MaplePrimes - answers and comments on Question, BesselY,BesselK

evalf,Int with Method

Scott03 — Thu, 07 Jun 2007 21:01:56 Z

I am not sure if I am using the correct method type but the following give roughly the same answer: >evalf(Int((1/4)*x^3*BesselY(0, x)*BesselK(0, x), x = 0 .. infinity, method = _CCquad)); 0.1591549431 if you evaluate the value that those authors say: > evalf(1/(2*Pi)); 0.1591549430 You can see that the answers are almost exact (not exact due to round off error when dealing with numerics instead of symbolic operations). I am not sure if there is another option that will give you a symbolic answer. Scott

Via Mellin transform

JacquesC — Thu, 07 Jun 2007 22:57:01 Z

>yy := (1/4)*x^3*BesselY(0,x)*BesselK(0,x):
> mel := inttrans[mellin](yy,x,s);

               s                 2                      Pi (s + 3)
              2  GAMMA(s/2 + 3/2)  GAMMA(s/4 + 3/4) cos(----------)
                                                            4
  mel := -1/4 -----------------------------------------------------
                                       Pi

> limit(mel,s=1);

                                  1
                                 ----
                                 2 Pi

This actually generalizes nicely, and one can get a closed-form with x^3 replaced by x^a. If a+1 is congruent to 0 mod 4, the answer simplifies a lot. The prettiest easy generalization along those lines (that I could find easily) is

> yy := (1/4)*x^a*BesselY(nu,x)*BesselK(nu,x):
> mel := inttrans[mellin](yy,x,s);

               (s + a - 3)                                     nu
  mel := -1/4 2            GAMMA(s/2 + a/2) GAMMA(s/2 + a/2 + ----)
                                                               2

                           nu       Pi (s + a - 2 nu)
        GAMMA(s/4 + a/4 - ----) cos(-----------------)/Pi
                           2                4

> eval(mel,s=1);

        (-2 + a)                                     nu
  -1/4 2         GAMMA(1/2 + a/2) GAMMA(1/2 + a/2 + ----)
                                                     2

                           nu       Pi (1 + a - 2 nu)
        GAMMA(1/4 + a/4 - ----) cos(-----------------)/Pi
                           2                4

Mgfun, etc

JacquesC — Thu, 05 Jul 2007 06:11:55 Z

[I am not reply-ing to the above message, because things have gotten too squeezed onto the RHS of the screen]. You need to set the variable 'libname' to include that directory. Take a look at the help pages for libname and LibraryTools.

Mgfun, libname

SandorSzabo — Thu, 05 Jul 2007 19:55:34 Z

Thanks. I read libname. There is only one problem, the priority. When I typed

> originallibname:=libname;

                "C:\Program Files\Maple 10/lib"
> with(Mgfun);

Error, (in with) invalid input: with expects its 1st argument, 
pname, to be of type {package, module}, but received Mgfun

> libname:="C:/Program Files/Maple 11/MyLib/Algolib";

           "C:/Program Files/Maple 11/MyLib/Algolib"

> with(Mgfun);

[MG_Internals, creative_telescoping, dfinite_expr_to_diffeq, 
  dfinite_expr_to_rec, dfinite_expr_to_sys, diag_of_sys, int_of_sys, 
  pol_to_sys, sum_of_sys, sys*sys, sys+sys]

> with(Groebner);

[Basis, FGLM, GB_Internals, HilbertDimension, HilbertPolynomial, 
  HilbertSeries, InterReduce, IsProper, IsZeroDimensional, 
LeadingCoefficient, 
  LeadingMonomial, LeadingTerm, MonomialOrder, MulMatrix, 
  MultiplicationMatrix, MultivariateCyclicVector, NormalForm, NormalSet, 
  Reduce, RememberBasis, SPolynomial, SetBasis, Solve, TestOrder, 
  ToricIdealBasis, UnivariatePolynomial, Walk, _pexports, fglm_algo, gbasis, 
  gsolve, hilbertdim, hilbertpoly, hilbertseries, init, inter_reduce, 
  is_finite, is_solvable, leadcoeff, leadmon, leadterm, normalf, 
  pretend_gbasis, reduce, spoly, termorder, testorder, univpoly]

So everything is OK. Almost. The problem is the order (priority). When I typed the following the GB_Internals disappeared.

> libname;
                "C:\Program Files\Maple 10/lib"
> libname:="C:/Program Files/Maple 11/MyLib/Algolib",libname;

"C:/Program Files/Maple 11/MyLib/Algolib", "C:\Program Files\Maple 10/lib"

> with(Mgfun);

[MG_Internals, creative_telescoping, dfinite_expr_to_diffeq, 
  dfinite_expr_to_rec, dfinite_expr_to_sys, diag_of_sys, int_of_sys, 
  pol_to_sys, sum_of_sys, sys*sys, sys+sys]

> with(Groebner);

[Basis, FGLM, HilbertDimension, HilbertPolynomial, HilbertSeries, InterReduce, 
  IsProper, IsZeroDimensional, LeadingCoefficient, LeadingMonomial, 
  LeadingTerm, MonomialOrder, MultiplicationMatrix, MultivariateCyclicVector, 
  NormalForm, NormalSet, Reduce, RememberBasis, SPolynomial, Solve, TestOrder, 
  ToricIdealBasis, UnivariatePolynomial, Walk]

> march('setattribute',"C:/Program Files/Maple 11/MyLib/Algolib",priority=2);

Error, (in march) individual archive file is expected for changing priority

How can I give higher priority to Algolib than Maple/lib? The help says For example, the commands: libname := "/usr/me/mylib/joe.lib","/usr/maple/lib"; myexpand; result in the two specified locations being searched for a definition of myexpand in a file or a member named myexpand.m. First the single library "joe.lib" in the directory "/user/me/mylib" is searched, and then, if necessary, the libraries in the directory "/usr/maple/lib" are searched. The search is terminated on its first success. In particular, this mechanism allows for easy additions or replacements to the main library. I need only (hopefully) a little help to go on.

libname

JacquesC — Thu, 05 Jul 2007 22:31:05 Z

libname is search in order (left to right), so that if you put Algobib in first, that is the one that will be picked up. The priority is only useful if you have multiple .mla libraries in the same directory, then you need to order them explicitly. You have to make sure that you change libname as the very first thing you do in a session (make sure you have no init files!). Otherwise any loading that happens will ``lock in'' where things are looked up. The way you did it above should have worked in a clean worksheet.

libname problem

SandorSzabo — Fri, 06 Jul 2007 11:40:20 Z

I use Maple 10 under xp. There is no init file. It seems to me there is something wrong with libname. I copy here the 2 examples. There is no anything else in these 2 mw files. 1)

> libname;
                "C:\Program Files\Maple 10/lib"
> with(Mgfun);

Error, (in with) invalid input: with expects its 1st argument, pname, 
to be of type {package, module}, but received Mgfun

> libname:="C:/Program Files/Maple 11/MyLib/Algolib";

           "C:/Program Files/Maple 11/MyLib/Algolib"

> with(Mgfun);

[MG_Internals, creative_telescoping, dfinite_expr_to_diffeq, 
  dfinite_expr_to_rec, dfinite_expr_to_sys, diag_of_sys, int_of_sys, 
  pol_to_sys, sum_of_sys, sys*sys, sys+sys]

> with(Groebner);

[Basis, FGLM, GB_Internals, HilbertDimension, HilbertPolynomial, 
  HilbertSeries, InterReduce, IsProper, IsZeroDimensional, LeadingCoefficient, 
  LeadingMonomial, LeadingTerm, MonomialOrder, MulMatrix, 
  MultiplicationMatrix, MultivariateCyclicVector, NormalForm, NormalSet, 
  Reduce, RememberBasis, SPolynomial, SetBasis, Solve, TestOrder, 
  ToricIdealBasis, UnivariatePolynomial, Walk, _pexports, fglm_algo, gbasis, 
  gsolve, hilbertdim, hilbertpoly, hilbertseries, init, inter_reduce, 
  is_finite, is_solvable, leadcoeff, leadmon, leadterm, normalf, 
  pretend_gbasis, reduce, spoly, termorder, testorder, univpoly]

Everything is OK. 2)

> libname;
                "C:\Program Files\Maple 10/lib"

> libname:="C:/Program Files/Maple 11/MyLib/Algolib",libname;

"C:/Program Files/Maple 11/MyLib/Algolib", 
"C:\Program Files\Maple 10/lib"

> with(Mgfun);

[MG_Internals, creative_telescoping, dfinite_expr_to_diffeq, 
  dfinite_expr_to_rec, dfinite_expr_to_sys, diag_of_sys, int_of_sys, 
  pol_to_sys, sum_of_sys, sys*sys, sys+sys]

> with(Groebner);

[Basis, FGLM, HilbertDimension, HilbertPolynomial, HilbertSeries, InterReduce, 
  IsProper, IsZeroDimensional, 
LeadingCoefficient, LeadingMonomial, 
  LeadingTerm, MonomialOrder, MultiplicationMatrix, MultivariateCyclicVector, 
  NormalForm, NormalSet, Reduce, RememberBasis, SPolynomial, Solve, TestOrder, 
  ToricIdealBasis, UnivariatePolynomial, Walk]

GB_Internals disappeared. Sorry, but I don't understand the reason of it. (Therefore I thought to force Maple by using priority.)