Re: Integration bug
- To: mathgroup at smc.vnet.net
- Subject: [mg38351] Re: Integration bug
- From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
- Date: Thu, 12 Dec 2002 01:37:31 -0500 (EST)
- References: <at4cv9$eta$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Thanks to Ray Gittings, I've investigated this bug a bit further,
and have found another entirely separate "curious behavior"
which I do not understand.
First, if we ask Mathematica (version 4.1) to
Integrate[Sin[x+d]/(x+d), {x, -Infinity, 0}]
we get, incorrectly, -Pi/2 + SineIntegral[d].
A correct result would have been +Pi/2 + SineIntegral[d].
The incorrect behavior above presumably explains, in some
sense, what I had noted in my previous post:
Integrate[Sin[x+d]/(x+d), {x, -Infinity, Infinity}] incorrectly yields 0.
Second, it seems that Mathematica does get correct
answers for such integrals if d is given as a specific value.
But sometimes the answer is quite needlessly complicated.
For example, Integrate[Sin[x+Pi]/(x+Pi), {x, -Infinity, 0}]
yields the messy result
(Pi*(-2 + Sqrt[2]*BesselJ[1/2, Pi])*Log[2])/2 -
(Pi*(-2 + Sqrt[2]*BesselJ[1/2, Pi])*Log[4])/4 +
(I/2)*(I*Pi + 2*Pi^2 - (2*I)*Pi*Log[Pi]) -
((-2 + Sqrt[2]*BesselJ[1/2, Pi])*(EulerGamma*Pi^(3/2) -
I*Pi^(5/2) - Pi^(3/2)*Log[Pi]))/(2*Sqrt[Pi]) +
(Pi^(3/2)*(-(((-2 + Sqrt[2]*BesselJ[1/2, Pi])*Log[2])/
Sqrt[Pi]) - (EulerGamma*Pi - 2*SinIntegral[Pi] +
SinIntegral[2*Pi])/Pi^(3/2) -
(-4*Sqrt[Pi] + 2*EulerGamma*Sqrt[Pi] +
2*Sqrt[Pi]*Log[4] - Sqrt[2*Pi]*
Derivative[1, 0][BesselJ][1/2, Pi])/(2*Pi)))/2
But Pi/2 + SineIntegral[Pi] is a much simpler correct answer!
BTW, asking for a numerical approximation of the messy
answer above, we get correctly 3.42273 + 1.77635*10^-15*I .
(Of course, the precise answer is purely real. The very small
imaginary component in the approximation is an understandable
artifact.)
Now here's the new behavior I can't understand:
Suppose that we ask Mathematica to Simplify the messy
result above, hoping that it will be smart enough to give us
something like Pi/2 + SineIntegral[Pi] . Instead, we get
Indeterminate
preceded by the comment
\!\(Power::"indet" \(\(:\)\(\ \)\)
"Indeterminate expression \!\(0``68.9889\^0\) encountered."\)
Now it's bad enough that Mathematica thinks the result is
Indeterminate, but what I really can't grasp is how floating-point
values could possibly be involved in this matter: The messy
expression to be simplified was _precise_! Floating-point values
should have nothing to do with its simplification.
Curiously,
David Cantrell
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