Talk:Risch algorithm: Difference between revisions

Revision as of 08:52, 8 April 2008

Revision as of 18:14, 18 April 2008

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This is simply not true. Both Mathematica and Maple can calculate the antiderivative (even not-so-recent versions). Anyone can try this at http://integrals.wolfram.com/ . (The Root[] objects represent roots of polynomials, i.e. numbers, and can be easily expanded (written in explicit form) using the ToRadicals[] command). I won't edit the article because I am not familiar with the topic (perhaps these programs don't use the Risch-algorithm for this specific problem?), but this needs to be cleaned up or rephrased ... <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/129.177.44.25|129.177.44.25]] ([[User talk:129.177.44.25|talk]]) 08:51, 8 April 2008 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

: Sorry to tell this, but this <b>is</b> correct. Try http://integrals.wolfram.com with input "x/sqrt(x^4+10*x^2-96*x-71)". You will receive large answer. But the problem is not in the Root[] objects (of course, you are right, they are just numbers, so the fuction is elementary even if it use such numbers - no matter whether they can be transferred to radicals or not). The problem that the answer contains "F", and "Π". F is which is elliptic ingegral of the first kind, and Π is elliptic integral of the third kind. Both F and Π cannot be written using elementary functions. And the question is not whether x/sqrt(x^4+10*x^2-96*x-71) has an antiderivative - the question is <i>whether this antiderivative can be written using elementary functions</i>. Now try to change 71 to 72 in the polynomial. Integrals.wolfram.com will give you the answer which will look near the same. This is because integrals.wolfram.com cannot understand, that there is very big difference between

: x/sqrt(x^4+10*x^2-96*x-71) and

: x/sqrt(x^4+10*x^2-96*x-72)

: Antiderivative of the first <i>can</i> be written using elementary functions, and antiderivative of the second cannot. This is because Galois groups of these polynomials are different:

: x^4+10*x^2-96*x-71 Galois group is D(4), e.g. generated by permutations (1 2 3 4) and (1 3), and contains 8 elements (same as in "x^4-2")

: x^4+10*x^2-96*x-72 Galois group is S(4), e.g. generated by permutations (1 2), (1 3), (1 4) and contains 24 elements

: So x^4+10*x^2-96*x-71 is very special case of quadric polynomials, and this is the reason why Risch algorithm in the 71 case gives the answer "yes", and in the 72 case gives the answer "no".

: I have tried Maple v. 11 with input:

: simplify(convert(int(x/sqrt(x^4+10*x^2-96*x-71),x),radical));

: The answer also contains "EllipticF" and "EllipticPi". So Maple also does not understand that antiderivative for x/sqrt(x^4+10*x^2-96*x-71) can be written using elementary fuctions.

: Do you agree with my argumentation?

: Sorry for my bad English.

[[User:Gaz v pol|Gaz v pol]] ([[User talk:Gaz v pol|talk]]) 18:14, 18 April 2008 (UTC)