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