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