^{-a} - x^{-b}}{\log(x)}\,dx$ convergent? How do I design a user friendly survey with more than 150 questions? I am designing UI/UX for a survey webapp which has more than 150 questions. Right now they are using 5 radio buttons for each question. Similar to the image below:      So the question is, how to make it super user friendly, easy and fun to do?   Further I am looking for a solution from responsive web design perspective too.  I hate designing it like the following image below but what are some other options. A new environment based on enumerate I have defined the following simple environment:  \newenvironment{renumerate}{\begin{enumerate}\raggedright}{\end{enumerate}}   I wonder how the above can be modified to use with, say, the following:  \begin{renumerate}[label=\color{blue}10.\theenumi] \item blah \item blah \end{renumerate}   The options given in [] are not known in advance. Name for the center amplitude of a sine wave I am trying to find the name for the value at the center of a sine wave $$ y(t) = A\sin (kx \pm \omega t+\varphi) + D $$ i.e. the offset $D$ of the wave in the $y$-direction.  calls it Center Amplitude, but further googling this name is difficult and does not provide any further sources. I also found the expression , but I'm not sure that it refers to the same thing. Is there a commonly used name for the $y$-direction-offset of a sine wave? I'm trying to split a string up into words and punctuation, adding the punctuation to the list produced by the split.  For instance:  &gt;&gt;&gt; c = "help, me" &gt;&gt;&gt; print c.split() ['help,', 'me']   What I really want the list to look like is:  ['help', ',', 'me']   So, I want the string split at whitespace with the punctuation split from the words.  I've tried to parse the string first and then run the split:  &gt;&gt;&gt; for character in c: ...     if character in ".,;!?": ...             outputCharacter = " %s" % character ...     else: ...             outputCharacter = character ...     separatedPunctuation += outputCharacter &gt;&gt;&gt; print separatedPunctuation help , me &gt;&gt;&gt; print separatedPunctuation.split() ['help', ',', 'me']   This produces the result I want, but is painfully slow on large files.  Is there a way to do this more efficiently? What is the function of "do" in this sentence? Rarely, though, do countries introduce a single reform alone.   What function does do have in this sentence? Dealing with suggested edits on question to be closed as duplicate There is a question on one of the SE sites that had a pending suggested edit: trust me, the way the original question was, it was needed. Before I even thought to approve the edit, I went to close it as a duplicate (as it was), but the suggested edit was still there.  If the question is being closed anyways (as a duplicate or otherwise) do I reject or approve the suggested edit? What is the proper etiquette? Why do objects in front and back view look reversed? Whenever I model objects that has different look at front and back. Viewing it from front perspective or orthographic the model displays back of the object. How is the no-cloning theorem compatible with the fact that fan-out gates work? I have some difficulty with understanding no-cloning theorem. Simply speaking, according to the theorem, it is not possible to copy a quantum state.  On the other hand, CNOT gate can be used as so-called fan-out gate which purpose is to copy one qubit to another one, previously in state $|0\rangle$.  It seems that these two facts negate each other.   My question: How is no-cloning theorem compatible with the fact that fan-out gate works? Why do shields in Star Trek "wear out"?