firstname.lastname@example.org wrote: > > Waldek Hebisch schrieb: > > > > email@example.com wrote: > > > > > > In my view, the simple, trivial, answer as to why such formulae > > > exist is that the "method of undetermined coefficients" works: In > > > other words, one may insert polynomial factors and ramp up their > > > degree until the linear system of equations that results after > > > differentiation becomes soluble. > > > > Do you have a proof that this works? It seem that if you incresase > > degree only on one side, then you end up with multiterm reccurences. > > If you increase degree on both sides you have more equations. > > You are overlooking that there are three sides to the equation: the > given integral, the transformed integral, and the integrated term. Extra > "numerator" polynomials are inserted into each: the coefficients of that > in the given integrand are assumed to be given, those of the other two > assumed to be unknown. For a recurrence formula, the polynomials in the > two integrands should have the same degree, and the highest admissible > degree of that in the integrated term will obviously correlate with > that. When the degrees are ramped up simultaneously, the numbers of > unknowns rises twice as fast as the number of knowns, and a solution > must eventually become feasible. This (fairly obvious) argument could no > doubt be formalized into a mathematical proof, but I doubt the author > would be interested in this.
What is obvoius is that you do not have a proof -- you deal with inhomogeneous system and merely counting number of equations is not enough to proof consistency of such system.
And there is quite trivial reason to be suspect of this argument. Namely, it does not use assumptions about roots.
> > > > > > [...] Also, there is very little practical interest in going beyond > > > elliptic integrals (or beyond Lauricella FD). > > > > Hmm, that is your opinion. Note however, that when integrating > > functions of form R*sqrt(P) where R is a rational function and P is > > of degree 3, the rules apply only if R is of degree 1. This seems to > > be quite serious limitation. Of course you can split R into partial > > fractions, but to get denominator which is power of linear factor may > > require extra algebraic extension (in particular may require going to > > complex numbers). > > Let's say, this summarizes my personal experience in science and > engineering. I agree with the author that the x^1 and x^3 terms in an > elliptic radicand are best annihilated right away to simplify the > problem. If the result involves elliptic integrals of the third kind, > Pi(x,n,k), the rational cofactor has to be split into partial fractions > anyway in order to extract the parameters n, which may therefore come in > complex conjugate pairs, etc.
There are lucky cases where reduction process completely solves the integral. In other cases, you may need algebraic extensions for logarithmic and Pi terms, but algebraic part can be still computed without extra extensions.
> > > > I do not have Gradshteyn-Ryzhik handy, [...] > > One of the pseudo-elliptic examples (here taken from Goursat's Cours > d'analyse) is given on page 50 of the booklet mentioned.
AFAICS this example is wrong -- FriCAS result indicates that there is no elementary integral in this case and I see no reasons why it should have one. OTOH, if one modifies the example to have
f/sqrt((1 - x^2)*(1 - k^4*x^2))
under integral, then FriCAS can handle it for particular f-s.
> > > > Well, there is more conditions beyond confluent roots, so just > > excluding them is not enough -- the user still have to check for > > division by zero. However, the point of my remark was that repeated > > blurb about confluent roots is more distraction than help for the > > reader. Of course, the author is free to ignore my opinion... > > I expect your opinion will be appreciated by the author. As far as I can > see, excluding additional confluences at any level of degeneracy is a > sufficient condition (though not a necessary condition) to prevent zero > division: Some confluences play no role (e.g. 4*d*f = e^2 doesn't affect > 8.1.1, 8.1.2, 8.1.3 in the paper at all) whereas others play a selective > role (e.g. 4*a*c = b^2 affects 8.1.1 and 8.1.3 in the paper but doesn't > affect 8.1.2). While simply excluding confluent roots then seems to be > enough indeed, it also does no harm since those integrands are dealt > with separately anyway.
AFAICS there are three kinds of conditions:
1) Nonzero leading terms (assumed to hold globally) 2) confluent roots 3) exceptional values of exponents, for example 3.3.2 contains factor (n + 1)*(2*n + 1), so -1 and -1/2 must be excluded.
In the intended range, that is increasing exponents toward -1 or decreasing them toward 0 all exponents are good. But outside of that range there are exceptional values.