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Erroneous Mathematica Kummer's (Hypergeometric1F1) function

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  • Subject: [mg81444] Erroneous Mathematica Kummer's (Hypergeometric1F1) function
  • From: "W. Brenig" <w.brenig at>
  • Date: Sun, 23 Sep 2007 21:14:25 -0400 (EDT)


This excerpt from a notebook demonstrates that Mathematica's
implementation of the confluent hypergeometric function is
erroneous. The integral representation of the Hypergeometric1F1[]
function is

In[115]:= f11[m_, n_, z_] := ((n - 1)!/((n - m - 1)! (m - 1)!))
   NIntegrate[Exp[ x z] x^(m - 1) (1 - x)^(n - m - 1), {x, 0, 1}];

Now, we use this representation to evaluate f11 for two
representative sets of parameters

In[116]:= f11[5, 106, 10 Pi I]
Out[116]= 0.101285+ 0.811816 \[ImaginaryI]

In[117]:= f11[4.6, 105.3, 10 Pi I]
Out[117]= 0.190397+ 0.80675 \[ImaginaryI]

These values have also been tested to result on use of the
routine CCHG(A,B,Z,CHG) for the confluent 1F1 hypergeometric
function, copyrighted to Zhang and Jin, authors of the book
'Computation of Special Functions', Wiley-Interscience, (1996).

Next we use Mathematica's Hypergeometric1F1[] function with
the same parameters

In[121]:= N[Hypergeometric1F1[5, 106, 10 Pi I]]
Out[121]= -5952. - 2512. \[ImaginaryI]

In[122]:= N[Hypergeometric1F1[4.6, 105.3, 10 Pi I]]
Out[122]= 0.190397+ 0.80675 \[ImaginaryI]

Obviously the 1st result is different from f11's value.
This is only one of many other examples, which show that
Mathematica's Hypergeometric1F1[] is broken - at least for
integer value input parameters.


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