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

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

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}]

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}]

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?

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