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some Limits

  • To: mathgroup at
  • Subject: [mg25794] some Limits
  • From: Otto Linsuain <linsuain+ at>
  • Date: Wed, 25 Oct 2000 03:53:50 -0400 (EDT)
  • Sender: owner-wri-mathgroup at

Hi all. I find the following results rather alarming.
As you probably know the Hypergeometric2F1[a,b,c,z] has a branch cut for 
z > 1. The function is real for real a,b,c and z < 1; but picks up an
imaginary part for z > 1. The function satisfies the Shwartz (spelling?)
reflection principle:

2F1[a,b,c,Conjugate[z]] = Conjugate[2F1[a,b,c,z]] for real a,b,c.

This works fine in Mathematica, but look what happens when you try to
find the Limits when z approaches the real axis from above and from
below to some value larger than one. The results of Limits are compared
with the results of evaluating slightly off the axis:

{a, b, c, z} = {0.7, 1.8, 2.1, 2.4};
Hypergeometric2F1[a, b, c, z]
{Limit[Hypergeometric2F1[a, b, c, z + I x], x -> 0, Direction -> -1], 
  Hypergeometric2F1[a, b, c, z + I 0.000001]}
{Limit[Hypergeometric2F1[a, b, c, z - I x], x -> 0, Direction -> -1], 
  Hypergeometric2F1[a, b, c, z - I 0.000001]}

-0.386677 - 0.923346 I

{-0.386677 - 0.923346 I, -0.386677 + 0.923346 I}

{-0.386677 - 0.923346 I, -0.386677 - 0.923346 I}

As you see, evaluating just at z=2.4 gives the same answer as evaluating
slightly below (a little awkward but OK)which gives the same answer as
taking the limit from below (OK) Which SHOULD NOT be the same as taking
the limit from above!! Evaluating slightly above gives the right answer:
the complex conjugate of evaluating slightly below.

 Well I guess Mathematica is not taking the Limit, but merely evaluating
at the point. Quite a shame! By the way, I just tried NLimit and it
gives correct answers, i.e. a positive sign for the imaginary part of
the first entry in Out[34] above, as it should. Otto Linsuain. 

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