Erroneous Mathematica Kummer's (Hypergeometric1F1) function
- To: mathgroup at smc.vnet.net
- Subject: [mg81444] Erroneous Mathematica Kummer's (Hypergeometric1F1) function
- From: "W. Brenig" <w.brenig at tu-bs.de>
- Date: Sun, 23 Sep 2007 21:14:25 -0400 (EDT)
Hi, 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. Regards, Wolfram
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