errr ... exponential integral ---> sum

Posted on 2008-03-28 18:00 By Axel Vogt ( Maple Rating 4

986)

Categories:

  restart: interface(version); Digits:=14:

    Classic Worksheet Interface, Maple 11.02, Windows, Nov 10 2007 Build ID 330022

  Ei(1,1/2*Pi*(1+2*k)): 
  %=convert(%,Sum);
  subs(k=0,%);
  evalf(%);

                    Pi (1 + 2 k)
              Ei(1, ------------) = GAMMA(Pi k + 1/2 Pi)
                         2

                             Pi            Pi
                      Ei(1, ----) = GAMMA(----)
                             2             2

                 0.090082461803865 = 0.89056089038154


and that mess not only for k=0 ...

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or simply

Comment on 2008-03-28 18:07 from Axel Vogt ( Maple Rating 4

986)

  Ei(1,x): 
  %=convert(%,Sum);
                         Ei(1, x) = GAMMA(x)

Strange bug

Comment on 2008-03-28 18:28 from Robert Israel ( Maple Rating 4

1553)

Interesting...

convert(Ei(1,x), GAMMA);

GAMMA(0,x)

(which is correct).

convert(Ei(1,x), FormalPowerSeries, x);

-gamma-ln(x)+Sum((-1)^k/(k+1)!*x^(k+1)/(k+1),k = 0 .. infinity)

(also correct, I think).

Why is the <maple> tag not working for me?

Concerning the maple tag

Comment on 2008-03-29 03:30 from John Fredsted ( Maple Rating 4

566)

It does not work for me either, and it has not worked since the mapleprimes "update" which, it seems, was more concerned with cosmestics than usability. See for instance The pre, code, and maple tags and Maple tag, still ....

perhaps ...

Comment on 2008-03-28 18:35 from Axel Vogt ( Maple Rating 4

986)

perhaps of this?
   
  GAMMA(a,z) = GAMMA(a) - z^a/a * `1F1`(a,1+a,-z);  # from the help page for GAMMA
  subs(a=0,%);

    Error, numeric exception: division by zero

The problem routine

Comment on 2008-03-29 11:35 from JacquesC ( Maple Rating 4

1699)

The core routine that does this transformation is

> showstat(`convert/GAMMA/IC/from`[Ei]);

`convert/GAMMA/IC/from`[Ei] := proc(f)
local a, z, G;
   1   if nargs = 0 then
   2     return 1, (a, z) -> if nargs = 2 then [[z <> 0]] end if
       end if;
   3   if nops(f) = 2 then
   4     a, z := op(f);
   5     if `tools/type`(z <> 0) then
   6       if `tools/type`(a,'integer') then
   7         G := ('GAMMA')(1-a,z)
           else
   8         G := GAMMA(1-a,z)
           end if;
   9       return z^(a-1)*G
         end if
       end if;
  10   f
end proc

which gives GAMMA(0,z) when given Ei(1,z) as input. What should it be instead?

Gamma( z ) vs Gamma( 0 , z )

Comment on 2008-03-29 14:42 from Axel Vogt ( Maple Rating 4

986)

converting Ei( 1 , z ) to Sum returns GAMMA( z ), not GAMMA( 0 , z )

BTW: strange that Maple knows about the (modified) exponential integral Ei( z ), but I can
not find it through the help (it should stand for the formal series in Robert's reply)

Edited for the "BTW": ok, should have looked closer at the help pages ... there it is, sorry

Why GAMMA?

Comment on 2008-03-29 15:23 from jakubi ( Maple Rating 4

943)

What I wonder is why

convert(Ei(1,z),Sum);

does not convert to a formal power series, as stated in ?convert,sum and something like this form is needed to get it:

convert(Ei(1,z), Sum, include = powers,dummy=n);

                 /infinity                 \
                 | -----        n  (n + 1) |
                 |  \       (-1)  z        |
-gamma - ln(z) + |   )     ----------------|
                 |  /      (n + 1)! (n + 1)|
                 | -----                   |
                 \ n = 0                   /

convert(Ei(1,z),Sum) OK in Maple10

Comment on 2008-03-29 15:33 from Alex Smith ( Maple Rating 3

300)

In Maple 10

convert(Ei(1,z),Sum);

does work as the help says.

But as you note, the same command does not work in Maple 11.

What I get

Comment on 2008-03-29 15:56 from jakubi ( Maple Rating 4

943)

in Maple 10.06 is different:

convert(Ei(1,z),Sum); 

infinity
 -----     (_k1)             (-1 + _k1)
  \      (D     )(GAMMA)(1) z
   )     ------------------------------
  /                   _k1!
 -----
_k1 = 0

Gamma(z) vs Gamma(0,z)

Comment on 2008-03-29 17:05 from JacquesC ( Maple Rating 4

1699)

Internally, it first returns Gamma(0,z), which some other routines then ``simplify'' to Gamma(z). Perhaps that is the actual bug?

Gamma(z) vs Gamma(0,z)

Comment on 2008-03-29 18:32 from Axel Vogt ( Maple Rating 4

986)

That might it be, GAMMA(0,z)/GAMMA(z); plot(%,z=0..6); shows it ...

errr ... exponential integral ---> sum

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