integrate

[Kjetil brinchmann Halvorsen]

Here is one other integral sympy cannot do, but which shouls not be to
difficult, it should be possible to do it symbolically,
the last integral in the following output:


In [1]: t1,t2,t3 = symbols('t1 t2 t3', real=True)

In [2]: beta=symbols('beta', real=True)

In [3]: integrate( (t1*t2*t3)**beta * (t1-t2)*(t1-t3)*(t2-t3),
(t1,0,1),(t2,0,1),(t3,0,1))
Out[3]: 0

In [4]: integrate( (t1*t2*t3)**beta * abs((t1-t2)*(t1-t3)*(t2-t3)),
(t1,0,1),(t2,0,1),(t3,0,1))
Out[4]:
1 1 1
⌠ ⌠ ⌠
⎮ ⎮ ⎮ β
⎮ ⎮ ⎮ (t₁⋅t₂⋅t₃) ⋅│(t₁ - t₂)⋅(t₁ - t₃)⋅(t₂ - t₃)│ d(t₁) d(t₂) d(t₃)
⌡ ⌡ ⌡
0 0 0


Kjetil

--
"If you want a picture of the future - imagine a boot stamping on the
human face - forever."

George Orwell (1984)

[Aaron Meurer]

Yes, unfortunately SymPy currently does not do so well with integrals
of absolute values.

Aaron Meurer

On Jun 23, 2012, at 6:47 PM, Kjetil brinchmann Halvorsen

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[Vinzent Steinberg]

Am Sonntag, 24. Juni 2012 07:10:05 UTC+2 schrieb Aaron Meurer:

Yes, unfortunately SymPy currently does not do so well with integrals
of absolute values.

Actually you can work around this limitation by using piecewise functions:

In [12]: myabs = lambda x: Piecewise((x, x>=0), (-x, x<0))

In [13]: integrate( (t1*t2*t3)**beta * myabs((t1-t2)*(t1-t3)*(t2-t3)), 

(t1,0,1),(t2,0,1),(t3,0,1)).doit()

Out[13]: 0

Maybe we should implement something like abs(x).rewrite(Piecewise).

Vinzent

[Vinzent Steinberg]

Am Montag, 25. Juni 2012 21:11:08 UTC+2 schrieb Vinzent Steinberg:

Am Sonntag, 24. Juni 2012 07:10:05 UTC+2 schrieb Aaron Meurer:

Yes, unfortunately SymPy currently does not do so well with integrals
of absolute values.

Actually you can work around this limitation by using piecewise functions:

In [12]: myabs = lambda x: Piecewise((x, x>=0), (-x, x<0))

In [13]: integrate( (t1*t2*t3)**beta * myabs((t1-t2)*(t1-t3)*(t2-t3)), 

(t1,0,1),(t2,0,1),(t3,0,1)).doit()

Out[13]: 0

(The .doit() is not necessary, sorry.)

Vinzent 

[Kjetil brinchmann Halvorsen]

see inline!

On Mon, Jun 25, 2012 at 3:11 PM, Vinzent Steinberg
<vinzent....@googlemail.com> wrote:
> Am Sonntag, 24. Juni 2012 07:10:05 UTC+2 schrieb Aaron Meurer:
>>
>> Yes, unfortunately SymPy currently does not do so well with integrals
>> of absolute values.
>
>
> Actually you can work around this limitation by using piecewise functions:
>
> In [12]: myabs = lambda x: Piecewise((x, x>=0), (-x, x<0))
>
> In [13]: integrate( (t1*t2*t3)**beta * myabs((t1-t2)*(t1-t3)*(t2-t3)),
> (t1,0,1),(t2,0,1),(t3,0,1)).doit()
> Out[13]: 0

But that answer 0 is wrong!

Kjetil

>
> Maybe we should implement something like abs(x).rewrite(Piecewise).
>
> Vinzent
>

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[Vinzent Steinberg]

Am Montag, 25. Juni 2012 21:14:38 UTC+2 schrieb kjetil1001:

> Actually you can work around this limitation by using piecewise functions:
>
> In [12]: myabs = lambda x: Piecewise((x, x>=0), (-x, x<0))
>
> In [13]: integrate( (t1*t2*t3)**beta * myabs((t1-t2)*(t1-t3)*(t2-t3)),
> (t1,0,1),(t2,0,1),(t3,0,1)).doit()
> Out[13]: 0

But that answer 0 is wrong!

You are right, it should not be 0, this is a bug. Do you know the correct value?

Vinzent

[Kjetil brinchmann Halvorsen]

see inline.

Yes. This is a special case of the so-called Selberg integral (named
for Atle Selberg) , see
http://en.wikipedia.org/wiki/Selberg_integral

The special case above has the value: (G is the gamma function)

S_3(beta+1,1,gamma=1/2) = \prod_{j=1}^3 \frac{ G(beta+1+(j-1)gamma)
G(1+(j-1)gamma) G(1+j gamma) }

{ G(beta+2+(3+j-2)gamma) G(1+gamma) }


Kjetil

> Vinzent
>
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[Vinzent Steinberg]

Am Freitag, 29. Juni 2012 16:11:01 UTC+2 schrieb kjetil1001:

> You are right, it should not be 0, this is a bug. Do you know the correct
> value?
>

Yes. This is a special case of the so-called Selberg integral (named
for Atle Selberg) ,  see
http://en.wikipedia.org/wiki/Selberg_integral

The special case above has the value: (G is the gamma function)

S_3(beta+1,1,gamma=1/2) = \prod_{j=1}^3  \frac{ G(beta+1+(j-1)gamma)
G(1+(j-1)gamma) G(1+j gamma) }

{ G(beta+2+(3+j-2)gamma) G(1+gamma) }

Thanks, I made this issue 3317 [1].

Vinzent

[1] http://code.google.com/p/sympy/issues/detail?id=3317&thanks=3317&ts=1341145588