Wikipedia:Reference desk/Archives/Mathematics/2007 January 18

= January 18 =

Derivatives and integrals of the factorial function

 * $$f(x)=x! \!$$
 * $$f'(x) = ? \!$$
 * $$ \int_{\,}^{\,}f(x) dx = ? \!$$

Thanks. -- Ķĩřβȳ ♥  Ťįɱé  Ø  07:15, 18 January 2007 (UTC)


 * The factorial in its strictest form can't be "calculused", because it only applies to integers. What you need is the gamma function. yandman  07:38, 18 January 2007 (UTC)


 * Also, the factorial can be approximated with Stirling's approximation, which can be easily differentiated. Dunno about an integral though. --Spoon! 07:40, 18 January 2007 (UTC)


 * Mathematica gives me an error when I try to put in Stirling's approximation. =( -- Ķĩřβȳ ♥  Ťįɱé  Ø  07:45, 18 January 2007 (UTC)


 * What did you input? When I input Sqrt[2*Pi*x]*x^x*Exp[-x] I get basically the same indefinite integral back, not an error message. For Gamma[x+1] the system reports: Mathematica could not find a formula for your integral. Most likely this means that no formula exists. That is what I expected, and I don't think the Stirling formula has an analytically expressible primitive either. --Lambiam Talk  10:00, 18 January 2007 (UTC)

Just to clarify something, if f(x) is the Gamma function, which equals (x-1)! for positive integers, then
 * $$f(x)= \Gamma(x) = \int_0^\infty t^{z-1} e^{-t}\,\mathrm{d}t $$
 * $$f'(x)= x^{z-1} e^{-x} \!$$

(from the Fundamental theorem of calculus) Dugwiki 21:50, 18 January 2007 (UTC)


 * No. $$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,dt$$ But notice how the integral is in $$t$$ and not in $$z$$. —The preceding unsigned comment was added by Spoon! (talk • contribs) 21:57, 18 January 2007 (UTC).


 * Yeah, be careful about the way you've written the Gamma function. There is no x in the expression; are you saying the function is constant?
 * Lastly, the derivative of the Gamma function is complicated and involves something called the polygamma function. –King Bee (T • C) 22:01, 18 January 2007 (UTC)
 * But that's only because the polygamma functions are defined in terms of the derivatives of the gamma function. So saying the derivative involves the polygamma functions is not useful. --Spoon! 00:03, 19 January 2007 (UTC)
 * It is useful in pointing out that the derivative of the above is incorrect. In any event, the derivative of the gamma function is not a simple application of the fundamental theorem of calculus. –King Bee (T • C) 00:17, 19 January 2007 (UTC)
 * D'oh! - "I am smart! S-M-R-T!" My bad, guys, sorry. :/  Dugwiki 18:40, 19 January 2007 (UTC)

how to do fraction notation
i am just a beginner how do you do fraction notation i am really stuck i am on online school so please just tell me the basics. —The preceding unsigned comment was added by 12.210.136.29 (talk) 19:08, 18 January 2007 (UTC).


 * Fraction (mathematics)?  x42bn6  Talk 21:26, 18 January 2007 (UTC)


 * A fraction is one number divided by another. The numerator and denominator may be separated by a slanting line called a solidus or slash, for example $3⁄4$, or may be written above and below a horizontal line called a vinculum, like so: $$\textstyle\frac{3}{4}$$. X ['Mac Davis '] (DESK |How's my driving? ) 23:24, 18 January 2007 (UTC)


 * I have a feeling the question is on the mixed fraction notation, such as 5 1/4 for 5.25. Then it's very easy, really. When you encounter a number in mixed notation, just add a + before the fraction, so a b/c becomes a + b/c. Then just use the arithmetic rules to get the actual number. &mdash; Kieff 03:56, 19 January 2007 (UTC)