From: Prof Brian Ripley <ripley_at_stats.ox.ac.uk>

Date: Fri, 07 Mar 2008 20:40:10 +0000 (GMT)

Date: Fri, 07 Mar 2008 20:40:10 +0000 (GMT)

On Fri, 7 Mar 2008, Max wrote:

> Prof Brian Ripley formulated on Friday :

*>> On Fri, 7 Mar 2008, Max wrote:
**>>
**>>> Dear UseRs,
**>>>
**>>> I'm curious about the derivative of n!.
**>>>
**>>> We know that Gamma(n+1)=n! So when on takes the derivative of
**>>> Gamma(n+1) we get Int(ln(x)*exp(-x)*x^n,x=0..Inf).
**>>>
**>>> I've tried code like
**>>>
**>>>> integrand<-function(x) {log(x)*exp(x)*x^n}
**>>>> integrate(integrand,lower=0,upper=Inf)
**>>>
**>>> It seems that R doesn't like to integrate for any n, and I was
**>>> wondering if anyone knew a way around this?
**>>
**>> ln(x) e^x x^n is not integrable on (0, Inf). You presumably slipped over
**>> a minus sign, but your definition of gamma(n) is wrong -- see ?gamma.
**>>
**>> integrate(function(x) exp(-x)*x^n, lower=0, upper=Inf)
**>>
**>> will work for gamma(n+1).
**>
**> I did miss a minus sign in the integration, which explains part of my
**> problems. The function of interest is the derivative of Gamma(n+1) with
**> respect to n, but obviously integrated over x from 0 to Infinity.
*

And you said n!, so n must be integer and you cannot differentiate a function of a integer argument.

If you are interested in the derivative of gamma(x), check out ?digamma.

> What happens now is:

*>
**>> integrand<-function(x) {log(x)*exp(-x)*x^n}
**>> integrate(integrand,lower=0,upper=Inf)
**> Error in f(x, ...) : object "n" not found
**>
**> Any ideas on how to get around this error?
*

Set 'n' in the evaluation environment.

E.g.

*> n <- 3
*

> integrate(integrand, lower=0, upper=Inf)

7.536706 with absolute error < 4.7e-06

*> digamma(4)*gamma(4)
*

[1] 7.536706

-- Brian D. Ripley, ripley_at_stats.ox.ac.uk Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/ University of Oxford, Tel: +44 1865 272861 (self) 1 South Parks Road, +44 1865 272866 (PA) Oxford OX1 3TG, UK Fax: +44 1865 272595 ______________________________________________ R-help_at_r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.Received on Fri 07 Mar 2008 - 20:43:45 GMT

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