An exact simplification challenge - 71 (EllipticF, EllipticPi)

[Vladimir Bondarenko]

Hello,

(1 + Sqrt[5]) (EllipticF[I ArcSinh[(Sqrt[5] - 1)/2],
(7 + 3 Sqrt[5])/2] - 2 EllipticPi[(3 + Sqrt[5])/2,
I ArcSinh[(Sqrt[5] - 1)/2], (7 + 3 Sqrt[5])/2]) I

?

Best wishes,

Vladimir Bondarenko

Co-founder, CEO, Mathematical Director

http://www.cybertester.com/ Cyber Tester, LLC

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"We must understand that technologies
like these are the way of the future."

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[Alec Mihailovs]

"Vladimir Bondarenko" <v...@cybertester.com> wrote

> (1 + Sqrt[5]) (EllipticF[I ArcSinh[(Sqrt[5] - 1)/2],
> (7 + 3 Sqrt[5])/2] - 2 EllipticPi[(3 + Sqrt[5])/2,
> I ArcSinh[(Sqrt[5] - 1)/2], (7 + 3 Sqrt[5])/2]) I
>
> ?

Obviously, arccosh(3/2) = ln((3+sqrt(5))/2)

Alec

[Alec Mihailovs]

"Alec Mihailovs" <al...@mihailovs.com> wrote in message
news:SIKuk.28421$9u1....@newsfe09.iad...

Now, how that could be done in Maple,

convert("(1 + Sqrt[5]) (EllipticF[I ArcSinh[(Sqrt[5] - 1)/2],

(7 + 3 Sqrt[5])/2] - 2 EllipticPi[(3 + Sqrt[5])/2,

I ArcSinh[(Sqrt[5] - 1)/2], (7 + 3 Sqrt[5])/2]) I",FromMma);
simplify(%);
convert(%,EllipticF);
convert(%,Int);
eval(%,1/2*I*(5^(1/2)-1)=(1/2*5^(1/2)-1/2)*I );
IntegrationTools:-Combine(%);
value(%);
simplify(%);
combine(%);
applyop(expand@rationalize,1,%);

Alec

[Vladimir Bondarenko]

On Sep 1, 8:04 am, "Alec Mihailovs" <a...@mihailovs.com> writes:

AM> Obviously, arccosh(3/2) = ln((3+sqrt(5))/2)

http://www.math.utah.edu/~cherk/mathjokes.html

In his lecture, ** formulated a theorem and said: "The proof is
obvious".

Then he thought for a minute, left the lecture room, returned
after 15 minutes and happily concluded: "Indeed, it is obvious!"

:)

[Alec Mihailovs]

It was the integrable case that I knew. I just looked at the Mathematica
functions site and found it there, too,

http://functions.wolfram.com/08.06.17.0004.01

It seems to be a slight problem with pattern matching preventing Mathematica
from using it, which may be fixed in the next release.

Alec

[Axel Vogt]

BTW and a bit off topic:

the former nice sites now only display sh.. in the lower parts
instead assuming visitors use standard browsers ... who wants
to see "Cell[BoxData[RowBox[List[RowBox[List["EllipticPi" ..." ?

[Vladimir Bondarenko]

On Sep 1, 4:26 pm, Axel Vogt <&nore...@axelvogt.de> writes:

AF> the former nice sites now only display sh.. in the
AF> lower parts instead assuming visitors use standard
AF> browsers ... who wants to see "Cell[BoxData[RowBox[
AF> List[RowBox[List["EllipticPi" ..." ?

Dear Axel,

You surpassed me in QA :)

I had no idea to have a look at functions.wolfram.com
to post this challenge. But you dared and won :))

Best wishes,

Vladimir

[Vladimir Bondarenko]

On Sep 1, 3:29 pm, "Alec Mihailovs" <a...@mihailovs.com> writes:

AM> It was the integrable case that I knew.

Dear Alec,

Soon after I met you several years ago, I always had an idea
that your knowledge is equal only to your heroic spirit.

Never I suspected that there are such cases as you quoted

http://functions.wolfram.com/08.06.17.0004.01

For me, to discover this challenge was a kind of shock.

Best wishes from Simferopol,

Vladimir

[Alec Mihailovs]

"Axel Vogt" <&nor...@axelvogt.de> wrote

>
> BTW and a bit off topic:
>
> the former nice sites now only display sh.. in the lower parts
> instead assuming visitors use standard browsers ... who wants
> to see "Cell[BoxData[RowBox[List[RowBox[List["EllipticPi" ..." ?

That can be copied and pasted into Mathematica. However, the Input Form also
can be copied and pasted there, so it is not exactly clear why they did
that. Also, the Rule Form is missing the Input Form equivalent.

Alec

[Alec Mihailovs]

"Vladimir Bondarenko" <v...@cybertester.com> wrote

Vladimir,

> For me, to discover this challenge was a kind of shock.
> Best wishes from Simferopol,

While standard elliptic integrals are not integrable in elementary
functions, some of their linear combinations are. Some of such cases are
(well-)known, but I don't know a good reference for that. In most cases the
answers are inverse trigonometric (or hyperbolic) functions. It could be an
interesting research problem - to describe all of such linear combinations.

I'm always glad to see your posts. If I owned a company producing a CAS, I
would definitely offer you a job there. Currently, my company, Mihailovs,
Inc., is offering consulting in mathematics and operations research, and
with the current economics situation in the US, we are doing pretty well,
but are not hiring at the moment (unless we get few more clients, but there
seem to be not that many start-ups around.)

Best wishes from Jackson, TN,
Alec