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An exact simplification challenge - 8

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Vladimir Bondarenko

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Jul 8, 2006, 2:06:33 PM7/8/06
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Hello the computer algebra fans,

Is there a person who can show us how to simplify
using some CAS operations, the following expression

a := 1+I:
b := conjugate(a):
c := abs(a):

a*c*EllipticPi(b/2, 1/c) -
a*c*EllipticPi(c, b/2, 1/c) +
b*c*EllipticPi(c, a/2, 1/c) -
b*c*EllipticPi(a/2, 1/c);


?

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

Axel Vogt

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Jul 8, 2006, 4:32:27 PM7/8/06
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Vladimir Bondarenko wrote:
>
> Hello the computer algebra fans,
>
> Is there a person who can show us how to simplify
> using some CAS operations, the following expression
>
> a := 1+I:
> b := conjugate(a):
> c := abs(a):
>
> a*c*EllipticPi(b/2, 1/c) -
> a*c*EllipticPi(c, b/2, 1/c) +
> b*c*EllipticPi(c, a/2, 1/c) -
> b*c*EllipticPi(a/2, 1/c);
>

It is a funny way to write Pi.

Rephrase the expression in terms of integrals, use the
range of the integrand to 'reduce' to the real case and
after changing variables (i did it not the elegant way)
arrive at Int(4*x^(1/2)/(1+x^2)/(1-x^2)^(1/2),x=0..1).
And now use:
changevar(x=(xi)^2,%,xi): subs(xi=x,%): #simplify(%);
changevar(1+x^4=(xi)^1,%,xi): subs(xi=x,%); #simplify(%);
value(%);

--
"please use mail at ... instead of test3 at ... to send me a mail"

C W

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Jul 13, 2006, 9:31:20 AM7/13/06
to

Axel Vogt wrote:
>
> Vladimir Bondarenko wrote:
> >
> > Hello the computer algebra fans,
> >
> > Is there a person who can show us how to simplify
> > using some CAS operations, the following expression
> >
> > a := 1+I:
> > b := conjugate(a):
> > c := abs(a):
> >
> > a*c*EllipticPi(b/2, 1/c) -
> > a*c*EllipticPi(c, b/2, 1/c) +
> > b*c*EllipticPi(c, a/2, 1/c) -
> > b*c*EllipticPi(a/2, 1/c);
> >
>
> It is a funny way to write Pi.

Maple (version V R4) computes the value numerically as

-3.15538673832555336190522019368

Chris

Peter Luschny

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Jul 13, 2006, 10:30:47 AM7/13/06
to
C W wrote:
> Axel Vogt wrote:
>> Vladimir Bondarenko wrote:

>>> Is there a person who can show us how to simplify
>>> using some CAS operations, the following expression
>>> a := 1+I:
>>> b := conjugate(a):
>>> c := abs(a):
>>> a*c*EllipticPi(b/2, 1/c) -
>>> a*c*EllipticPi(c, b/2, 1/c) +
>>> b*c*EllipticPi(c, a/2, 1/c) -
>>> b*c*EllipticPi(a/2, 1/c);

>> It is a funny way to write Pi.

> Maple (version V R4) computes the value numerically as
> -3.15538673832555336190522019368

So they will tell you to upgrade :)
But wait a moment! Let's see:

Maple V Release 5, IBM INTEL NT, Jun 16 1998
-3.1553867383255533619052201936751

Hmm, this will not help ;-)

However you can simplify the expression --
via EllipticK for example.

Peter

C W

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Jul 13, 2006, 12:40:20 PM7/13/06
to
Transforming into integral, changing variables and valuating gives :

8*2^(1/2)*EllipticPi(-3+2*2^(1/2),3-2*2^(1/2))/(2^(1/2)+1)+2*(2*2^(1/2)-4)*EllipticK(3-2*2^(1/2));

,which appears to be Pi.


Chris

Axel Vogt

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Jul 13, 2006, 1:43:10 PM7/13/06
to


Here is my hack (I should have changed the int bounds at the end):

restart; interface(version);

Classic Worksheet Interface, Maple 10.04, Windows, May 30 2006 Build ID 233114

a := 1+I:
b := conjugate(a):
c := abs(a):

a*c*EllipticPi(b/2, 1/c) -
a*c*EllipticPi(c, b/2, 1/c) +
b*c*EllipticPi(c, a/2, 1/c) -

b*c*EllipticPi(a/2, 1/c):

S8:=%:
evalf(S8,40);

3.141592653589793238462643383279502884195-.121857397313417128471e-18*I

'convert(S8,Int)';
combine(%): subs(_alpha1=x, %): normal(%);

J:=integrand(%);

J := 32*I*(-1+x^2)/(4-2*x^2)^(1/2)/(2-x^2+x^2*I)/(2-2*x^2)^(1/2)/(-2+x^2+x^2*I)

assume( 1 < x, x < sqrt(2)); getassumptions(x);

evalc(J): simplify(%):
Int(%,x=1..sqrt(2));

Int(-16*(-1+x^2)/(2-2*x^2+x^4)/(-2+2*x^2)^(1/2)/(4-2*x^2)^(1/2),x = 1 .. 2^(1/2))

changevar(1-x^2=eta,%,eta);

Int(8*eta/(2*eta+(1-eta)^2)/(-2*eta)^(1/2)/(2+2*eta)^(1/2)/(1-eta)^(1/2),eta = -1 .. 0)

changevar(1-eta=xi+1,%,xi);

Int(-4*xi^(1/2)/(xi^2+1)*2^(1/2)/(2-2*xi)^(1/2)/(xi+1)^(1/2),xi = 0 .. 1)

evalf(%,40);

-3.141592653589793238462643383279502884197

Robert Israel

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Jul 13, 2006, 3:07:11 PM7/13/06
to
In article <44B68323...@ns.sympatico.ca>,

C W <sylve...@ns.sympatico.ca> wrote:
>
>
>Axel Vogt wrote:
>>
>> Vladimir Bondarenko wrote:
>> >
>> > Hello the computer algebra fans,
>> >
>> > Is there a person who can show us how to simplify
>> > using some CAS operations, the following expression
>> >
>> > a := 1+I:
>> > b := conjugate(a):
>> > c := abs(a):
>> >
>> > a*c*EllipticPi(b/2, 1/c) -
>> > a*c*EllipticPi(c, b/2, 1/c) +
>> > b*c*EllipticPi(c, a/2, 1/c) -
>> > b*c*EllipticPi(a/2, 1/c);
>> >
>>
>> It is a funny way to write Pi.
>
>Maple (version V R4) computes the value numerically as
>
>-3.15538673832555336190522019368

Maple 10.04 (using Digits:= 50) computes it as
3.1415926535897932384626419083070865997433942587131+.3e-48*I
which agrees pretty well with Pi (an error of about 1.5*10^(-24)).

Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Peter Luschny

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Jul 13, 2006, 4:00:59 PM7/13/06
to
C W wrote:

> Transforming into integral, changing variables and valuating gives :
> 8*2^(1/2)*EllipticPi(-3+2*2^(1/2),3-2*2^(1/2))/(2^(1/2)+1)+2*(2*2^(1/2)-4)*EllipticK(3-2*2^(1/2));
> ,which appears to be Pi.

Yes.
x := 2*sqrt(2)-3;
2*(x-1)*(EllipticK(-x)-2*EllipticPi(x,-x));

Peter

C W

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Jul 13, 2006, 6:21:57 PM7/13/06
to

Robert Israel wrote:
>
> In article <44B68323...@ns.sympatico.ca>,
> C W <sylve...@ns.sympatico.ca> wrote:
> >
> >
> >Axel Vogt wrote:
> >>
> >> Vladimir Bondarenko wrote:
> >> >
> >> > Hello the computer algebra fans,
> >> >
> >> > Is there a person who can show us how to simplify
> >> > using some CAS operations, the following expression
> >> >
> >> > a := 1+I:
> >> > b := conjugate(a):
> >> > c := abs(a):
> >> >
> >> > a*c*EllipticPi(b/2, 1/c) -
> >> > a*c*EllipticPi(c, b/2, 1/c) +
> >> > b*c*EllipticPi(c, a/2, 1/c) -
> >> > b*c*EllipticPi(a/2, 1/c);
> >> >
> >>
> >> It is a funny way to write Pi.
> >
> >Maple (version V R4) computes the value numerically as
> >
> >-3.15538673832555336190522019368
>
> Maple 10.04 (using Digits:= 50) computes it as
> 3.1415926535897932384626419083070865997433942587131+.3e-48*I
> which agrees pretty well with Pi (an error of about 1.5*10^(-24)).

So what is
-3.15538673832555336190522019368

Chris

isr...@math.ubc.ca

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Jul 13, 2006, 8:04:39 PM7/13/06
to

C W wrote:
> Robert Israel wrote:
> >
> > In article <44B68323...@ns.sympatico.ca>,
> > C W <sylve...@ns.sympatico.ca> wrote:
> > >
> > >
> > >Axel Vogt wrote:
> > >>
> > >> Vladimir Bondarenko wrote:
> > >> >
> > >> > Hello the computer algebra fans,
> > >> >
> > >> > Is there a person who can show us how to simplify
> > >> > using some CAS operations, the following expression
> > >> >
> > >> > a := 1+I:
> > >> > b := conjugate(a):
> > >> > c := abs(a):
> > >> >
> > >> > a*c*EllipticPi(b/2, 1/c) -
> > >> > a*c*EllipticPi(c, b/2, 1/c) +
> > >> > b*c*EllipticPi(c, a/2, 1/c) -
> > >> > b*c*EllipticPi(a/2, 1/c);
> > >> >
> > >>
> > >> It is a funny way to write Pi.
> > >
> > >Maple (version V R4) computes the value numerically as
> > >
> > >-3.15538673832555336190522019368
> >
> > Maple 10.04 (using Digits:= 50) computes it as
> > 3.1415926535897932384626419083070865997433942587131+.3e-48*I
> > which agrees pretty well with Pi (an error of about 1.5*10^(-24)).
>
> So what is
> -3.15538673832555336190522019368

Evidently, a former bug in EllipticPi in earlier versions of Maple.

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