Re: Did you hear about Euler-Mascheroni integrals?
- From: Badger <invalid@xxxxxxxxxxxxxxx>
- Date: Wed, 21 Feb 2007 17:42:20 -0500
On Wed, 21 Feb 2007 11:12:03 +0100, Joachim Selke <selke@xxxxxx>
wrote:
Hi!
While doing some computation tasks I recently came across an interesting
family of integrals, defined by (for natural numbers n)
I(n) = (-1)^n * \int_0^\infty [ln(z)]^n * exp(-z) dz.
The (-1)^n factor is there because the integral is positive for even n
and negative otherwise.
Let's look at some values of I(n): I(0) is 1, I(1) is the
Euler-Mascheroni constant gamma. Additionally, I(n) seems to converge to
n!, for large n. To be more precise, by experiments I guess that
I(n) / n! tends to 1.
Also note that I(n) is equal to the absolute value of Gamma[n](1), where
Gamma[n] is the n-th derivative of the Gamma function.
By chance, I found a Mathworld entry that deals with I(n) and calls it
Euler-Mascheroni integrals:
<http://mathworld.wolfram.com/Euler-MascheroniIntegrals.html>.
Unfortunately, this entry does not mention who coined this name for I(n)
and gives no further references. I already asked the MathWorld team a
few weeks ago but received no answer yet.
I'm interested in more details about this family of integrals but was
not able to find anything, except for some web pages I describe below. I
even looked in some mathematical encyclopedias but still did not find
anything.
This is where you come into play: Maybe you have heard the name
"Euler-Mascheroni integrals" before. If so, then you can make me happy
by sending me references to books or scientific articles that either
mention this term or deal with some properties of I(n), for n >= 2.
Please let me know. :-)
Thanks,
Joachim
I can't help with references to the specific term "Euler-Mascheroni
integrals", but I can give you some pointers to relevant material.
"Irresistible Integrals: Symbolics, Analysis and Experiments in the
Evaluation of Integrals" by Boros and Moll, ISBN 0521796369.
Chapter 10 discusses "Eulerian Integrals: The Gamma and Beta
Functions". The authors do not refer specifically to the integral you
are calling I(n), but their discussion of the Psi Function does
include (page 213) essentially the same values as one would find for
I(0) through I(5).
They also give (page 214) a recursion for what would be your I(n+1),
and refer to a book that contains the values up to what would be your
I(10), namely "Series Associated with the Zeta and Related Functions"
by Srivastava and Choi.
At his web site, Victor Moll has a link to his on-going work that
provides proofs for a number of the integral formulas that appear in
G&R. <http://www.math.tulane.edu/~vhm/Table.html>
Proofs that are related to the integral you asked about include:
<http://www.math.tulane.edu/~vhm/web_html/gamma1web.pdf>
<http://www.math.tulane.edu/~vhm/web_html/explog1web.pdf>
HTH
.
- Follow-Ups:
- Re: Did you hear about Euler-Mascheroni integrals?
- From: Joachim Selke
- Re: Did you hear about Euler-Mascheroni integrals?
- References:
- Did you hear about Euler-Mascheroni integrals?
- From: Joachim Selke
- Did you hear about Euler-Mascheroni integrals?
- Prev by Date: Re: ZFC? countable?uncountable?
- Next by Date: Re: Did you hear about Euler-Mascheroni integrals?
- Previous by thread: Re: Did you hear about Euler-Mascheroni integrals?
- Next by thread: Re: Did you hear about Euler-Mascheroni integrals?
- Index(es):
Relevant Pages
|
