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

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January 18

Derivatives and integrals of the factorial function

• $f(x)=x!$
• $f^{\prime }(x)=?$
• $\underset{}{\overset{}{\int }}f(x)dx=?$

Thanks. --Ķĩřβȳ♥ŤįɱéØ 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♥ŤįɱéØ 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

• $f(x)=\mathrm{\Gamma }(x)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{z-1}{e}^{-t}\mathrm{d}t$
• $f^{\prime }(x)=x^{z-1}{e}^{-x}$

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

No. $\mathrm{\Gamma }(z)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{z-1}{e}^{-t}dt$ 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 Template:Fraction, or may be written above and below a horizontal line called a vinculum, like so: $\frac{3}{4}$. 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. — Kieff 03:56, 19 January 2007 (UTC)