
Codemist Standard Lisp 3.54 for DEC Alpha: May 23 1994
Dump file created: Mon May 23 10:39:11 1994
REDUCE 3.5, 15-Oct-93 ...
Memory allocation: 6023424 bytes

+++ About to read file ndotest.red


%
%  More Tests for REDUCE Special Functions Package
%
%  Winfried Neun, ZIB Berlin, February 1993
%

load_package specfn;


(specfn)


% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
%     1. Binomial Coefficients and Stirling numbers
% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

Binomial (2,1/2);


  16
------
 3*pi


on rounded;


Binomial (2.1,2);


1.155

off rounded;



Binomial (n,2);


  gamma(n + 1)
----------------
 2*gamma(n - 1)


1/Binomial (49,6);


    1
----------
 13983816
 % for those who play Lotto in Germany.

Stirling1(10,5);


-269325


Stirling2(10,5);


42525


% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
%     2. Bernoulli Polynomials
% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

BernoulliP (4,x);


     4       3       2
 30*x  - 60*x  + 30*x  - 1
---------------------------
            30


BernoulliP (4,1/2);


  7
-----
 240


% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
%     4. Laguerre Polynomials
% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

LaguerreP(3,x);


     3      2
  - x  + 9*x  - 18*x + 6
-------------------------
            6


LaguerreP(2,1,x);


  2
 x  - 6*x + 6
--------------
      2


% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
%     5. Legendre and Jacobi Polynomials
% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

LegendreP (2,x);


    2
 3*x  - 1
----------
    2


LegendreP (3,x);


       2
 x*(5*x  - 3)
--------------
      2


LegendreP (6,3,x);


              2             4       2
 315*sqrt( - x  + 1)*x*(11*x  - 14*x  + 3)
-------------------------------------------
                     2


JacobiP (2,1,3/4,x);


      2
 437*x  + 38*x - 91
--------------------
        128


% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
%     6. Chebychev Polynomials
% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

ChebyshevT(2,x);


   2
2*x  - 1


ChebyshevT(3,x);


      2
x*(4*x  - 3)


ChebyshevU(2,x);


   2
4*x  - 1


ChebyshevU(3,x);


        2
4*x*(2*x  - 1)


% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
%     7. Hermite Polynomials
% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

HermiteP (2,x);


      2
2*(2*x  - 1)


HermiteP (3,x);


        2
4*x*(2*x  - 3)


% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
%     8. Gegenbauer Polynomials
% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

GegenbauerP(2,5,x);


       2
5*(12*x  - 1)


GegenbauerP(3,2,x);


        2
4*x*(8*x  - 3)


% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
%     9. Some well-known Infinite Sums
% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

off exp;



sum(1/m^4,m,1,infinity);


   4
 pi
-----
 90


sum((-1)^m/m^4,m,1,infinity);


     4
 7*pi
-------
  720


sum(1/(2*m-1)^4,m,1,infinity);


   4
 pi
-----
 96


sum((-1)^m/(2*m-1)^3,m,1,infinity);


   3
 pi
-----
 32


on exp;



sum((-1)^m/(2*m-1)^2,m,1,infinity);


catalan



% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
%    10. Euler Numbers and Polynomials
% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

Euler(6);


-61


EulerP(4,x);


    3      2
x*(x  - 2*x  + 1)


EulerP(4,1/2);


 5
----
 16
 

% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
%    11. Integral Functions
% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

on rounded;



Shi(3.4);


6.50998313882


Si(3.4);


1.84191398333


Chi(3.4);


*** ROUNDBF turned on to increase accuracy 

6.50209216532


Ci(3.4);


 - 0.00451807793074


Ei(Pi);


10.9283743893


erfc(2.0);


0.00467773498105


Fresnel_C(3.1);


0.561593902462


Fresnel_S(3.1);


0.581815868171


off rounded;



df(Si(z),z);


 sin(z)
--------
   z

 
limit(Si(x),x,infinity);


 pi
----
 2


limit(Fresnel_S(x),x,infinity);


 1
---
 2


s_i(x);


 2*si(x) - pi
--------------
      2


defint(cos(t)/t,t,X,INFINITY);


 - ci(x)


% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
%    12. Misc Functions
% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=


defint(log(t)/(1-t),t,1,x);


dilog(x)


on rounded;



dilog(3.1);


 - 1.49114561815


off rounded;


end;
(TIME:  specfmor 4034 4234)


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