Integral of x^x: Does It Exist?

[gulsen]

Is there a solution for \int x^x dx, I wonder...

 

[dextercioby]

It is, but not in closed form. However, we're pretty sure that

\int \left(ln x +1) x^{x} \ dx

exists in closed form...Even though neither of the 2 integrals involved in it exist in closed form.

Daniel.

 

[GoldNow]

This integral doesn't have a solution . Say will be great to know the solution of this integral :!)
Function x^x
This function cannot be integrated.
Most likely it's a nonelementary integral. from http://www.numberempire.com/integralcalculator.php.

 

[arildno]

This integral doesn't have a solution . Say will be great to know the solution of this integral :!)
Function x^x
This function cannot be integrated.
Most likely it's a nonelementary integral. from http://www.numberempire.com/integralcalculator.php.

1. Please, do not do paleontological research here at PF. This thread is 4 years old
2. x^x is an integrable function, because it is continuous

 

[GoldNow]

I am new here and can you give me this function ? I love it to see it .

 

[arildno]

Indeed.
It is called F(x), and can be represented as follows:
F(x)=\int_{0}^{x}t^{t}dt+1

A number of other representations of it is possible, none of them in terms of a finite combination of elementary functions.

 

[Pere Callahan]

I remember from my calc III course that there are some algorithms to determine if a given function's integral is expressible in terms of elementary function. Unfortunately I don't remember the details.
Does someone know how to prove for a given function that it will never show up as a derivative of an elementary function?

Of course, "elementary function" can be defined in different ways, but that's not my point.

And sorry for hijacking the thread, this just came to my mind when I read your posts.

 

[JJacquelin]

Does someone know how to prove for a given function that it will never show up as a derivative of an elementary function?
Of course, "elementary function" can be defined in different ways, but that's not my point.

Many functions which aren't "derivatives of elementary functions" (as commonly said) can be expressed in terms of an infinite series of elementary functions. But don't split hairs !
Proving for a given function that it will never show up as a derivative of a FINITE number of elementary functions is a different ball game. It was done in some particular and limited cases, but not in general. Present mathematical theory in this field isn't advanced enough.
For French speakers only : an essay on special functions, presented as a scientific review for general public, is avalable through the link :
http://www.scribd.com/people/documents/10794575-jjacquelin
Then, select the e-paper: "Safari au pays des fonctions spéciales"
.
By the way, an integral of x^x is knows as a special function, namely the "Sophomore's dream" function.

 

[Gib Z]

http://en.wikipedia.org/wiki/Risch_algorithm" is used to determine if an antiderivative can be evaluated in terms of elementary functions and if so, it evaluates them.

Also, the "Sophomores Dream" usually refers to the result \int^1_0 \frac{1}{x^x} dx = \sum_{n=1}^{\infty} \frac{1}{n^n} also sometimes it is used to refer to \int^1_0 x^x dx = - \sum_{n=1}^{\infty} (-n)^{-n}. It never refers to a function.

 

[JJacquelin]

"Somophores dream" :

It never refers to a function.

Well, if not, why not the "Masters dream function" ?

 

[JJacquelin]

The discussion on this topic inspired an article entitled "Sophomore's Dream Function", now published on Scribd :
http://www.scribd.com/people/documents/10794575-jjacquelin

 

[JJacquelin]

URL changed. The new link is :
http://www.scribd.com/JJacquelin/documents

 

[lamarche]

JJ, You seem to know a lot about this "int x^x" function - does it have poles or zeros on the complex plane? does it have the same branch cuts as x ln(x) (or ln(x)) ??
That's the real importance of creating a new function ... to know the pole/zero/cut structure
I am asking because integral(-inf,+inf) exp(-|w|/2) w^(i/pi w ln |w|+i w x) seems to have a pole at -pi^2/6... but I don't know why it would have a pole there...

 

[JJacquelin]

JJ, You seem to know a lot about this "int x^x" function - does it have poles or zeros on the complex plane? does it have the same branch cuts as x ln(x) (or ln(x)) ??
That's the real importance of creating a new function ... to know the pole/zero/cut structure

In fact, I don't know a lot about this "int x^x" function, as it is written : "My own contribution will here appear so modest in the light of all that still remains to be done", page 3 in the paper "The Sophomores Dream Function",
http://www.scribd.com/JJacquelin/documents
Especially, the section 11 Complex arguments, is quite empty. Study was not advanced enough. Only in case of more available results they could be added in a new edition. Moreover, I expect that more publications will be made by other authors in order to increase the background. As it is written "it will depend on many contributors".

I am asking because integral(-inf,+inf) exp(-|w|/2) w^(i/pi w ln |w|+i w x) seems to have a pole at -pi^2/6... but I don't know why it would have a pole there...

The question is interesting. Unfortunately I am not sure to correctly read the integral. I suppose that dw is missing. Also w^(i/pi w ln |w|+i w x) looks confusing for me. Would you like rewrite the whole integral more clearly ?

 

[lamarche]

Hi! JJ,
The integral I am seeking to evaluate is

int_{-\inf}^{+\inf) exp(-|w|/2) exp( i w [ln(|w|)/pi-x]) dw

a definite integral which is a function of x...

I don't know why, but this function seems to have a singularity at -pi^2/6

This does have a physical application: it would be the minimum phase impulse
response of a cable having losses linear with frequency,,,

 

[HallsofIvy]

Hi! JJ,
The integral I am seeking to evaluate is

int_{-\inf}^{+\inf) exp(-|w|/2) exp( i w [ln(|w|)/pi-x]) dw

a definite integral which is a function of x...

I don't know why, but this function seems to have a singularity at -pi^2/6

This does have a physical application: it would be the minimum phase impulse
response of a cable having losses linear with frequency,,,

Since this has no relation to the integral of x^x I recommend you start a new thread rather than add onto this one.

 

[lamarche]

https://www.physicsforums.com/showthread.php?p=3025068#post3025068"
You're right, I posted my question at the above link