Ok, so there is some sort of bug.
Here is what Matlab (maple) 2008a gives:
>> int((8*pi^2*f^2*w^2+w^4)/(16*pi^4*f^4+w^4), f, 0, inf)
ans =
PIECEWISE([NaN, And(0 < (w^4)^(1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^
(1/2),(w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2) < 0,(w^4)^
(1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2) < 0,0 < (w^4)^(1/4)*2^
(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2))],[NaN, And(0 < (w^4)^(1/4)*2^
(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2),(w^4)^(1/4)*2^(1/2)*pi+pi*(-2*csgn
(w^2)*w^2)^(1/2) < 0,0 < (w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^
(1/2))],[NaN, And((w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2) <
0,(w^4)^(1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2) < 0,0 < (w^4)^
(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2))],[NaN, And((w^4)^(1/4)*2^
(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2) < 0,0 < (w^4)^(1/4)*2^(1/2)*pi-pi*
(-2*csgn(w^2)*w^2)^(1/2))],[NaN, And(0 < (w^4)^(1/4)*2^(1/2)*pi+pi*
(-2*csgn(w^2)*w^2)^(1/2),(w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^
(1/2) < 0,(w^4)^(1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2) < 0)],
[NaN, And(0 < (w^4)^(1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2),(w^4)^
(1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2) < 0)],[NaN, And((w^4)^
(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2) < 0,(w^4)^(1/4)*2^(1/2)*pi
+pi*(-2*csgn(w^2)*w^2)^(1/2) < 0)],[NaN, (w^4)^(1/4)*2^(1/2)*pi+pi*
(-2*csgn(w^2)*w^2)^(1/2) < 0],[NaN, And(0 < (w^4)^(1/4)*2^(1/2)*pi+pi*
(-2*csgn(w^2)*w^2)^(1/2),(w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^
(1/2) < 0,0 < (w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2))],
[NaN, And(0 < (w^4)^(1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)*w^2)^(1/2),0 <
(w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2))],[NaN, And((w^4)^
(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2) < 0,0 < (w^4)^(1/4)*2^
(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2))],[NaN, 0 < (w^4)^(1/4)*2^(1/2)
*pi-pi*(-2*csgn(w^2)*w^2)^(1/2)],[NaN, And(0 < (w^4)^(1/4)*2^(1/2)*pi
+pi*(-2*csgn(w^2)*w^2)^(1/2),(w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)
*w^2)^(1/2) < 0)],[NaN, 0 < (w^4)^(1/4)*2^(1/2)*pi+pi*(-2*csgn(w^2)
*w^2)^(1/2)],[NaN, (w^4)^(1/4)*2^(1/2)*pi-pi*(-2*csgn(w^2)*w^2)^(1/2)
< 0],[1/8*2^(1/2)*w^2*(2+csgn(w^2))/(w^4)^(1/4), otherwise])
And here is what Mathematica's web integral gives (with 'x' in place
of 'f'):
Integrate[(8*Pi^2*x^2*w^2 + w^4)/(16*Pi^4*x^4 + w^4), x] ==
(w*(-6*ArcTan[1 - (2*Sqrt[2]*Pi*x)/w] + 6*ArcTan[1 + (2*Sqrt[2]*Pi*x)/
w] + Log[-w^2 + 2*Sqrt[2]*Pi*w*x - 4*Pi^2*x^2] - Log[w^2 + 2*Sqrt[2]
*Pi*w*x + 4*Pi^2*x^2]))/ (8*Sqrt[2]*Pi)
They don't let you do definite integrals there. And the computation
timed out on Wolfram Alpha.
Anybody actually *know* what this integral should be?
~Luke
On May 26, 5:25 pm, Robert Kern <robert.k...@gmail.com> wrote:
> On Tue, May 26, 2009 at 19:22, Luke <hazelnu...@gmail.com> wrote:
>
> > I'm using the latest pull from git://git.sympy.org/sympy.git, and this
> > is the response I get:
> > In [1]: from sympy import *
>
> > In [2]: f, w = symbols('fw')
>
> > In [3]: s = 2*pi*I*f
>
> > In [4]: ia = (-2*s**2*w**2 + w**4)/(s**4 + w**4)
>
> > In [5]: simplify(integrate(ia, (f, 0, infty)))
> > ---------------------------------------------------------------------------
> > NameError Traceback (most recent call
> > last)
>
> > /home/luke/lib/python/sympy/<ipython console> in <module>()
>
> > NameError: name 'infty' is not defined
>
> > In [6]: simplify(integrate(ia, (f, 0, oo)))
> > Out[6]: 0
>
> > In [10]: oo.__class__
> > Out[10]: <class 'sympy.core.numbers.Infinity'>
>
> > I'm guessing 'infty' is something you've defined on your own machine
> > for convenience.
>
> No, I'm just dumb. That's from numpy.
>
> In [3]: from sympy import *
>
> In [4]: %sym -r f w
> Adding real variables:
> f
> w
>
> In [5]: s = 2*pi*I*f
>
> In [6]: ia = (-2*s**2*w**2 + w**4)/(s**4 + w**4)
>
> In [7]: simplify(powsimp(integrate(ia, (f, 0, oo))))
> Out[7]: 0
>
> > What exactly does it mean to be using the 'trunk'?
>
> "the latest pull from git://git.sympy.org/sympy.git"
>
> --
> Robert Kern
>
> "I have come to believe that the whole world is an enigma, a harmless
> enigma that is made terrible by our own mad attempt to interpret it as
> though it had an underlying truth."
> -- Umberto Eco
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