#1033 taylor at infinity for non algebraic

closed
nobody
3
2007-10-19
2006-11-29
No

Consider:

(%i1) log(x) + exp(x^2)$
(%i2) e : diff(%,x)/%;
(%o2) (2*x*%e^x^2+1/x)/(log(x)+%e^x^2)

Towards infinity, a good approximation to e is 2 x. But

(%i3) taylor(e,x,inf,2);
(%o3) 4/x+...

And

(%i4) taylor(e,x,inf,5);
(%o4) 240/x^5+...

And

(%i5) taylor(ratsimp(e),x,inf,5);
1/(x*log(x)+x*%e^x^2)
Assumed to be zero in `taylor'
(%o5) 0+...

It would be better if taylor just gave up. All
the results are not right.

Barton

Discussion

  • Barton Willis

    Barton Willis - 2006-12-01

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    Two related problems:

    (%i1) log(log(x)) + exp(-x)$
    (%i2) diff(%,x)/%$
    (%i3) taylor(%,x,inf,2);
    Invalid call to var-expand

    (%i4) log(log(x)) + x$
    (%i5) diff(%,x)/%$
    (%i6) taylor(%,x,inf,2);
    (%o6) 1/x+(1/log(x)+(-log(log(x))+zeroa+...)+...)/x^2+...

    Isn't this unsimplified? I don't know what taylor
    is trying to do with expressions similar to %o5.
    Again, maybe it would be better if taylor gave up.

     
  • Dan Gildea

    Dan Gildea - 2007-10-04

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    These seem ok in current cvs.

    (%i7) log(x) + exp(x^2)$

    (%i8) e : diff(%,x)/%;
    (%o8) (2*x*%e^x^2+1/x)/(log(x)+%e^x^2)

    (%i9) taylor(e,x,inf,2);
    (%o9) +2*x+((+(-2*log(x)))*x+1/x)*%e^-x^2
    +((+2*log(x)^2)*x+(+(-log(x)))/x)*(%e^-x^2)^2

    (%i10) taylor(e,x,inf,5);
    (%o10) +2*x+((+(-2*log(x)))*x+1/x)*%e^-x^2
    +((+2*log(x)^2)*x+(+(-log(x)))/x)*(%e^-x^2)^2
    +((+(-2*log(x)^3))*x+(+log(x)^2)/x)*(%e^-x^2)^3
    +((+2*log(x)^4)*x+(+(-log(x)^3))/x)*(%e^-x^2)^4
    +((+(-2*log(x)^5))*x+(+log(x)^4)/x)*(%e^-x^2)^5

    (%i11) taylor(ratsimp(e),x,inf,5);
    (%o11) +2*x+((+(-2*log(x)))*x+1/x)*%e^-x^2
    +((+2*log(x)^2)*x+(+(-log(x)))/x)*(%e^-x^2)^2
    +((+(-2*log(x)^3))*x+(+log(x)^2)/x)*(%e^-x^2)^3
    +((+2*log(x)^4)*x+(+(-log(x)^3))/x)*(%e^-x^2)^4
    +((+(-2*log(x)^5))*x+(+log(x)^4)/x)*(%e^-x^2)^5

    (%i12) log(log(x)) + exp(-x)$

    (%i13) diff(%,x)/%;
    (%o13) (1/(x*log(x))-%e^-x)/(log(log(x))+%e^-x)

    (%i14) taylor(%,x,inf,2);
    (%o14) +(+(+1/log(log(x)))/log(x))/x+(+(-1/log(log(x)))
    +(+(+(-1/log(log(x))^2))/log(x))/x)
    *%e^-x+(+1/log(log(x))^2)*(%e^-x)^2

    (%i15) log(log(x)) + x$

    (%i16) diff(%,x)/%;
    (%o16) (1/(x*log(x))+1)/(log(log(x))+x)

    (%i17) taylor(%,x,inf,2);
    (%o17) 1/x+(+(-log(log(x)))+1/log(x))/x^2

     
  • Dan Gildea

    Dan Gildea - 2007-10-04
    • status: open --> pending
     
  • SourceForge Robot

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    This Tracker item was closed automatically by the system. It was
    previously set to a Pending status, and the original submitter
    did not respond within 14 days (the time period specified by
    the administrator of this Tracker).

     
  • SourceForge Robot

    • status: pending --> closed
     

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