vladimir bondarenko

Hello the computer algebra adorers,

Neither Mathematica nor Maple can calculate this integral
straightforwardly.

Is there a person who can show how to get to the exact value
of the integral using some computer algebra system's commands

int(arctan(z)*arctan(2*z)^2/z^3, z= 0..infinity);

?


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

it obviously has a singularity error. And to transmitt the answer is
the question?

A singular composition of all integral is possible given the transform
of all
integrals in computer lanuguage terms.

Find the singular derivative always. And then the exact integral
become a sum of the singluar series.

the exact singular is at z where all functional representation become a
divergent solution. Making the divergent seris as opposed to the
convergent one the test of closure.

And to integral nonclosure functional system is the true question here
I guess.

How to get maple or mathematica to easily close. And to detect
singluar series is the answer. Write a code to iterate the integral z,
steped by step and series detect integral validity. And the apply the
series.

A closure function was to be the final answer here, but to trick the
current program is also allowed.

A renormal as the solution appears the answer in higher mathematics. A
porper usage in maple of renormalization allows all integral now a
days. BUt beware of contrarian mathematicians who do not want proper
renormlaization amongst the masses.

jason pawloski

I can get Maple to calculate a decimal expansion. And it can calculate
a decimal expansion to arbitrary precision, bounded only by
computational and memory limitations.

Considering that I could not do that integral, and I do not own an
integral table, Maple would have served as a great boon had I come
across this in, say, my job.

(The answer I got, by the way, was 3.225783513)

Jason

vladimir bondarenko

Jason Pawloski writes:

JP> Considering that I could not do that integral, and I
JP> do not own an integral table, Maple would have served
JP> as a great boon had I come across this in, say, my job.

I share your feelings with all my heart and soul.

For the math newbies, here comes some interesting initial
data on definite integration...

http://mathworld.wolfram.com/DefiniteIntegral.html

.... and some of its difficulties ...

The Difficulties of Definite Integration by James Davenport
http://www-calfor.lip6.fr/~rr/Calculemus03/davenport.pdf#search=%22theory%20definite%20integration%22

As a CAS customer, I really want to get correct results
automatically, and get them, preferably, very quickly,
say, within 1-2 seconds for all the integration/limit
challenges published. If some masochists want invalid
results coming in hours or crashes, it's their choice -
but we have many reasons to believe that the overwhelming
majority of CAS customers, be they human beings or talking
math-loving, maple-using cuttlefish share our choice of
correctness and speed.

Quite a number of definite integrals can be calculated via
Marichev*Adamchik Mellin transform methods. As a rule, the
answers come in terms of Meijer G functions; then one can
convert those Meijers into hypergeometrics applying the
Slater's theorem, and simplify the output.

Small wonder, this approach works not always; even more, its
current implementation has defects. For example, none of the
current CASs, alas, can get this (relatively simple) integral
directly.

This is why we keep working on this challenging issue of
high practical importance - and of divine beauty.

axel vogt

a quite constructive contribution ...

mate

Do you know to compute it by hand?

Other suggestion(s) for your challanges:

1. take an uggly function f,
2. take some nasty a,b,
3. define g(x) = f(x-a) / (f(x-a) + f(b-x))

and ask for

int( g(x), x=a..b)

[you can optionally apply a strange change of variable first]

Mate

vladimir bondarenko

Mate writes:

M> Other suggestion(s) for your challenges:

Thank you for your suggestion(s).

(Stage direction, aside:

Having read the suggestion(s), VB, gripping his head, in a
moaning voice, pronounces loudly enough, Oh woe is me! -
after which words he roughs his hair tangibly, but not too
hastily, and, then, tragically, holds up his hands to heaven,
in full despair)

By the way, in BPM, vol 1, one could find quite a number
of general fornulas of the type you have proposed.

More tiny improvements to your method?

M> [you can optionally apply a strange change of variable first]

....apply *several* strange change of variable first ?

M> take an uggly function f,

take an ugggggly function f ?! (ugh! pah! ough!)




So, how about Maple step-by-step calculation of

int(arctan(z)*arctan(2*z)^2/z^3, z= 0..infinity);

?

mate

J2:=Int(arctan(z)*arctan(2*z)^2/z^3,z = 0 .. infinity):
ans :=
-Pi*ln(2)-3/16*Pi^3+3/4*Pi*dilog(2/3)+2*Pi*arctanh(1/2)+2*Pi*ln(3):
evalf[50]([J2,ans]);
[3.225783512517288644464486565373898372455115887237 5,
3.225783512517288644464486565373898372455115887237 0]

Mate

vladimir bondarenko

Mate writes:

M> Pi*ln(27/2)-3/16*Pi^3+3/4*Pi*dilog(2/3)

Excellent, Sir!

Could we please enjoy with your skillful processing?

Thanks.

(Also, I take seriously your comments on beauty of the
challenges and hope to get to this important point one
fine day; again, please read the word "challenge" as
"challenge for a CAS/CASs" rather than "challenge for
a human being"; one of our lifetime goals is to help
building a new generation CAS which can beat all these
challenges - and FAR much more - without human help.)