Re: Incorrect symbolic improper integral
- To: mathgroup at smc.vnet.net
- Subject: [mg103636] Re: Incorrect symbolic improper integral
- From: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>
- Date: Wed, 30 Sep 2009 07:41:14 -0400 (EDT)
The integral you tried is a classical one. It is always calculated in the textbooks on application of
complex variables to calculation of integrals. Its exact value is therefore, well-known. Evaluate this please:
HoldForm[\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(-\[Infinity]\), \
\(\[Infinity]\)]\(
FractionBox[\(Cos[a\ x]\), \(
\*SuperscriptBox[\(b\), \(2\)] +
\*SuperscriptBox[\(x\), \(2\)]\)] \[DifferentialD]x\)\) = \[Pi]/
b Exp[-a b]]
assuming a>0 and b>0. Its evaluation at a=b=1 yields:
In[5]:= \[Pi]/b Exp[-a b] /. {a -> 1, b -> 1}
Out[5]= \[Pi]/\[ExponentialE]
which is obviously the same as the solution returned by Mathematica that you showed. Another point that when I evaluated your
integral with parameter
In[6]:= Integrate[Cos[a x]/(1 + x^2), {x, -\[Infinity], \[Infinity]},
Assumptions -> a \[Element] Reals]
Out[6]= \[ExponentialE]^-Abs[a] \[Pi]
it returned (by my machine Math 6.0, Windows XP)
\[ExponentialE]^-Abs[a] \[Pi]
that is in line with the above exact solution, rather than with the value
\[Pi] Cosh[a]
that you report. So may be you still have a problem.
Alexei
Below is a definite integral that Mathematica does incorrectly.
Thought someone might like to know:
In[62]:= Integrate[Cos[x]/(1 + x^2), {x, -\[Infinity], \[Infinity]}]
Out[62]= \[Pi]/E
What a pretty result--if it were true. The correct answer is \[Pi]*Cosh
[1], which can be checked by adding a new parameter inside the
argument of Cos and setting it to 1 at the end:
In[61]:= Integrate[Cos[a x]/(1 + x^2), {x, -\[Infinity], \[Infinity]},
Assumptions -> a \[Element] Reals]
Out[61]= \[Pi] Cosh[a]
Regards,
Jason Merrill
--
Alexei Boulbitch, Dr., habil.
Senior Scientist
IEE S.A.
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11, rue Edmond Reuter
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Luxembourg
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