Talk:Risch algorithm

message: This text is in the article: "...For the function f*e^g, where f and g [what?], we have..."
I don't know what's missing... "are elementary functions"?  Thought I should point it out. 71.191.51.134 19:04, 29 July 2007 (UTC)
thread_title:  typo of some sort? 

message: The last example says that no software is known to be able to solve it, however it appears sympy(which uses the Risch algorithm), can: http://dpaste.com/81951/  —Preceding unsigned comment added by 128.113.195.217 (talk) 16:52, 2 October 2008 (UTC)
replies: text: I profiled the code and it seems that it is indeed the rich algorithm that computes this one (i.e., it is not some special case in the code).  This is interesting, because SymPy cannot even do the other one on this page (maybe because it is not so easy in the elementary case and it doesn't know about the special functions that Maple and Mathematica produce).  I will update the article. Asmeurer (talk  ♬ contribs) 16:09, 9 October 2009 (UTC)
thread_title:  Unsolved Integral 

message: The section says:
:"Liouville formulated the problem solved by the Risch algorithm.   Liouville proved by analytical means that if there is an elementary solution g to the equation g ′ = f then for constants αi and elementary functions ui and v the solution is of the form
:: f = \sum_{i<n} \alpha_i \frac{u_i^{\prime}}{u_i} + v^\prime \,."
#f should be already known, we seek g (see also above).
#What does n (the upper bound for the sum index) stand for?? I assume it just indicates that the index set is finite, but in that case I don't understand the following: "Risch developed a method that allows one to only consider a finite set of elementary functions of Liouville's form."
Can anybody enlighten me on this? --Berntie (talk) 20:19, 4 October 2008 (UTC)
thread_title:  Issues with section [[Risch algorithm#Description|Description]]? 

message: But isn’t |x|=\sqrt{x^2} elementary presentable? --Chricho ∀ (talk) 23:07, 26 June 2012 (UTC)
replies: text: The problem is that there are 2 square roots to each number, and the choice between them cannot be controlled. So \sqrt{x^2} could also mean x, or -x--77.126.235.228 (talk) 14:34, 13 May 2013 (UTC)
replies: text: I agree with Chriko. Nowadays the \sqrt{} symbol is most widely used to mean "the positive square root of". An example of this is the quadratic formula. If the symbol were used to signify both the positive and negative square root we wouldn't need the plus or minus sign before it. Also \sqrt{x} would not be a function if that symbol were used for both the positive and negative square root.  — Preceding unsigned comment added by 98.232.93.206 (talk) 05:22, 20 October 2013 (UTC)
thread_title:  Decidability unknown for elementary functions 

message: It's possible we need a new algorithm. Since this topic about symbolic integration algorithms is connected to differential Galois theory, maybe we need one that uses Galois groups and it's generated permutations. Galois group would be relatively easy to implement, perhaps using set from C++ would be a good idea. We also need to generate a constructor which tests that E is an extension to F. For more information look at this      Gave232 (talk) 22:49, 15 July 2013 (UTC)
thread_title:  A New Algorithm 

message: "the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions."
Unless they are including complex logarithms, you also need to consider inverse tangents, which are the result of irreduceable quadratic factors in the denominator.
For example, the integral of (x^3+x-1)/(x^4+x^2) is ln|x|+1/x+arctanx+C
98.232.93.206 (talk) 05:14, 20 October 2013 (UTC)
replies: text: Complex logarithms are included. They are also allowed in the definition of an elementary function. — Pt (T) 13:20, 20 April 2015 (UTC)
thread_title:  Inaccurate statement 

message: An adequate description of the Risch Algorithm seems to be missing ... from the article about the Risch Algorithm. Bah, minor detail. 88.130.29.105 (talk) 07:27, 18 April 2015 (UTC)
replies: text: Can you use AGI level GPT 4 to create small outline of the algorithm? 2A00:1370:8184:1CE9:15B1:85BD:FE6A:1AB6 (talk) 20:48, 24 March 2023 (UTC)
thread_title:  Missing Algorithm