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what is integral of x^x = ?
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drhu...@gmail.com
Feb 20, 2021, 7:51:54 PM2/20/21
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what is integral of x^x = ?
mathHand.com
mathHand.com
Nasser M. Abbasi
Feb 20, 2021, 8:41:25 PM2/20/21
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A known trick is to write x^x as exp(x*ln x), then
use exp(u) = sum u^n/n!, n=0..infinity, and integrate
term by term, and sum. So you get
sum( integrate( (x*ln x)^n/n! ,x) , n=0..infinity)
For one term only, Rubi gives
oneterm = Int[ x^n*Log[x]^n/n!, x];
(Gamma[1 + n, -((1 + n) Log[x])] Log[x]^n (-((1 + n) Log[x]))^-n)/((1 + n) n!)
But this has no closed form sum
Sum[oneterm, {n, 0, Infinity}];
--Nasser
use exp(u) = sum u^n/n!, n=0..infinity, and integrate
term by term, and sum. So you get
sum( integrate( (x*ln x)^n/n! ,x) , n=0..infinity)
For one term only, Rubi gives
oneterm = Int[ x^n*Log[x]^n/n!, x];
(Gamma[1 + n, -((1 + n) Log[x])] Log[x]^n (-((1 + n) Log[x]))^-n)/((1 + n) n!)
But this has no closed form sum
Sum[oneterm, {n, 0, Infinity}];
--Nasser
Richard Fateman
Feb 21, 2021, 1:54:52 PM2/21/21
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On Saturday, February 20, 2021 at 5:41:25 PM UTC-8, Nasser M. Abbasi wrote:
> On 2/20/2021 6:51 PM, drhu...@gmail.com wrote:
> > what is integral of x^x = ?
> > ........
> On 2/20/2021 6:51 PM, drhu...@gmail.com wrote:
> > what is integral of x^x = ?
>
> But this has no closed form sum
>
> Sum[oneterm, {n, 0, Infinity}];
>
> --Nasser
on the other hand, for any particular integer n, you can integrate x^n*log(x)^n. (at least Maxima has no problem)
> But this has no closed form sum
>
> Sum[oneterm, {n, 0, Infinity}];
>
> --Nasser
so you can generate as many terms as you wish, in elementary form, rational in x and log(x).
RJF
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