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 Literature for x^x and Sophomore's Dream
03/02/2010, 02:33 AM
Post: #1
 Ztolk Junior Fellow Posts: 21 Joined: Mar 2010
Literature for x^x and Sophomore's Dream
Hello Tetration Forum.

I'm interested in the second order tetration function, x^x. The integral from 0 to 1 of this function is called the Sophomore's Dream. I was wondering which papers, journals, etc cover either the function or its integral. I've read the respective sections in Dunham as well as Bailey and Borwein, but it's difficult to search for because "x^x" in google scholar just returns everything with "xx," and sophomore's dream just brings up poetry and the like.

Most of the articles on tetration by Galidakis, Kouznetsov, etc are about iterative properties and analytical extension, but not really about the function I'm interested in.

Does anyone have some papers in mind, or know which journal to browse through?

Thanks
03/02/2010, 10:00 AM
Post: #2
 Gottfried Long Time Fellow Posts: 693 Joined: Aug 2007
RE: Literature for x^x and Sophomore's Dream
(03/02/2010 02:33 AM)Ztolk Wrote:  Hello Tetration Forum.

I'm interested in the second order tetration function, x^x. The integral from 0 to 1 of this function is called the Sophomore's Dream. I was wondering which papers, journals, etc cover either the function or its integral. I've read the respective sections in Dunham as well as Bailey and Borwein, but it's difficult to search for because "x^x" in google scholar just returns everything with "xx," and sophomore's dream just brings up poetry and the like.

Most of the articles on tetration by Galidakis, Kouznetsov, etc are about iterative properties and analytical extension, but not really about the function I'm interested in.

Does anyone have some papers in mind, or know which journal to browse through?

Thanks

I've an old article. title "WexZal", which deals specifically with x^x on a not-too-advanced level. I think I've put it into Henryk's database (he provided a link for us tetration-forumers) . If you can't find it there I can send you a copy, just ask me per email, the author has given his permission to do so.

Gottfried

Gottfried Helms, Kassel
03/02/2010, 01:03 PM (This post was last modified: 03/02/2010 01:06 PM by bo198214.)
Post: #3
 bo198214 Administrator Posts: 1,365 Joined: Aug 2007
RE: Literature for x^x and Sophomore's Dream
Hello Ztolk,

what particularly do you want to know about that function?
I dont think there is an article that covers that function and its integral completely.
Rather perhaps particular problems are proven in scattered articles.
E.g. whether there is an expression in elementary functions for the function, and similar questions (though I think that problem is unsolved, only practically nobody found one).

PS: The link to the article Gottfried mentiones is here.
03/02/2010, 03:30 PM (This post was last modified: 03/02/2010 03:32 PM by Ztolk.)
Post: #4
 Ztolk Junior Fellow Posts: 21 Joined: Mar 2010
RE: Literature for x^x and Sophomore's Dream
(03/02/2010 01:03 PM)bo198214 Wrote:  Hello Ztolk,

what particularly do you want to know about that function?
I dont think there is an article that covers that function and its integral completely.
Rather perhaps particular problems are proven in scattered articles.
E.g. whether there is an expression in elementary functions for the function, and similar questions (though I think that problem is unsolved, only practically nobody found one).

PS: The link to the article Gottfried mentiones is here.

Thank you for that book. I'll have a look through it. Was it published anywhere or is it just a work of love?

I'm an amateur writing a paper about some related integrals, and I wanted to do some background research and see what else was done in the area, but I wasn't able to find much.
03/02/2010, 04:51 PM
Post: #5
 bo198214 Administrator Posts: 1,365 Joined: Aug 2007
RE: Literature for x^x and Sophomore's Dream
(03/02/2010 03:30 PM)Ztolk Wrote:  Thank you for that book. I'll have a look through it. Was it published anywhere or is it just a work of love?

Ya, a work of love. But you will also find some references of published articles/books in this book.
03/03/2010, 03:28 PM
Post: #6
 Ztolk Junior Fellow Posts: 21 Joined: Mar 2010
RE: Literature for x^x and Sophomore's Dream
(03/03/2010 04:44 AM)Ansus Wrote:  By the way, you can use Russian Rambler search engine to search for this function:

http://nova.rambler.ru/srch?query=%22x^x%22

Or you can use Uniquotation - special mathematical formula search engine which searches TEX formulas:

http://uniquation.ru/ru/solution-tex.aspx?query=x^x

http://uniquation.ru/ru/solution-tex.asp...int+x^x+dx

Thanks for the links. Now I have to learn Russian...
03/04/2010, 12:05 AM
Post: #7
 tommy1729 Ultimate Fellow Posts: 982 Joined: Feb 2009
RE: Literature for x^x and Sophomore's Dream
(03/02/2010 11:24 PM)Ansus Wrote:  Multiplicative integral and discrete multiplicative integral for this function are known to be

$
e^{-\frac{1}{4}x^2+\frac{1}{2}x^2\ln x}
$

and

$
e^{\frac{z-z^2}{2}+\frac z2 \ln (2\pi)-\psi^{(-2)}(z)}
$

respectively.

Conventional integral and discrete integral are not known.

just like log(x)/log(x+1) , dx / log(x) , exp(x^3) , exp(x^2) , exp(x)/x , sin(x)/x , cos(x)/x , arctan(x)/x , x^x has no conventional integral in terms of elementary functions.

however you may want to look at lagrange inversion , Barnes G , Lambert W and perhaps Meijer G since they relate strongly.

AFAIK Sophomore's dream has never been extended , but lagrange inversion is strongly related.

in fact i believe the extention of Sophomore's dream is more important and relevant to tetration than x^x is !!

as for equating sums to integrals , euler and maclaurin ( and perhaps newton ) have intresting formula's , yet i dont think they help much at first sight ...
( i think lagrange inversion is more important at first sight )

regards

tommy1729
03/04/2010, 12:13 AM
Post: #8
 tommy1729 Ultimate Fellow Posts: 982 Joined: Feb 2009
RE: Literature for x^x and Sophomore's Dream
(03/03/2010 03:28 PM)Ztolk Wrote:
(03/03/2010 04:44 AM)Ansus Wrote:  By the way, you can use Russian Rambler search engine to search for this function:

http://nova.rambler.ru/srch?query=%22x^x%22

Or you can use Uniquotation - special mathematical formula search engine which searches TEX formulas:

http://uniquation.ru/ru/solution-tex.aspx?query=x^x

http://uniquation.ru/ru/solution-tex.asp...int+x^x+dx

Thanks for the links. Now I have to learn Russian...

not really , just do a website / paper translate. ( do it online )

current ( free ! ) technology allows that.
03/04/2010, 05:30 PM (This post was last modified: 03/04/2010 05:33 PM by bo198214.)
Post: #9
 bo198214 Administrator Posts: 1,365 Joined: Aug 2007
RE: Literature for x^x and Sophomore's Dream
You can also develop the selfpower $x^x$ into a powerseries at 1:
$
x^x = \sum_{n=0}^\infty a_n (x-1)^n,\quad |x|<1\\
a_n = \frac{1}{n!}\sum_{k=0}^n \operatorname{lc}(n,k)
$

where lc is the so called Lehmer-Comtet numbers:
$\operatorname{lc}(n,k) = \sum_{l=k}^n \left(l\\ k\right) k^{l-k} s(n,l)$
where $s(n,l)$ are the stirling numbers of the first kind.

The sequence of the first few derivatives ( $a_n n!$) is
1, 1, 2, 3, 8, 10, 54, -42, 944, -5112, 47160, -419760, 4297512, ... (see sloane A005727)

The coefficients of the indefinite integral at 1 are then:
$\frac{1}{n!}\sum_{k=0}^{n-1} \operatorname{lc}(n-1,k)$
accordingly the sequence of derivatives of the integral is shifted 1 to the right.
03/04/2010, 09:36 PM
Post: #10
 Ztolk Junior Fellow Posts: 21 Joined: Mar 2010
RE: Literature for x^x and Sophomore's Dream
I have another method with some interesting results that I'm writing up in a paper.

Any ideas where to submit?
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