05/11/2014, 08:28 PM

Because of recent events here, I felt the need to talk about the inverse gamma function.

Im talking about the entire approximation of exp^[1/2] and the Taylor coefficients being O ( 1/ Gamma( n (arc2sinh(n) - 1) ) ).

Cleary from the viewpoint of numerical analysis it seems usefull to have a good approximation of the inverse gamma function.

The functional inverse that is.

Now my honesty forces me to quote the source of an intresting approximation of the inverse gamma function :

David W Cantrell

http://mathforum.org/kb/message.jspa?messageID=342552

In case that page gets removed I post the main formula here :

A is the positive zero of the digamma function ( 1.4616... )

B = sqrt(2 pi) / e - Gamma(A). ( 0.0365... )

L(x) = ln( (x+c) / sqrt(2 pi) )

W(x) is the Lambert W function. ( the functional inverse of x exp(x))

Gamma^[-1](x) = approx = 1/2 + L(x)/W(L(x)/e)

The error term goes to 0 as x goes to +oo.

Together with the approximation for the Lambert W :

LambertW(x) = ln(x) - ln(ln(x)) + ln(ln(x)) / ln(x)

this gives a practical way to compute the inverse gamma function.

I think there is also a page by wolfram about the inverse gamma but I cant find it ? ( And no, I dont mean the inverse regularized gamma or the statistical inverse gamma distribution )

I would like to see an integral representation of the inverse gamma function too.

regards

tommy1729

Im talking about the entire approximation of exp^[1/2] and the Taylor coefficients being O ( 1/ Gamma( n (arc2sinh(n) - 1) ) ).

Cleary from the viewpoint of numerical analysis it seems usefull to have a good approximation of the inverse gamma function.

The functional inverse that is.

Now my honesty forces me to quote the source of an intresting approximation of the inverse gamma function :

David W Cantrell

http://mathforum.org/kb/message.jspa?messageID=342552

In case that page gets removed I post the main formula here :

A is the positive zero of the digamma function ( 1.4616... )

B = sqrt(2 pi) / e - Gamma(A). ( 0.0365... )

L(x) = ln( (x+c) / sqrt(2 pi) )

W(x) is the Lambert W function. ( the functional inverse of x exp(x))

Gamma^[-1](x) = approx = 1/2 + L(x)/W(L(x)/e)

The error term goes to 0 as x goes to +oo.

Together with the approximation for the Lambert W :

LambertW(x) = ln(x) - ln(ln(x)) + ln(ln(x)) / ln(x)

this gives a practical way to compute the inverse gamma function.

I think there is also a page by wolfram about the inverse gamma but I cant find it ? ( And no, I dont mean the inverse regularized gamma or the statistical inverse gamma distribution )

I would like to see an integral representation of the inverse gamma function too.

regards

tommy1729