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MathGroup Archive 1994

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Improper Integral

  • To: mathgroup at yoda.physics.unc.edu
  • Subject: Improper Integral
  • From: Richard Mercer <richard at rmercer.wright.edu>
  • Date: Thu, 03 Mar 1994 11:47:17 -0500

>  	     When I integrate the function f[x]=1/x ( x
>  	     is Real) the result is Log[Abs[x]]. But,
>  	     when I integrate using Mathematica, the
>  	     result obtained is Log[x]. This is not
>  	     correct if the function is real with real
>  	     variables.
>  	      For instance, Integrate[1/x,{x,-a,a}]=0.
>  	      But Mathematica answer:
>  

>  

>  In[29]:= Integrate[1/x,{x,-a,a}]
>  

>  Out[29]= -Log[-a] + Log[a]
>  

>  

>  	       This causes problems to me, when I am
>  	       calculating definite multiple integrals.
>  	       How could I solve this problem? I think
>  	       I have to say to Mathematica that my
>  	       functions and variables are real. But I
>  	       do not know to do that.

Neither -Log[-a] + Log[a] nor 0 is a satisfactory output for this integral. The  
ideal answer would be "Divergent" (or "ComplexInfinity" or "Indeterminate")  
because this definite integral does not exist.

The only way you can get an answer of zero is as a Cauchy principal value by  
taking the limit as eps -> 0 of the integrals over the intervals (-a,-eps) and  
(eps,a). Mathematica should not give an answer which depends on that approach.





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