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MathGroup Archive 1993

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Erroneous factor Pi/4 in Integrate[ 2F1[ 1/2, 3/2, 2, x^2 ], x ]

  • To: mathgroup at yoda.physics.unc.edu
  • Subject: Erroneous factor Pi/4 in Integrate[ 2F1[ 1/2, 3/2, 2, x^2 ], x ]
  • From: "NELSON M. BLACHMAN" <blachman at gtewd.mtv.gtegsc.com>
  • Date: Sun, 14 Nov 1993 13:31:14 EST

Mathematica 2.2 for DOS 387
Copyright 1988-93 Wolfram Research, Inc.

In[1]:= Integrate[(2/Pi)^2 ArcSin[r Cos[t]] Cos[t], {t,0,Pi}]

On::none: Message SeriesData::csa not found.

General::intinit: Loading integration packages -- please wait.

On::none: Message SeriesData::csa not found.

        4 Integrate[ArcSin[r Cos[t]] Cos[t], {t, 0, Pi}]
Out[1]= ------------------------------------------------
                                2
                              Pi

  SINCE MMA THUS FAR SEEMS UNABLE TO CARRY OUT THIS INTEGRATION, 
I ``SIMPLIFIED'' THE PROBLEM BY EXPRESSING THE ArcSin AS AN INTEGRAL 
IN PREPARATION FOR INTERCHANGING THE ORDER OF THE THEN TWO INTEGRATIONS:

In[7]:= Integrate[r(2/Pi)^2 /Sqrt[1 - (x r Cos[t])^2] Cos[t]^2, {t,0,Pi}]

On::none: Message SeriesData::csa not found.

                              1  3      2  2
        2 r Hypergeometric2F1[-, -, 2, r  x ]
                              2  2
Out[7]= -------------------------------------
                         Pi

  NOW WE CAN CARRY OUT THE FIRST INTEGRATION:  

In[8]:= Integrate[%, {x,0,1}]

On::none: Message SeriesData::csa not found.

                            1  1      2
        r Hypergeometric2F1[-, -, 2, r ]
                            2  2
Out[8]= --------------------------------
                       2

  NOTICE THAT, WHEN MMA INTEGRATED Out[7] WITH RESPECT TO x TO GIVE ME THE 
INTEGRAL I WANTED, IT LOST A FACTOR 4/Pi, AS IS EASILY SEEN BY MAKING r 
SMALL IN Out[7].  THE COEFFICIENT HERE SHOULD HAVE BEEN 2/Pi RATHER THAN 1/2: 

In[14]:= %7 /. r -> .5

                              1  3           2
         1. Hypergeometric2F1[-, -, 2, 0.25 x ]
                              2  2
Out[14]= --------------------------------------
                           Pi

In[15]:= NIntegrate[%, {x,0,1}]

Out[15]= 0.329333

In[16]:= %8 /. r -> .5 // N

Out[16]= 0.258658

Out[8] WAS EVIDENTLY WRONG--WRONG BY THE FACTOR Pi/4:

In[18]:= 4 %16 / Pi // N

Out[18]= 0.329333

  WHY WAS Out[8] WRONG, AND WHAT RELATED ERRORS CAN WE EXPECT FROM MMA?

   ////\\\	
  /  _ _  \			Nelson M. Blachman, Mail Stop 3G35 
(|---o-o---|)			GTE Government Systems Corporation  
 |    L    |			Mountain View, California 94039-7188  
  \ (---) /			(415)-966-2247  blachman at gtewd.mtv.gtegsc.com  
   \\\|///			                blachman%gtewd.dnet at gte.com
    \\|//                       		blachman at itd.nrl.navy.mil  	





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