It has long been known that the gamma function is an extension (shift) of the factorial function defined on integers.   There were an infinite numbers of ways to continue the factorial, so what property does the gamma function have that makes/made us use it? (Also, this question is not about history, but the unique features of a gamma function.)
There are lots (an infinitude) of smooth functions that coincide with $f(n)=n!$ on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^{-t} dt$ is the "best"?  In particular, I'm looking for reasons that I can explain to first-year calculus students.
Excessive fields in biblatex could not be removed if using \fullcite Combination of biblatex-chicago and Zotero produces many excessive fileds in the bibliography. This problem could be solved by adding to the preambule caveats like this:  \AtEveryBibitem{% \ifentrytype{online} {} {\clearfield{urlyear}\clearfield{urlmonth}\clearfield{urlday}}}   (Described in detail in the answer to my previous question: .)    This solution worked well for me, but now I have to produce a syllabus using \fullcite command. Unfortunately, this solution does not solve the problem for full bibliographic entries outside of the bibliography itself.  Here is the code with bibliography inside (to get the author's first and last name in the right order I use the following solution: ):    \begin{filecontents*}{database.bib}     @book{dalton_apartisan_2012,         title = {The Apartisan American: Dealignment and Changing Electoral Politics},         isbn = {9781452216942},         url = {http://books.google.com/books?id=eYkczUyX5wMC},         shorttitle = {The Apartisan American},         pagetotal = {241},         publisher = {{CQ} Press},         author = {Dalton, Russell J.},         urldate = {2014-04-03},         date = {2012-02-22},         langid = {english},         keywords = {Political Science / Political Process / Elections, Political Science / Public Policy / General}     } \end{filecontents*} \documentclass[11pt]{article} \usepackage[hmargin=3cm,vmargin=3cm]{geometry} \usepackage[onehalfspacing]{setspace} \usepackage{amssymb,amsmath,amsthm} \usepackage{booktabs,graphicx} \usepackage{paralist} \usepackage{cancel,soul} \usepackage{enumitem} \usepackage[authordate,backend=biber,bibencoding=utf8,bookpages=false,doi=only,isbn=false,footmarkoff]{biblatex-chicago} \usepackage[colorlinks, pdfstartview={XYZ null null 1.25},bookmarksopen=true,bookmarksopenlevel=\maxdimen,citecolor={blue},urlcolor={blue}]{hyperref} \addbibresource{database.bib} \DeclareCiteCommand{\fullcite} {\usebibmacro{prenote}} {\usedriver     {}     {\thefield{entrytype}}} {\multicitedelim} {\usebibmacro{postnote}} \AtEveryBibitem{%     \ifthenelse{\ifentrytype{article}\OR\ifentrytype{book}\OR\ifentrytype{collection}\OR\ifentrytype{incollection}\OR\ifentrytype{mvbook}\OR\ifentrytype{mvcollection}\OR\ifentrytype{mvincollection}}     {\clearfield{month}\clearfield{url}\clearfield{doi}\clearfield{urlyear}\clearfield{urlmonth}\clearfield{urlday}}}   \begin{document} \begin{center} {\huge Partisanship in Old, New and Non-Democracies}\\ \end{center}  \vspace{30pt} \noindent\textbf{Instructor: }  \hfill \textbf{Time and Location:} TBA\\ \textbf{Contact:}  \hfill \hfil \textbf{Office Hours:} TBA\\  \vspace{-20pt}    \subsection*{Partisanship in the United States}   \begin{enumerate}     \item \parencite{dalton_apartisan_2012}     \item \fullcite{dalton_apartisan_2012} \end{enumerate}  \printbibliography[heading=bibintoc]  \end{document}    Here is the output. Note the excessive fileds in the in-text full citation (they are absent in the References):    Any help, especially simple acessible solutions would be greatlly apreciated. The use of "exact same" Misplaced noalign after rowcolor in multicolumn How can I uninstall a nvidia driver completely ? As expected, Nvidia drivers have reduced my customizations. I can't login to Unity 3D session. I can't find any content to help me to uninstall the driver and I don't know what to do as I have never uninstalled them before. When I import text file into R, I get a special character appended to the first value of the first column How to tell the difference between linear and non-linear regression models? I was reading the following link on non linear regression . My understanding from reading the first section "Nonlinear Regression vs. Linear Regression" was that the equation below is actually a linear regression, is that correct? If so why?  $$y = b_1x^3 + b_2x^2 + b_3x + c$$  Am I also to understand that in non linear regression multicollinearity isn't an issue? I know that multicollinearity can be an issue in linear regression so surely if the model above is in fact a linear regression there would be multicollinearity? What is the simplest way to generate a dodecahedron? I've been thinking of animating some dice and while most are easy to make, I'm having some trouble figuring out a way to generate a regular dodecahedron. I could hand build it but it would be potentially uneven. Would using the base of a 5 sided cylinder and then some variation on spin be the best option or is there a better way? Kinetic energy is simple, right?! from the equation it seems as though if you know the speed and mass you'll know the work. Let's try to prove it. let's assume we know the acceleration and distance for now. $$w=mda$$ So the distance is $$d= 1/2 *at^2$$ Then  $$t = \sqrt{2d/a}$$ So $a*t$, which is the speed, is  $$a*\sqrt{2d/a} = a* \sqrt{2d}/\sqrt{a} = \sqrt{2da} = \sqrt{2w/m}$$ So regardless of the distance or acceleration, if you know the work and the mass, you can calculate the speed which is  $$v=\sqrt{2w/m}$$  What if you know the speed and the mass? You can easily calculate the work. $$v = \sqrt{2w/m}$$ Then  $$v^2 = 2w/m$$  That would mean  $$m*1/2*v^2=W $$ Rearrange it and  $$W= 1/2 *m *v^2$$ would be the equation.  What if you have an initial speed and a new speed and want calculate the amount of work done after an initial speed to get the new speed? You'll just subtract the total work from the initial speeds work.  $$1/2mv^2 - 1/2ms^2$$  where s is  the initial speed and v is the new speed. However if you do it the other way which is  $$d= 1/2at^2 + vt$$ and solve it the answer would be $$1/2ms^2  + 1/2msv$$ Why aren't they equal? How to efficiently count the number of keys/properties of an object in JavaScript? What's the fastest way to count the number of keys/properties of an object?  It it possible to do this without iterating over the object?  i.e. without doing  var count = 0; for (k in myobj) if (myobj.hasOwnProperty(k)) count++;   (Firefox did provide a magic __count__ property, but this was removed somewhere around version 4.) I've tried to find any solution or proof for $$\langle x| x' \rangle=\delta(x-x'),$$ but I only came to this post:   So I got the information, that the vector $|x\rangle$ form a dirac-normalized basis for the Hilbert Space.   I know that the dirac-delta distribution is defined like this: $$\delta(x-x') = \begin{cases} 0 &amp;\mbox{if } x\neq x' \\ \infty &amp; \mbox{if } x=x' \end{cases},$$ this means that my x' is a point on my x axis where I have my infinite high peak. And also $$\int_{-\infty}^{\infty}dx\cdot \delta(x-x')=1.$$  But how actually correlate this with the scalar product of vectors x, x' in the Hilbert Space that form a so-called 'diracl-normailzed" basis of it?  Can you give me some tips on this please? Or maybe you actually know a link, where this is explained. 3D software like GeoGebra Does it exist a free interactive geometry software, like GeoGebra, which works for 3D geometry? I would be able to draw spheres, great circles, and so on. Pull down resistor on logic gate input I would like to use a pull-down resistor on the input of a NOT logic gate. I am using a 5V microcontroller, and I want to ensure that the input of the gate will never be floating (eg. microcontroller damage).  General information:  The logic gate I am using is a HD74LS04 inverter whose datasheet can be found . I am testing the output by placing my positive multi-meter lead on R4 and the negative lead on GND. With the digital pin floating, the output of 1Y is 4.33V. This is the expected behavior.  The logic gate requires a minimum of 2.0V for Vih. If I understand this circuit correctly, I expect the voltage at pin 1A to be 3.76V since R2 and R3 will form a voltage divider. This value seems plenty to signify a 'high' in the logic gate.  R2 exists to protect the digital pin. Ideally, I would have liked R2 to be 1KΩ. Increasing the value of R2 means I have to increase the value of R3 to maintain the same voltage divider. During testing I found that if R3 was any value over 3.3KΩ, the pin 1A would not get pulled down all the way if I left the digital pin floating.   Tests: If I disconnect everything except GND and Vcc and place the positive lead of my multi-meter on pin 1A (now floating) and the negative lead to GND, I read 1.49V. Placing a pull-down resistor (R3) on pin 1A to GND with a value of 100Ω changes the reading to 0.02V. Increasing the resistor's value to 330Ω increases the reading to 0.07V. At the value I need, 3.3KΩ, the reading increases to 0.60V. Finally, changing the resistor's value to 1MΩ increases the reading to 1.47V and actually changes the output of pin 1Y to low.  The test that uses the 3.3KΩ resistor is concerning. The reading I took (0.60V) seems to be approaching the Vil maximum of the logic gate which is 0.80V.   Questions:  This must be a trivial question, but why does an R3 value of 3.3KΩ not pull pin 1A all the way to GND? Is there a formula for determining the correct pull-down resistor value? Other answers I found on here simply state 10KΩ as a 'good' value for a pull-up resistor. I'm not sure where that value comes from or if it is applicable to a pull-down resistor.         &ndash; Schematic created using   Edit: Although Ignacio's answer technically answers the question, I thought I would share some information I found for others who may come across this question.  User 'Papabravo' on the  states that the calculation for a pull down resistor can be calculated by taking the Vil (0.8V) and dividing it by the "standard TTL load" of 1.6mA. As Ignacio pointed out, my specific logic gate can source up to 0.4mA, so the calculation would be 0.8/0.0004A which equals 2KΩ. For safety, I performed the calculation at 0.5V in order to give myself some headroom.  Other Resources: How to convert java.util.Date to java.sql.Date? I am trying to use a  as input and then creating a query with it - so I need a .    I was surprised to find that it couldn't do the conversion implicitly or explicitly - but I don't even know how I would do this, as the Java API is still fairly new to me. PreparedStatement IN clause alternatives? How to change the href for a hyperlink using jQuery How can you change the href for a hyperlink using jQuery? How can I make a single event handler handle ALL Button.Click events? How to remove headers and footers for pages between chapters? Where is .bashrc? More than 1000 gallons of paint is/are sold each day Possible Duplicate:             Can anyone help me determine the correct verb in this sentence? I am not sure what to do. If it were not such a complex introductory phrase, it would be more obvious. The general consensus of my friends who are not professional writers is that the verb should be are.   To me the question is whether or not the subject is singular (i.e., “a large collective volume of paint”, perhaps in a tank) or plural (a lot of the individual gallon containers of paint).  If simplified to other options, it would be like these:   Paint is sold. Gallons of paint are sold. More paint is sold. More gallons of paint are sold. More than 1000 gallons of paint is sold. [emphasis on total volume] More than 1000 gallons of paint are sold. [emphasis on individual containers] Is any quotient of a Euclidean domain by a prime ideal a Euclidean domain? Let $R$ be a Euclidean domain, i.e., a ring with a norm $N : R \rightarrow \mathbb N$ such that for any $a,b\in R$ with $b\not=0$, we may write $a = qb + r$ for some $q,r \in R$ with $N(r) &lt; N(b)$. Let $I$ be a prime ideal of $R$, so $R/I$ is a domain. Must $R/I$ in fact be a Euclidean domain?   I know that this is true if we replace "Euclidean domain" with "PID", and false if we replace it with "UFD", so there is no clear intuition to be gained from similar concepts. My first attempts at a proof was to borrow a construction from analysis and define a norm on $R/I$ by  $N(a + I) = \inf\{N(a+i) : i\in I\}$  but this didn't really get me anywhere.   So, this fact true? If so, how can I prove it? If not, whats a counterexample?