Mathematica : Weird Integral

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  • #1
Hepth
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How is MM doing the integral? Is this something with a definite solution?

Limit[Integrate[(1 - 2 x)/x^2, {x, a, b}], a -> 0]
Integrate[(1 - 2 x)/x^2, {x, 0, b}]

The first one gives infinity, the second a closed solution, though technically they should be the same, right?
 

Answers and Replies

  • #2
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Which version of Mma are you using and what is the closed solution the second one gives you?

The integrand goes like 1/x^2 + 1/x as x->0 and so diverges, and as Wolfram|Alpha says, the Cauchy principle value is +Infinity.

In Mma8 the first expression returns
ConditionalExpression[Indeterminate, b >= 0]
(which seems like a mistake to me... what's b>=0 got to do with anything?)
While the second gives a warning about nonconvergence and keeps the expression unevaluated.

ConditionalExpression is new in Mma8. So Mma7 generates conditions for when the integral over {x, a, b} converges in the form of an If[] statement. I gave up waiting for it to take the limit of the resulting expression. Mma7 has the same behaviour as Mm8 for the integral over {x, 0 b}.

And yes, assuming that b is Real, both expressions should yield +Infinity
 
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  • #3
Dale
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I get Simon's results also. Essentially both methods diverge.
 
  • #4
Hepth
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Strange, I must have had some weird assumptions. (Version 7)
So when I run it from a fresh start, I get that they both diverge. If I do some of my other code, NONE OF WHICH uses a,b, or x, I get this:

vpztkm.png


And if I change x-> something random and b-> something random, it does the SAME thing.

EDIT::

I Think I found the problem, there must be something in Feyncalc. If I do:

<< HighEnergyPhysics`FeynCalc`;

just to load it, then do the integrals , it gives the weird responses... I wonder if there is something in its definitions to handle the N-dimensional divergent integrals that is messing with this.
 
  • #5
Hepth
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Here's what I get:

2z6xwmt.png
 
  • #6
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Probably somewhere some assumptions have been set in FeynCalc, or the option GenerateCondtions->False has been set.

The output you obtain after loading FeynCalc is the same as

Integrate[(1 - 2 x)/x^2, {x, 0, b}, GenerateConditions -> False]
 
  • #7
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I just downloaded the latest version of the http://www.feyncalc.org/" [Broken] package and did a quick

find . -exec grep -l "GenerateConditions" {} \;

and the only files that turned up in the latest version of FeynCalc was the FeynCalcBook.nb and /fctables/Integrate3.m. But it only applied to a specific PolyLog integral... So I'm not sure what's causing the problem.

You can similarly find every file that contains the string "Integrate" - but a quick look at these does not show anything obvious....

Finally, I ran the same sequence of commands you gave in your screenshots in both Mma7 and Mma8 and did not reproduce your results...
Can you load FeynCalc and run "Options[Integrate, GenerateConditions]" to see if it returns the default "GenerateConditions->Automatic"?
I assume that it must be set to False, because otherwise your Out[8] would have the same conditions attached to it as your Out[3]. Then it's just a matter of tracking down what setting you've made in you FeynCalc install that would do this. Try running the "find ..." command I gave above in your FeynCalc directory.
 
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  • #8
Hepth
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yeah thats it.

In[1]:= Options[Integrate,GenerateConditions]

Out[1]= {GenerateConditions->Automatic}
In[2]:= <<HighEnergyPhysics`FeynCalc`;
During evaluation of In[2]:= Set::wrsym: Symbol MonomialList is Protected. >>
During evaluation of In[2]:= DumpGet::bgbf: File C:\Program Files\Wolfram Research\Mathematica\7.0\AddOns\Applications\HighEnergyPhysics\Tarcer\tarcer25.mx cannot be loaded, it is corrupted or is written on a different machine. >>
In[3]:= Options[Integrate,GenerateConditions]

Out[3]= {GenerateConditions->False}
 

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