sci.physics

On Aug 7, 11:05 pm, Robert Israel
wrote:
> Hannu Poropudas writes:
> > On Aug 7, 10:59 am, Eric Gisse wrote:
> > > On Aug 4, 11:21 pm, Hannu Poropudas wrote:
> > > [...]
>
> > > /
>
> > If I use series approximation of order 4 from the above reference:
>
> > x^x = 1+ln(x)x+(1/2)ln(x)^2 x^2+(1/6)ln(x)^3 x^3+O(x^4),
>
> > then I have the following series approximation of order 5
> > for the integral function of x^x:
>
> > int(x^x dx)= x+(1/2)ln(x)x^2-(1/4)x^2+(1/6)ln(x)^2 x^3-(1/9)x^3 ln(x)+
> > + (1/27)x^3 + (1/24)ln(x)^3 x^4 -(1/32)x^4 ln(x)^2+(1/64)x^4 ln(x)+
> > -(1/256)x^4 + O(x^5).
>
> Actually it isn't really O(x^5), it's O(x^5 (1 + ln(x)^4)).
> Maple's "O" doesn't mean the same as the mathematical "O".
>

Yes and above formulae seems to be OK.

> > These all are calculated with Maple 9. I just wonder why Maple gives
> > two different series for x^x ? Are they both correct or do Maple 9
> > have some BUG here ?
>
> What other series are you referring to?
>

That 5 Aug 2007 posted was some BUG of Maple 9 or most probably there
were some user (my) error in using Maple 9.

> BTW, this series can be used to get one of my favourite formulas:
>
> int_0^1 x^x dx = - sum_{n=1}^infty (-n)^(-n)

This is a very nice result. I found more one general result:

1-y/(2^2)+y^2 /(3^3)-y^3 /(4^4)+y^4 /(5^5)- ... infinity =

= int_0^1 x^(xy) dx .

This formula is nro 1002 in pages 184-185 in the reference:

Jolley, . (collected by), 1961.
Summation of Series.
edition, Dover Publications Inc., New York.
Manufactored in the United States of America.
251 pages, pages 184-185.

Hannu


> --
> Robert Israel isr...@
> Department of Mathematics /~israel
> University of British Columbia Vancouver, BC, Canada