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Science Forum Index » Math - Symbolic Forum » An exact 1-D integration challenge - 41
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| Author |
Message |
| Vladimir Bondarenko |
 Posted: Mon Jun 04, 2007 9:51 pm |
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Guest
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Hello computer algebra artists,
Is there a soul who can show in CAS steps how
to get to the exact value of this integral
int((sqrt(4-3*z^2)*sqrt(9-12*z^2)
+ I*sqrt(4-z^2)*sqrt(9-12*z^2)
- I*sqrt(4-z^2)*sqrt(12-9*z^2) )
/sqrt(144-480*z^2+555*z^4-255*z^6+36*z^ ,
z= 0..1);
?
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing |
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| Thomas Richard |
| Posted: Tue Jun 05, 2007 9:49 am |
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Guest
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Vladimir Bondarenko <vb@cybertester.com> wrote:
Quote: Hello computer algebra artists,
Is there a soul who can show in CAS steps how
to get to the exact value of this integral
int((sqrt(4-3*z^2)*sqrt(9-12*z^2)
+ I*sqrt(4-z^2)*sqrt(9-12*z^2)
- I*sqrt(4-z^2)*sqrt(12-9*z^2) )
/sqrt(144-480*z^2+555*z^4-255*z^6+36*z^ ,
z= 0..1);
A:=int((sqrt(4-3*z^2)*sqrt(9-12*z^2)
+ I*sqrt(4-z^2)*sqrt(9-12*z^2)
- I*sqrt(4-z^2)*sqrt(12-9*z^2) )
/sqrt(144-480*z^2+555*z^4-255*z^6+36*z^,
z= 0..1);
F:=factor(A);
J:=convert(F,'Int');
combine(J); # optionally
value(%);
--
Thomas Richard
Maple Support
Scientific Computers GmbH
http://www.scientific.de |
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| Mate |
| Posted: Tue Jun 05, 2007 10:37 am |
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Guest
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On Jun 5, 5:51 am, Vladimir Bondarenko <v...@cybertester.com> wrote:
Quote:
int((sqrt(4-3*z^2)*sqrt(9-12*z^2)
+ I*sqrt(4-z^2)*sqrt(9-12*z^2)
- I*sqrt(4-z^2)*sqrt(12-9*z^2) )
/sqrt(144-480*z^2+555*z^4-255*z^6+36*z^ ,
z= 0..1);
ans := 0;
Hint: Int(f,0..1) = Int(f,0..sqrt(3)/2) + Int(f,sqrt(3)/2..1)
Mate |
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| dimitris |
| Posted: Tue Jun 05, 2007 2:14 pm |
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Hi.
$VersionNumber
5.2
In[14]:=
Integrate[(Sqrt[4 - 3*z^2]*Sqrt[9 - 12*z^2] + I*Sqrt[4 - z^2]*Sqrt[9 -
12*z^2] - I*Sqrt[4 - z^2]*Sqrt[12 - 9*z^2])/
Sqrt[144 - 480*z^2 + 555*z^4 - 255*z^6 + 36*z^8], {z, 0, 1}]
Developer`ZeroQ[%]
Out[14]=
(1/6)*(2*I*Sqrt[3]*EllipticF[Pi/3, 4/3] - 3*I*EllipticF[ArcSin[2/
Sqrt[3]], 3/4] + 3*EllipticK[1/4])
Out[15]=
True
Dimitris
/ Vladimir Bondarenko :
Quote: Hello computer algebra artists,
Is there a soul who can show in CAS steps how
to get to the exact value of this integral
int((sqrt(4-3*z^2)*sqrt(9-12*z^2)
+ I*sqrt(4-z^2)*sqrt(9-12*z^2)
- I*sqrt(4-z^2)*sqrt(12-9*z^2) )
/sqrt(144-480*z^2+555*z^4-255*z^6+36*z^ ,
z= 0..1);
?
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing |
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| Daniel Lichtblau |
| Posted: Tue Jun 05, 2007 3:08 pm |
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Guest
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On Jun 5, 2:14 pm, dimitris <dimmec...@yahoo.com> wrote:
Quote: Hi.
$VersionNumber
5.2
In[14]:=
Integrate[(Sqrt[4 - 3*z^2]*Sqrt[9 - 12*z^2] + I*Sqrt[4 - z^2]*Sqrt[9 -
12*z^2] - I*Sqrt[4 - z^2]*Sqrt[12 - 9*z^2])/
Sqrt[144 - 480*z^2 + 555*z^4 - 255*z^6 + 36*z^8], {z, 0, 1}]
Developer`ZeroQ[%]
Out[14]=
(1/6)*(2*I*Sqrt[3]*EllipticF[Pi/3, 4/3] - 3*I*EllipticF[ArcSin[2/
Sqrt[3]], 3/4] + 3*EllipticK[1/4])
Out[15]=
True
Dimitris
The zero testing uses numeric approximation. So this gives a
plausibility argument but not a rigorous proof.
Daniel Lichtblau
Wolfram Research
Quote: / Vladimir Bondarenko :
Hello computer algebra artists,
Is there a soul who can show in CAS steps how
to get to the exact value of this integral
int((sqrt(4-3*z^2)*sqrt(9-12*z^2)
+ I*sqrt(4-z^2)*sqrt(9-12*z^2)
- I*sqrt(4-z^2)*sqrt(12-9*z^2) )
/sqrt(144-480*z^2+555*z^4-255*z^6+36*z^ ,
z= 0..1);
?
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing |
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| dimitris |
| Posted: Tue Jun 05, 2007 3:35 pm |
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Guest
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Hi Daniel.
I was prety sure about that.
In fact in another similar response I point
out also myself it!
But having spent at least two hours "playing" with these,
and failed to simplify to zero (A really hard task!)
I tried Developer`ZeroQ[..] due to my desperate
situation!
Anyway...I give up!
Dimitris
/ Daniel Lichtblau :
Quote: On Jun 5, 2:14 pm, dimitris <dimmec...@yahoo.com> wrote:
Hi.
$VersionNumber
5.2
In[14]:=
Integrate[(Sqrt[4 - 3*z^2]*Sqrt[9 - 12*z^2] + I*Sqrt[4 - z^2]*Sqrt[9 -
12*z^2] - I*Sqrt[4 - z^2]*Sqrt[12 - 9*z^2])/
Sqrt[144 - 480*z^2 + 555*z^4 - 255*z^6 + 36*z^8], {z, 0, 1}]
Developer`ZeroQ[%]
Out[14]=
(1/6)*(2*I*Sqrt[3]*EllipticF[Pi/3, 4/3] - 3*I*EllipticF[ArcSin[2/
Sqrt[3]], 3/4] + 3*EllipticK[1/4])
Out[15]=
True
Dimitris
The zero testing uses numeric approximation. So this gives a
plausibility argument but not a rigorous proof.
Daniel Lichtblau
Wolfram Research
/ Vladimir Bondarenko :
Hello computer algebra artists,
Is there a soul who can show in CAS steps how
to get to the exact value of this integral
int((sqrt(4-3*z^2)*sqrt(9-12*z^2)
+ I*sqrt(4-z^2)*sqrt(9-12*z^2)
- I*sqrt(4-z^2)*sqrt(12-9*z^2) )
/sqrt(144-480*z^2+555*z^4-255*z^6+36*z^ ,
z= 0..1);
?
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing |
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| Axel Vogt |
| Posted: Tue Jun 05, 2007 3:55 pm |
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Guest
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Daniel Lichtblau wrote:
Quote: On Jun 5, 2:14 pm, dimitris <dimmec...@yahoo.com> wrote:
Hi.
$VersionNumber
5.2
In[14]:=
Integrate[(Sqrt[4 - 3*z^2]*Sqrt[9 - 12*z^2] + I*Sqrt[4 - z^2]*Sqrt[9 -
12*z^2] - I*Sqrt[4 - z^2]*Sqrt[12 - 9*z^2])/
Sqrt[144 - 480*z^2 + 555*z^4 - 255*z^6 + 36*z^8], {z, 0, 1}]
Developer`ZeroQ[%]
Out[14]=
(1/6)*(2*I*Sqrt[3]*EllipticF[Pi/3, 4/3] - 3*I*EllipticF[ArcSin[2/
Sqrt[3]], 3/4] + 3*EllipticK[1/4])
Out[15]=
True
Dimitris
The zero testing uses numeric approximation. So this gives a
plausibility argument but not a rigorous proof.
Daniel Lichtblau
Wolfram Research
Do not know MMA, but can you 'imitate' Thomas Richard's solution?
Quote:
/ Vladimir Bondarenko :
Hello computer algebra artists,
Is there a soul who can show in CAS steps how
to get to the exact value of this integral
int((sqrt(4-3*z^2)*sqrt(9-12*z^2)
+ I*sqrt(4-z^2)*sqrt(9-12*z^2)
- I*sqrt(4-z^2)*sqrt(12-9*z^2) )
/sqrt(144-480*z^2+555*z^4-255*z^6+36*z^ ,
z= 0..1);
?
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing
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