Numerical Integration in 1D

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.

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?

n <- 3
integrate(integrand, lower=0, upper=Inf)

digamma(4)*gamma(4)