MMA Integration Warnings

• To: mathgroup at yoda.physics.unc.edu
• Subject: MMA Integration Warnings
• From: bert at netcom.com (Roberto Sierra)
• Date: Thu, 18 Nov 1993 02:37:59 -0800

```I tried to evaluate the following definite integral in Mathematica
(Mac v2.03), and encountered some unexpected problems.  As I have a
fairly old version of MMA, I'd be curious to find out if the trouble
still manifests itself in the current version.  Does this indicate
that there are typos in the v2.03 integration package?

Here's the integral (from a posting to sci.math on USENET):

+1
_
/ i
i
i
i	   -0.98     -0.98
i  (1-x)	(1+x)	   dx
i
i_/

-1

In[1]:=	\$Version
Out[1]=	Macintosh 2.0 (September 3, 1991)

In[2]:=	f[x_] := (1-x)^(-98/100) (1+x)^(-98/100)

[Note rational exponents]

In[3]:=	g = Integrate[f[x],{x,-1,1}]

Syntax::bktwrn:
Warning: "f (b+a x^dg)" should probably be "f [b+a x^dg]".
(line 169 of "Integrate`mainalgorithm`")

Syntax::bktwrn:
Warning: "f (Denominator[a]^Abs[b])" should probably be
"f [Denominator[a]^Abs[b]]".
(line 1227 of "Integrate`mainalgorithm`")

[What gives with these warnings?  I've never seen MMA warnings like
these before.  Do these suggest that there are typos in Integrate.m?]

Out[3]=
51
25 Sqrt[Pi] Gamma[--]
50
---------------------
13
Gamma[--]
25

In[4]:=	N[g,30]   (* numeric value to 30 digits *)
Out[4]=	25.686418380599329666080166119

The answer *looks* about right, but I haven't been able to verify it
by hand or by performing a numerical integration.  NIntegrate gives
a vastly different answer, and seems to have trouble due to the
singularities at +1 and -1 -- I haven't figured out how to get around
that yet by adjusting RecursionLimit or WorkingPrecision.

In[5]:=	NIntegrate[f[x],{x,-1,1}]

NIntegrate::ncvb:
NIntegrate failed to converge to prescribed accuracy after 7
recursive bisections in x near x = -1..

Out[5]=	51.3739

Another thing I checked was to see what would happen if I used real
exponents (-0.98) instead of rational exponents (98/100), to see if
the answer was the same.  Interestingly, though the same internal
warnings are generated, the results are defined in terms of the
HypergeometricPFQ function, for which I can find no documentation.

In[6]:=	f[x_] := (1-x)^(-0.98) (1+x)^(-0.98)

[Note real exponents]

In[7]:=	g = Integrate[f[x],{x,-1,1}]

Syntax::bktwrn:
Warning: "f (b+a x^dg)" should probably be "f [b+a x^dg]".
(line 169 of "Integrate`mainalgorithm`")

Syntax::bktwrn:
Warning: "f (Denominator[a]^Abs[b])" should probably be
"f [Denominator[a]^Abs[b]]".
(line 1227 of "Integrate`mainalgorithm`")

Out[7]=	50. HypergeometricPFQ[{1, 0.98}, {1.02}, -1]

In[8]:=	N[g,30]
Out[8]=	49.9999999999999999 HypergeometricPFQ[{1.,
0.9800000000000000001}, {1.02}, -1.]

[Why can't MMA evaluate a numerical solution here?]

Also, I can find no mention of the HypergeometricPFQ function anywhere
in the doc, except for what little MMA has to say when you ask about it:

In[9]:=	??HypergeometricPFQ
Out[9]=	HypergeometricPFQ[numlist, denlist, z] gives the generalized
hypergeometric function pFq where numlist is a list of the p
parameters in the numerator  and denlist is a list of the q
parameters in the denominator.

Does anyone know what this function is and how it is used??  How
do I get a numeric solution from it?  What about those integration
warnings?  Have they been fixed in the current version of MMA?

Thanks in advance for any replies...

--
\\|//                         "Television is a medium -- it is
- -                           neither rare nor well done."
o o                                            -- Ernie Kovacs
J   roberto sierra
O   tempered microdesigns    NOTICE:
\_/  san francisco, ca        The ideas and opinions expressed
bert at netcom.com          herein are not those of the author.

```

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