From 285c14a37dc1f04617f859cd1e251edc08a03a2b Mon Sep 17 00:00:00 2001 From: Brent Baccala Date: Fri, 27 Apr 2018 12:28:09 -0400 Subject: [PATCH] move some more loose files into git --- .gitignore | 27 +++ CAS | 153 ++++++++++++++ cyclotomic.gp | 34 +++ references/elemint.pdf | Bin 0 -> 315496 bytes references/maxima-integral | 157 ++++++++++++++ references/nonelem_integr2 | 418 +++++++++++++++++++++++++++++++++++++ 6 files changed, 789 insertions(+) create mode 100644 CAS create mode 100755 cyclotomic.gp create mode 100644 references/elemint.pdf create mode 100644 references/maxima-integral create mode 100644 references/nonelem_integr2 diff --git a/.gitignore b/.gitignore index e71a1b1..d6ff536 100644 --- a/.gitignore +++ b/.gitignore @@ -1,7 +1,34 @@ GRAPH*.pdf 04-RATIONALS-EXAMPLE*.pdf +ModernIntegration.pdf + +# a copy of Chapter 1, page 1 +wwtbam.pdf + +# directory with pythontex's autogenerated files +pythontex-files-ModernIntegration + +# Auto-generated from 'maximacommon' blocks +common.mac + +# Symlinks to auto-generated files +maxima.mac +chebyshev.mac +sage.sage + +# LaTeX's comment package +comment +comment.sty + +# symlink to /home/baccala/texmf/fvextra/fvextra.sty +fvextra.sty + +# symlink to /home/baccala/src/maxima-code/share/ +share + # TeX litter +ModernIntegration.pytxcode ModernIntegration.aux ModernIntegration.log ModernIntegration.out diff --git a/CAS b/CAS new file mode 100644 index 0000000..8e6d2a1 --- /dev/null +++ b/CAS @@ -0,0 +1,153 @@ +Comparison of open source computer algebra systems + +Axiom + doesn't use readline + builtin hyperdoc is way behind HTML + couldn't figure how to parse: + integrand := x*((x^2*exp(2*x^2)-log(x+1)^2)^2) / ((x+1)*(log(x+1)^2 - x^2*exp(2*x^2))^2) + + +Maxima + uses LISP-style function definition syntax + + MAXIMA function tex_displ(MLABEL %O14 EXPR) + - fetch tex-environment - either "" (strings) OR tex-environment property on car OR *tex-environment-default* + - print tex1(LABEL %O14) (minus 14 chars because of trailing \mathbb{false}) + - then print env around tex1(EXPR) + + MAXIMA function print(EXPR) calls displa for output + + +Sage + struggling to do polynomial long division + calls Maxima or Singular to do this + implements algebraic closure of Q (QQbar) + + + fix latex printing on power series + - subs should raise NotImplementedError if element not callable + - subs should work recursively on complicated rings + - option to specify how QQbar elements print + - can't type convert between two slightly different NumberRings: + Number Field in a with defining polynomial a^7 - 18 + Number Field in a with defining polynomial y^7 - 18 + - add map_coefficients method to Laurent series (Taylor, too?) + + +Macaulay 2 + doesn't implemente fraction fields over complicated (non-Z, non-Q) constant fields + +Singular + can't compute Riemann-Roch spaces in characteristic p + can compute Hamber-Noether expansions (can this be converted to a Puiseux expansion?) + +CoCoA + C++ library + + +Macaulay 2 - Riemann-Roch space calculator (Noether-Brill algorithm ??) + +completeLinearSystemOnNodalPlaneCurve=method() +completeLinearSystemOnNodalPlaneCurve(Ideal,List):=(J,D)->( + singJ:=saturate(ideal jacobian J+J); + -- adjoint ideal + H:=ideal (mingens ideal(gens intersect(singJ,D_0)%J))_(0,0); + -- a curve passing through singJ and D_0 + E0:=((J+H):D_0):(singJ^2); -- residual divisor + if not(degree J *degree H - degree D_0 -2*degree singJ==degree E0) + then error"residual divisor of has wrong degree"; + L1:=mingens ideal (gens truncate(degree H, intersect(E0,D_1,singJ)))%J; + h0D:=(tally degrees source L1)_{degree H}; -- h^0 O(D) + L:=L1_{0..h0D-1}; -- matrix of homogeneous forms, L/H =L(D) subset K(C) + (L,(gens H)_(0,0))) + +http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.11/share/doc/Macaulay2/Macaulay2Doc/html/___Tutorial_co_sp__Divisors.html + + + + + +a=QQ['a'].0 +aRing = NumberField(a^2 + 1, 'a') + +b=QQ['b'].0 +bRing = NumberField(b^2 + 1, 'a') + +aRing is bRing + +aa=aRing.0 +bb=bRing.0 + +bRing(aa) + +def ringmap(x): + # First map x from the polynomial ring down to its coefficent ring (a NumberField) + x = x.parent()(x) + # Then convert it to a polynomial and map to into numfield + return x.polynomial()(numfield.gens()[0]) + + +HOW TO CREATE A MAP BETWEEN TWO POLYNOMIAL RINGS INDUCED BY A MAP ON THEIR COEFFICIENTS + +R. = QQ[] +aRing = NumberField(a^2 + 1, 'a') +R. = QQ[] +bRing = NumberField(b^2 + 1, 'a') + +aRing2 = aRing['x'] +bRing2 = bRing['x'] + +h = aRing.hom(bRing.gens()) + +h2 = aRing2.hom(h, bRing2) + +DOESN'T WORK FOR LAURENT SERIES + + + +BUG REPORT TO SHOW HOW SINGULAR RETURNS RESULTS IN WRONG FIELD + +R. = QQ[] +S = NumberField(b^7 - 18, 'a') + +R. = S[] +f = x + y + +singular.lib('hnoether.lib') +hne = f._singular_().hnexpansion().sage() + +hne[0][0].base_ring() is f.parent() + + + +SAGE - SINGULAR IDEALS + +rings/polynomial/multi_polynomial_ideal.py + lines 558-592 + create Singular ideal + lift generators into parent ring before str-ifying them for Singular + +interfaces/singular.py + lines 1957-1959 + converts Singular ideal to sage ideal + + 1958 R = R or self.sage_global_ring() + 1959 return R.ideal([p.sage_poly(R) for p in self]) + + if R is a quotient ring, + p.sage_poly(R) can't be lifted into the base ring + + sage_poly: + + 1779 p = MPolynomial_polydict(R,PolyDict(sage_repr,force_int_exponents=False,force_etuples=False)) + 1780 if isinstance(R, MPolynomialRing_polydict): + 1781 return p + 1782 else: + 1783 return QuotientRingElement(R,p,reduce=False) + + what we get back is a QuotientRingElement containing an MPolynomial_polydict + + self.sage_global_ring() will return a quotient ring different from the one we started with + its generators will be named 'x', 'y', 'z', instead of 'xbar', 'ybar', 'zbar' + + subs doesn't work because _call_ isn't defined on QuotientRingElement's + (subs in RingElement returns self if _call_ isn't defined) diff --git a/cyclotomic.gp b/cyclotomic.gp new file mode 100755 index 0000000..7894ac3 --- /dev/null +++ b/cyclotomic.gp @@ -0,0 +1,34 @@ +#!/bin/sh +# +# gnuplot script to plot roots of an equation + +equation=$1 +if [ "x$equation" = "x" ]; then equation="z**6-1"; fi + +/usr/bin/gnuplot -geometry 500x500 -persist < cyclo.out <b ? b : a + +set isosamples 500,500 + +set view 0,0 + +set key off +set size 1.4,1.4 +set origin -0.2,-0.2 + +strength = 1.0 +poly(z) = (strength * ($equation)) + +splot [-2:2] [-2:2] [5:100] min(abs(1/poly(x+{0,1}*y)),10) + +bind Up "strength = strength * 2; replot" +bind Down "strength = strength / 2; replot" + +EOF diff --git a/references/elemint.pdf b/references/elemint.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3a6c98ce52d72bb7eac305379d40ba07c5f2d08a GIT binary patch literal 315496 zcma&sQ+FnSwx!|Nwr$(CZQHhuic_&|+qUz?NyWD9-rb|`&b~P7YW;=x8IxQ|OoE=7 zfgOe%uspgB!%D1IX5%Fe`0^#2ZqQNr5J&D@2EQNqsH&0Ngf 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zq=$dGCX%r4VRqz1;^&alIT`(1?9hjUMb+5k6Qne@>N1J>olw|0YqZ%agJfr{dx=^~ zAf631+cZ%CerrlPZGZlvRo^N<&~kx(-0ME@azSqTm2=6`Cje|1>a-(tLcS)g;u2O( z;&h19CGnw?y-23zdtXH0cjq1)ZN>G_gsBSR9@^xM2h@qR)H3{PTiJOa zE}s%#5RdYAUylO^hl;O*fAepGp)co(^KfmLN{GGC-71KE(|A?+Jzu$syg_Y--(ln0 u%!u~?+6H)Dt^Wa2b_wVH_ZQCD#n8#c!|7*n{cyc-va!IBk%`KS!Tf)u&=W=g literal 0 HcmV?d00001 diff --git a/references/maxima-integral b/references/maxima-integral new file mode 100644 index 0000000..2643555 --- /dev/null +++ b/references/maxima-integral @@ -0,0 +1,157 @@ +A bunch of responses: + +1. Thanks for providing a brief way of producing this error message. +2. If you really want an answer, you could try + +integrate_use_rootof:true; + +integrate(1/(x^(5/2)+3*x^(1/3)),x); + +though I'm not sure the answer returned is correct or +even sensible. + +It returns +6*lsum(log(x^(1/6)-%r1)/(13*%r1^9),%r1,rootsof(x^(13/6)+3)) + +Mathematica produces an answer, but one which is quite large. +Wolfram Alpha produces an answer that is different but is +also large, and runs out of time, it seems, in analyzing the +expression. Incidentally, it apparently allows the Maxima syntax :) + +I'm not sure the answer is correct or even sensible. The +Mathematica system 10 could not simplify the derivative back +to the input, at least not before I lost patience. + + + +3. My understanding of algebraic:true is that it uses +only roots of integers (absent in your expression) +%i (also absent), and tellrat() info (also absent). +So setting algebraic to true is maybe not the right thing to do. + +Introducing both sqrt(x) and x^(1/3) by a single algebraic +extension of degree 6 is possible via tellrat. I'm not +sure this gets you to where you want. + +Or even if you really want this integral or are just poking at Maxima. + +4. If you really want to try to patch this bug in the handling of +algebraic:true, ++ integrate + ratsimp, not by fixing the bug, but by catching it in a +different +place, you can look at code in src/rat3a.lisp where you can see that the +polynomial division in the function pquotient, at some point says, in +effect, +HEY! you want me to divide exactly -- that's what pquotient does -- but +the remainder is not zero. There's no bug in pquotient, but there is a bug +in someone who called it on bad argumemtns. (This is probably because +the GCD routine in use fails when there are algebraically dependent +indeterminates, but this handling of algebraics is touchy. maybe because +of comment 1. (thanks) someone can find this??? Anyway, this +error-catch of the throw in pquotient is what you would have to change, so +that +it doesn't just vomit all the way to the top, but to some other err-catch. +Maybe you can figure out enough from the comments and the code to +do this. The main body of code (but not all the comments) are almost +50 years old, circa 1966 I think. + +5. I am not sure what you mean by "throw to the outside" and how this makes +life easier. Perhaps you are resisting the "easy" way -- which is to use +the Maxima read-meval-print loop to call your functionality -- and +calling +Maxima via some other language at the end of a pipe? +(Why easy? load into Maxima some lisp function from a file foo, by +load("c:/..../foo.lisp"); + +and in the file foo.lisp + +(defun $sherfgen(x) (call_whatever_you_want_next x)) + +;; call_whatever .... can link to fortran, python, jscript, java, most any + windows dll, typical unix .o files, libraries etc + +And the load() command can be put into a maxima initialization file so +it is loaded up without user attention. + + +Anyway. I hope this info is useful, and you take the opportunity to learn +more lisp and +Maxima stuff. + +RJF + + +On Mon, Sep 7, 2015 at 12:54 AM, David Scherfgen wrote: + +> Hello, +> +> the following integral results in an error "quotient by zero": +> +> integrate(ev(ratsimp(1/(x^(5/2)+3*x^(1/3))),algebraic),x); +> +> Unfortunately, I can't catch the error using errcatch. It goes right +> through it, i.e. the whole evaluation is aborted, and I don't get any +> result (usually, errcatch would give me an empty list if it caught an +> error). +> +> I've had a similar problem with questions being asked by Maxima - there is +> no official way to intercept them. This is really bad if you run Maxima +> "automatedly" - without anybody being there to answer questions. +> +> But luckily, there is Robert Dodier's noninteractive package that +> re-defines the Maxima functions that are responsible for asking those +> questions. The re-defined functions don't ask the questions, but throw +> them, so that it can be caught from the outside. +> +> I tried to do something similar with the function that's responsible for +> generating the error, the function merror in merror.lisp. +> +> This is my attempt at it: +> +> ; If this is enabled, errors will be intercepted and thrown. +> (defmvar $no_errors t) +> +> ; A wrapper around MERROR in merror.lisp. +> (defvar *real-merror* (symbol-function 'merror)) +> (defun merror (sstring &rest l) +> (if $no_errors +> (meval `(($throw) '(($merror) ,sstring ,@ l))) +> (apply *real-merror* sstring l))) +> +> In a simple test case, it seems to work: +> +> (%i1) catch(1/0); +> (%o1) merror("expt: undefined: 0 to a negative exponent.") +> +> It throws the error instead of aborting the evaluation, just as I want. +> +> But in my original problem, it doesn't work, but crashes Maxima instead: +> +> (%i1) catch(integrate(ev(ratsimp(1/(x^(5/2)+3*x^(1/3))),algebraic),x)); +> Maxima encountered a Lisp error: +> Condition in MEVAL [or a callee]: INTERNAL-SIMPLE-ERROR: Bind stack +> overflow. +> Automatically continuing. +> To enable the Lisp debugger set *debugger-hook* to nil. +> +> Unfortunately, my LISP knowledge is basically nonexistent. +> +> Does anyone of you know why this doesn't work? Why can't errcatch catch +> the error in the first place? Is there something wrong with my merror +> wrapper? +> +> Thanks for any help! +> +> Best regards, +> David +> +> +> ------------------------------------------------------------------------------ +> +> _______________________________________________ +> Maxima-discuss mailing list +> Maxima-discuss at lists.sourceforge.net +> https://lists.sourceforge.net/lists/listinfo/maxima-discuss +> +> diff --git a/references/nonelem_integr2 b/references/nonelem_integr2 new file mode 100644 index 0000000..76a29d0 --- /dev/null +++ b/references/nonelem_integr2 @@ -0,0 +1,418 @@ +[This post updates a 1995 post by Wiener also available at this site. --djr] +============================================================================== +From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener) +Subject: Re: Repost: Integral of x^x +Date: 30 Nov 1997 20:42:28 GMT +Newsgroups: sci.math +Keywords: Functions without elementary antiderivative + +What's the antiderivative of exp(-x^2)? of sin(x)/x? of x^x? +------------------------------------------------------------------------ +These, and some similar problems, can't be done. + +More precisely, consider the notion of "elementary function". These are +the functions that can be expressed in terms of exponentals and logarithms, +via the usual algebraic processes, including the solving (with or without +radicals) of polynomials. Since the trigonometric functions and their +inverses can be expressed in terms of exponentials and logarithms using +the complex numbers C, these too are elementary. + +The elementary functions are, so to speak, the "precalculus functions". + +Then there is a theorem that says certain elementary functions do not +have an elementary antiderivative. They still have antiderivatives, +but "they can't be done". The more common ones get their own names. +Up to some scaling factors, "erf" is the antiderivative of exp(-x^2) +and "Si" is the antiderivative of sin(x)/x, and so on. +------------------------------------------------------------------------ +For those with a little bit of undergraduate algebra, we sketch a proof +of these, and a few others, using the notion of a differential field. +These are fields (F,+,.,1,0) equipped with a derivation, that is, a +unary operator ' satisifying (a+b)'=a'+b' and (a.b)'=a.b'+a'.b. Given +a differential field F, there is a subfield Con(F)={a:a'=0}, called the +_constants_ of F. We let I(f) denote an antiderivative. We ignore +cs. + +Most examples in practice are subfields of M, the meromorphic functions +on C (or some domain). Because of uniqueness of analytic extensions, one +rarely has to specify the precise domain. + +Given differential fields F and G, with F a subfield of G, one calls G +an algebraic extension of F if G is a finite field extension of F. + +One calls G a logarithmic extension of F if G=F(t) for some transcendental +t that satisfies t'=s'/s, some s in F. We may think of t as log s, but +note that we are not actually talking about a logarithm function on F. We +simply have a new element with the right derivative. Other "logarithms" +would have to be adjoined as needed. + +Similarly, one calls G an exponential extension of F if G=F(t) for some +transcendental t that satisfies t'=t.s', some s in F. Again, we may +think of t as exp s, but there is no actual exponential function on F. + +Finally, we call G an elementary differential extension of F if there is +a finite chain of subfields from F to G, each an algebraic, logarithmic, +or exponential extension of the next smaller field. + +The following theorem, in the special case of M, is due to Liouville. The +algebraic generality is due to Rosenlicht. More powerful theorems have +been proven by Risch, Davenport, and others, and are at the heart of +symbolic integration packages. + +A short proof, accessible to those with a solid background in undergraduate +algebra, can be found in Rosenlicht's AMM paper (see references). It is +probably easier to master its applications first, which often use similar +techniques, and then learn the proof. +------------------------------------------------------------------------ +MAIN THEOREM. Let F,G be differential fields, let a be in F, let y be in G, +and suppose y'=a and G is an elementary differential extension field of F, +and Con(F)=Con(G). Then there exist c_1,...,c_n in Con(F), u_1,...,u_n, v +in F such that + u_1' u_n' + a = c_1 --- + ... + c_n --- + v'. + u_1 u_n + +That is, the only functions that have elementary antiderivatives are the +ones that have this very specific form. In words, elementary integrals +always consist of a function at the same algebraic "complexity" level as +the starting function (the v), along with the logarithms of functions at +the same algebraic "complexity" level (the u_i 's). +------------------------------------------------------------------------ +This is a very useful theorem for proving non-integrability. Because +this topic is of interest, but it is only written up in bits and pieces, +I give numerous examples. (Since the original version of this FAQ from +way back when, two how-to-work-it write-ups have appeared. See Fitt & +Hoare and Marchisotto & Zakeri in the references.) + +In the usual case, F,G are subfields of M, so Con(F)=Con(G) always holds, +both being C. As a side comment, we remark that this equality is necessary. +Over R(x), 1/(1+x^2) has an elementary antiderivative, but none of the above +form. + +We first apply this theorem to the case of integrating f.exp(g), with f +and g rational functions. If g=0, this is just f, which can be integrated +via partial fractions. So assume g is nonzero. Let t=exp(g), so t'=g't. +Since g is not zero, it has a pole somewhere (perhaps out at infinity), +so exp(g) has an essential singularity, and thus t is transcendental over +C(z). Let F=C(z)(t), and let G be an elementary differential extension +containing an antiderivative for f.t. + +Then Liouville's theorem applies, so we can write + + u_1' u_n' + f.t = c_1 --- + ... + c_n --- + v' + u_1 u_n + +with the c_i constants and the u_i and v in F. Each u_i is a ratio of +two C(z)[t] polynomials, U/V say. But (U/V)'/(U/V)=U'/U-V'/V (quotient +rule), so we may rewrite the above and assume each u_i is in C(z)[t]. +And if any u_i=U.V factors, then (U.V)'/(U.V)=U'/U+V'/V and so we can +further assume each u_i is irreducible over C(z). + +What does a typical u'/u look like? For example, consider the case of u +quadratic in t. If A,B,C are rational functions in C(z), then A',B',C' +are also rational functions in C(z) and + + (A.t^2+B.t+C)' A'.t^2 + 2At(gt) + B'.t + B.(gt) + C' + ------------- = ------------------------------------- + A.t^2+B.t+C A.t^2 + B.t + C + + (A'+2Ag).t^2 + (B'+Bg).t + C' + = ----------------------------- . + A.t^2 + B.t + C + +(Note that contrary to the usual situation, the degree of a polynomial +in t stays the same after differentiation. That is because we are +taking derivatives with respect to z, not t. If we write this out +explicitly, we get (t^n)' = exp(ng)' = ng'.exp(ng) = ng'.t^n.) + +In general, each u'/u is a ratio of polynomials of the same degree. We +can, by doing one step of a long division, also write it as D+R/u, for +some D in C(z) and R in C(z)[t], with deg(R)oo but sqrt(z)*h(z)->0 as z->0. No rational function +does this. + +Or one can assume h has a partial fraction decomposition. Obviously no h' +term will give 1/z, so 1/z must be present in h already. But (1/z)'=-1/z^2, +and this is part of h'. So there is a 1/z^2 in h to cancel this. But +(1/z^2) is -2/z^3, and this is again part of h'. And again, something +in h cancels this, etc etc etc. This infinite regression is impossible. + +** sin(z)/z [sin(exp(z))] +** sin(z^2) [sqrt(z).sin(z),sin(z)/sqrt(z)] + +Since sin(z)=%[exp(iz)-exp(-iz)] (where %=1/2i), we merely rework the +above f.exp(g) result. Let f be rational, let t=exp(iz) (so t'/t=i) and +let T=exp(iz^2) (so T'/T=2iz) and we want an antiderivative of either +%f.(t-1/t) or T-1/T. For the former, the same partial fraction results +still apply in identifying %f.t=(h.t)'=(h'+ih).t, which can't happen, as +above. In the case of sin(z^2), we want %T=(h.T)'=(h'+2izh).T, and again, +this can't happen, as above. + +Although done, we push this analysis further in the f.sin(z)/z case, as +there are extra terms hanging around. This time around, the conclusion +gives an additional k/t term inside v, so we have -%f/t=(k/t)'=(k'-ik)/t. +So the antiderivative of %f*(t-1/t) is h.t+k/t. + +If f is even and real, then h and k (like t=exp(iz) and 1/t=exp(-iz)) are +parity flips of each other, so (as expected) the antiderivative is even. +Letting C=cos(z), S=sin(z), h=H+iF and k=K+iG, the real (and only) part +of the antiderivative of f is (HC-FS)+(KC+GS)=(H+K)C+(G-F)S. So over +the reals, we find that the antiderivative of (rational even).sin(x) is +of the form (rational even).cos(x)+ (rational odd).sin(x). + +A similar result holds for (odd).sin(x), (even).cos(x), (odd).cos(x). And +since a rational function is the sum of its (rational) even and odd parts, +(rational).sin integrates to (rational).sin + (rational).cos, or not at all. + +Let's backtrack, and apply this to sin(x)/x directly, using reals only. +If it has an elementary antiderivative, it must be of the form E.S+O.C. +Taking derivatives gives (E'-O).S+(E+O').C. As with partial fractions, +we have a unique R(x)[S,C] representation here (this is a bit tricky, +as S^2=1-C^2: this step can be proven directly or via solving for t,1/t +coefficients over C). So E'-O=1/x and E+O'=0, or O''+O=-1/x. Expressing +O in partial fraction form, it is clear only (-1/x) in O can contribute +a -1/x. So there is a -2/x^3 term in O'', so there is a 2/x^3 term in +O to cancel it, and so on, an infinite regress. Hence, there is no such +rational O. + +** arcsin(z)/z [z.tan(z)] + +We consider the case where F=C(z,Z)(t) as a subfield of the meromorphic +functions on some domain, where z is the identify function, Z=sqrt(1-z^2), +and t=arcsin z. Then Z'=-z/Z, and t'=1/Z. We ask in the main theorem +result if this can happen with a=t/z and some field G. t is transcendental +over C(z,Z), since it has infinite branch points. + +So we consider the more general situation of f(z).arcsin(z) where f(z) +is rational in z and sqrt(1-z^2). By letting z=2w/(1+w^2), note that +members of C(z,Z) are always elementarily integrable. + +Because x^2+y^2-1 is irreducible, C[x,y]/(x^2+y^2-1) is an integral domain, +C(z,Z) is isomorphic to its field of quotients in the obvious manner, and +C(z,Z)[t] is a UFD whose field of quotients is amenable to partial fraction +analysis in the variable t. What follows takes place at times in various +z-algebraic extensions of C(z,Z) (which may not have unique factorization), +but the terms must combine to give something in C(z,Z)(t), where partial +fraction decompositions are unique, and hence the t term will be as claimed. + +Thus, if we can integrate f(z).arcsin(z), we have f.t = sum of (u'/u)s +and v', by the main theorem. + +The u terms can, by logarithmic differentiation in the appropriate +algebraic extension field (recall that roots are analytic functions of +the coefficients, and t is transcendental over C(z,Z)), be assumed to +all be linear t+r, with r algebraic over z. Then u'/u=(1/Z+r')/(t+r). +When we combine such terms back in C(z,Z), they don't form a t term +(nor any higher power of t, nor a constant). + +Partial fraction decomposition of v gives us a polynomial in t, with +coefficients in C(z,Z), plus multiples of powers of linear t terms. +The latter don't contribute to a t term, as above. + +If the polynomial is linear or quadratic, say v=g.t^2 + h.t + k, then +v'=g'.t^2 + (2g/Z+h').t + (h/Z+k'). Nothing can cancel the g', so g +is just a constant c. Then 2c/Z+h'=f or I(f.t)=2c.t+I(h'.t). The +I(h'.t) can be integrated by parts. So the antiderivative works out +to c.(arcsin(z))^2 + h(z).arcsin(z) - I(h(z)/sqrt(1-z^2)), and as +observed above, the latter is elementary. + +If the polynomial is cubic or higher, let v=A.t^n+B.t^(n-1)+...., then +v'=A'.t^n + (n.A/Z+B').t^(n-1) +.... A must be a constant c. But then +nc/Z+B'=0, so B=-nct, contradicting B being in C(z,Z). + +In particular, since 1/z + c/sqrt(1-z^2) does not have a rational in +"z and/or sqrt(1-z^2)" antiderivative, arcsin(z)/z does not have an +elementary integral. + +** z^z + +In this case, let F=C(z,l)(t), the field of rational functions in z,l,t, +where l=log z and t=exp(z.l)=z^z. Note that z,l,t are algebraically +independent. Then t'=(l+1).t, so for a=t in the main theorem, the +partial fraction analysis shows that the only possibility is for +v=w.t+... to be the source of the t term on the left, with w in C(z,l). + +So this means, equating t coefficients, 1=w'+(l+1)w. This is a first +order ODE, whose solution is w=I(z^z)/z^z. So we must prove that no +such w exists in C(z,l). So suppose w=P/Q, with P,Q in C[z,l] and no +common factors. Then z^z=(z^z*P/Q)'=z^z*[(1+l)PQ+P'Q-PQ']/Q^2, or +Q^2=(1+l)PQ+P'Q-PQ'. So Q|Q', meaning Q is a constant, which we may +assume to be one. So we have it down to P'+P+lP=1. + +Let P=Sum[P_i l^i], with P_i, i=0...n in C[z]. But then in our equation, +there's a dangling P_n l^(n+1) term, a contradiction. +------------------------------------------------------------------------ +On a slight tangent, this theorem of Liouville will not tell you that +Bessel functions are not elementary, since they are defined by second +order ODEs. This can be proven using differential Galois theory. A +variant of the above theorem of Liouville, with a different normal form, +does show however that J_0 cannot be integrated in terms of elementary +methods augmented with Bessel functions. +======================================================================== +What follows is a fairly complete sketch of the proof of the Main Theorem. +First, I just state some easy (if you've had Galois Theory 101) lemmas. +Throughout the lemmas F is a differential field, and t is transcendental +over F. + +Lemma 1: If K is an algebraic extension field of F, then there exists a +unique way to extend the derivation map from F to K so as to make K into +a differential field. + +Lemma 2: If K=F(t) is a differential field with derivation extending F's, +and t' is in F, then for any polynomial f(t) in F[t], f(t)' is a polynomial +in F[t] of the same degree (if the leading coefficient is not in Con(F)) +or of degree one less (if the leading coefficient is in Con(F)). + +Lemma 3: If K=F(t) is a differential field with derivation extending F's, +and t'/t is in F, then for any a in F, n a positive integer, there exists +h in F such that (a*t^n)'=h*t^n. More generally, if f(t) is any polynomial +in F[t], then f(t)' is of the same degree as f(t), and is a multiple of +f(t) iff f(t) is a monomial. + +These are all fairly elementary. For example, (a*t^n)'=(a'+at'/t)*t^n +in lemma 3. The final 'iff' in lemma 3 is where transcendence of t comes +in. Lemma 1 in the usual case of subfields of M is an easy consequence +of the implicit function theorem. +-------------------------------------------------------------------------- +MAIN THEOREM. Let F,G be differential fields, let a be in F, let y be in G, +and suppose y'=a and G is an elementary differential extension field of F, +and Con(F)=Con(G). Then there exist c_1,...,c_n in Con(F), u_1,...,u_n, v +in F such that + u_1' u_n' + a = c_1 --- + ... + c_n --- + v'. + u_1 u_n + +In other words, the only functions that have elementary antiderivatives +are the ones that have this very specific form. +------------------------------------------------------------------------ +Proof: + +By assumption there exists a finite chain of fields connecting F to G +such that the extension from one field to the next is given by performing +an algebraic, logarithmic, or exponential extension. We show that if the +form (*) can be satisfied with values in F2, and F2 is one of the three +kinds of allowable extensions of F1, then the form (*) can be satisfied +in F1. The form (*) is obviously satisfied in G: let all the c's be 0, the +u's be 1, and let v be the original y for which y'=a. Thus, if the form +(*) can be pulled down one field, we will be able to pull it down to F, +and the theorem holds. + +So we may assume without loss of generality that G=F(t). + +Case 1: t is algebraic over F. Say t is of degree k. Then there are +polynomials U_i and V such that U_i(t)=u_i and V(t)=v. So we have + + U_1(t)' U_n(t)' + a = c_1 ------ + ... + c_n ------ + V(t)'. + U_1(t) U_n(t) + +Now, by the uniqueness of extensions of derivatives in the algebraic case, +we may replace t by any of its conjugates t_1,..., t_k, and the same equation +holds. In other words, because a is in F, it is fixed under the Galois +automorphisms. Summing up over the conjugates, and converting the U'/U +terms into products using logarithmic differentiation, we have + + [U_1(t_1)*...*U_1(t_k)]' + k a = c_1 ----------------------- + ... + [V(t_1)+...+V(t_k)]'. + U_1(t_1)*...*U_n(t_k) + +But the expressions in [...] are symmetric polynomials in t_i, and as +they are polynomials with coefficients in F, the resulting expressions +are in F. So dividing by k gives us (*) holding in F. + +Case 2: t is logarithmic over F. Because of logarithmic differentiation +we may assume that the u's are monic and irreducible in t and distinct. +Furthermore, we may assume v has been decomposed into partial fractions. +The fractions can only be of the form f/g^j, where deg(f)