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Re: Why does this lead to an answer with complex numbers?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71481] Re: Why does this lead to an answer with complex numbers?
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Mon, 20 Nov 2006 06:17:15 -0500 (EST)
  • Organization: The Open University, Milton Keynes, UK
  • References: <ejosmm$n3k$1@smc.vnet.net> <ejrmr9$97b$1@smc.vnet.net>

aaronfude at gmail.com wrote:
> Hi,
> 
> Thanks for all the answers. They were all very useful, even though I
> have done my best to confuse everyone by leaving a beta in there which
> had nothing to do with the problem.
> 
> So I understand that the answer may be complex and the complex part is
> constant which is in a certain sense valid for a indefinite integral.
> But I very much need a real answer and I still can't quite extract.
> Consider the following:
> 
> \!\(\(\(\ \)\(Assuming[x > 0 && A > 0 && B > 0 && \ B < 1, \
>     FullSimplify[Integrate[Log[\@\(A^2 + x\^2\) - B*x\ ], \ x]]]\)\)\)
> 
> The answer that I get is correct, but not very useful since it is
> appears complex and I could find a way to determine the real part. Do
> you have any suggestions?
> 
> 
> Thank you!
> 
> Aaron Fude
> 

Hi Aaron,

What version of Mathematica do you use? With Mathematica 5.2 I do not 
get any complex numbers for your integral.

In[1]:=
Assuming[x > 0 && A > 0 && B > 0 && B < 1,
   FullSimplify[Integrate[Log[Sqrt[A^2 + x^2] - B*x], x]]]

Out[1]=
        1                      2                           2    2
--------------- (2 Sqrt[-1 + B ] x (-1 + Log[-B x + Sqrt[A  + x ]]) +
              2
2 Sqrt[-1 + B ]

                             2
                  Sqrt[-1 + B ] x
     A (2 ArcTanh[---------------] - 4 Log[A] - 4 Log[B] +
                         A

                                               2                2    2
              4 (-1 + B) (1 + B) (A Sqrt[-1 + B ] - x + B Sqrt[A  + x ])
 
Log[-(----------------------------------------------------------)] +
                                         2           2
                            A Sqrt[-1 + B ] + (-1 + B ) x

                   2                2                2    2
            (-1 + B ) (A Sqrt[-1 + B ] + x + B Sqrt[A  + x ])
        Log[-------------------------------------------------]))
                                    2         2
                       A Sqrt[-1 + B ] + x - B  x

In[2]:=
FreeQ[%, I]

Out[2]=
True

In[3]:=
Assuming[x > 0 && A > 0 && B > 0 && B < 1,
   Integrate[Log[Sqrt[A^2 + x^2] - B*x], x]]

Out[3]=
                                                              2
        1                       2                  Sqrt[-1 + B ] x
--------------- (-2 Sqrt[-1 + B ] x + 2 A ArcTanh[---------------] +
              2                                           A
2 Sqrt[-1 + B ]

                  2                     2    2
     2 Sqrt[-1 + B ] x Log[-B x + Sqrt[A  + x ]] +

                      2                2                2    2
             2 (-1 + B ) (A Sqrt[-1 + B ] - x + B Sqrt[A  + x ])
     A Log[-(---------------------------------------------------)] +
                     2  2               2           2
                    A  B  (A Sqrt[-1 + B ] + (-1 + B ) x)

                    2                2                2    2
           2 (-1 + B ) (A Sqrt[-1 + B ] + x + B Sqrt[A  + x ])
     A Log[---------------------------------------------------])
                    2  2               2         2
                   A  B  (A Sqrt[-1 + B ] + x - B  x)

In[4]:=
FreeQ[%, I]

Out[4]=
True

In[5]:=
Integrate[Log[Sqrt[A^2 + x^2] - B*x], x]

Out[5]=
                                                              2
        1                       2                  Sqrt[-1 + B ] x
--------------- (-2 Sqrt[-1 + B ] x + 2 A ArcTanh[---------------] +
              2                                           A
2 Sqrt[-1 + B ]

                  2                     2    2
     2 Sqrt[-1 + B ] x Log[-B x + Sqrt[A  + x ]] +

                      2                2                2    2
             2 (-1 + B ) (A Sqrt[-1 + B ] - x + B Sqrt[A  + x ])
     A Log[-(---------------------------------------------------)] +
                     2  2               2           2
                    A  B  (A Sqrt[-1 + B ] + (-1 + B ) x)

                    2                2                2    2
           2 (-1 + B ) (A Sqrt[-1 + B ] + x + B Sqrt[A  + x ])
     A Log[---------------------------------------------------])
                    2  2               2         2
                   A  B  (A Sqrt[-1 + B ] + x - B  x)

In[6]:=
FreeQ[%, I]

Out[6]=
True

In[7]:=
Assuming[x > 0 && A > 0 && B > 0 && B < 1, FullSimplify[%% == %%%% == 
%%%%%%]]

Out[7]=
True

In[8]:=
$Version

Out[8]=
5.2 for Microsoft Windows (June 20, 2005)

Regards,
Jean-Marc


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