[R] Numerical Integration in 1D

[Prof Brian Ripley]

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