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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input huang.input}
+\author{The Axiom Team}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{License}
+<<license>>=
+--Copyright The Numerical Algorithms Group Limited 1996.
+@
+<<*>>=
+<<license>>
+
+-- this is here strictly for documentation purposes, nothing is executed
+--
+--	Here are some problems that Maple and Mathematica cannot
+--solve, but SymbMath can do.
+--	The following examples came from news on the sci.math.symbolic
+--newsgroup in 1991, and were run in Maple V, Mathematica 2.0, or
+--SymbMath 2.1.
+--
+--************************ Example 1 ******************************
+--	Maple:
+--> int(exp(-a * x^2), x=0..infinity);
+--
+--                             infinity
+--                                /
+--                               |               2
+--                               |      exp(- a x ) dx
+--                               |
+--                              /
+--                              0
+--
+--# unevaluated.
+--# Declare 'a' non-negative:
+--
+--> signum(a) := 1;
+--
+--                                 signum(a) := 1
+--
+--# The same integral is now evaluated fully:
+--
+--> int(exp(-a * x^2), x=0..infinity);
+--
+--                                         1/2
+--                                       Pi
+--                                   1/2 -----
+--                                         1/2
+--                                        a
+--
+--	SymbMath :
+--	Input:
+--inte(exp(-a*x^2), x from 0 to inf)
+--assume(sqrt(a) > 0)
+--inte(exp(-a*x^2), x from 0 to inf)
+--	Output:
+--1/2*a^(-0.5)*sqrt(pi)*erf(inf*sgn(sqrt(a)))
+--assumed
+--1/2*a^(-0.5)*sqrt(pi)
+--
+--************************** Example 2 *********************************
+--	Maple:
+--# Despite the fact that 'n' is declared non-negative...
+--> signum(n) := 1;
+--                                 signum(n) := 1
+--
+--# ...this simple proper definite integral (that any freshman calculus
+--# student can evaluate!) is left unevaluated:
+--> int(x^n, x=0..1);
+--
+--                                     1
+--                                     /
+--                                    |   n
+--                                    |  x  dx
+--                                    |
+--                                   /
+--                                   0
+--
+--
+--	SymbMath :
+--	Input:
+--assume(n > -1)
+--inte(x^n, x from 0 to 1)
+--	Output:
+--assumed
+--1/(1 + n)
+--
+--************************** Example 3 *****************************
+--	Maple:
+--> int(x^n, x=eps..1);
+--
+--                                          (n + 1)
+--                                 1     eps
+--                               ----- - ----------
+--                               n + 1      n + 1
+--
+--# ...but the one-sided limit...
+--
+--> limit(", eps=0, right);
+--
+--                                               (n + 1)
+--                                      1     eps
+--                          limit     ----- - ----------
+--                          eps -> 0+ n + 1      n + 1
+--
+--# ...remains unevaluated...
+--
+--> simplify(");
+--
+--                                               (n + 1)
+--                                      - 1 + eps
+--                          limit     - ----------------
+--                          eps -> 0+         n + 1
+--
+--
+--# ...no matter what we do!
+--
+--> eval(");
+--
+--                                               (n + 1)
+--                                      - 1 + eps
+--                          limit     - ----------------
+--                          eps -> 0+         n + 1
+--
+--	SymbMath:
+--	Input:
+--assume(n > -1)
+--inte(x^n, x from eps to 1)
+--subs(eps=0 to last)
+--	Output:
+--assumed
+--1/(1 + n) - eps^(1 + n)/(1 + n)
+--1/(1 + n)
+--
+--
+--*************************** Example 4 ***************************
+--	Maple:
+--> 0^n;
+--            0
+--
+--# Maple flags the error only when 'n' is replaced by the constant 0:
+--
+--> 0^0;
+--Error, 0^0 is undefined
+--
+--	SymbMath:
+--	Input:
+--assume(n > 0)
+--0^n
+--0^-n
+--0^0
+--	Output:
+--assumed
+--0
+--discont
+--undefined
+--
+--**************************** Example 5 ******************************
+--	Maple:
+--> int(x^k, x);
+--
+--                                     (k + 1)
+--                                    x
+--                                    --------
+--                                      k + 1
+--
+--	SymbMath:
+--	Input:
+--inte(x^k*d(x))
+--subs(k=-1 to last)
+--	Output:
+--constant + x^(1 + k)/(1 + k)
+--discont
+--
+--**************************** Example 6 *****************************
+--	Maple:
+--# The following limit is left unevaluated:
+--> limit(x^k/exp(x), x=infinity);
+--
+--                                               k
+--                                              x
+--                              limit         ------
+--                              x -> infinity exp(x)
+--
+--
+--# We might ask if Maple knows this result for specific (constant) values
+--# of the symbolic parameter 'k'.  The answer is a QUALIFIED "yes".
+--# Maple knows the result for 'k' equal to 10^8...
+--
+--> limit(x^(10^8)/exp(x), x=infinity);
+--
+--                                       0
+--
+--
+--# ...and Maple knows the result for 'k' equal to 10^9...
+--
+--> limit(x^(10^9)/exp(x), x=infinity);
+--
+--                                       0
+--
+--# ...BUT, Maple seems to FORGET the result when 'k' equals 10^10...
+--
+--> limit(x^(10^10)/exp(x), x=infinity);
+--
+--                                          10000000000
+--                                         x
+--                           limit         ------------
+--                           x -> infinity    exp(x)
+--
+--	SymbMath:
+--	Input:
+--lim(x=inf, x^k/exp(x))
+--lim(x=inf, x^(10^10)/exp(x))
+--lim(x=inf, x^(10^10000)/exp(x))
+--	Output:
+--0
+--0
+--0
+--
+--****************************** Example 7 ****************************
+--	Maple:
+--> int(x^m * exp (-b * x), x=0..infinity);
+--
+--                            infinity
+--                               /
+--                              |       m
+--                              |      x  exp(- b x) dx
+--                              |
+--                             /
+--                             0
+--
+--
+--# As expected, the integral remains unevaluated.  Now declare the signs
+--# of the symbolic parameters:
+--
+--> signum(b) := 1;
+--
+--                                 signum(b) := 1
+--
+--> signum(m) := 1;
+--
+--                                 signum(m) := 1
+--
+--
+--# Upon attempting to compute the integral a second time...
+--
+--> int(x^m * exp (-b * x), x=0..infinity);
+--
+--                            infinity
+--                               /
+--                              |       m
+--                              |      x  exp(- b x) dx
+--                              |
+--                             /
+--                             0
+--
+--# ...THE INTEGRAL CONTINUES TO REMAIN UNEVALUATED, despite the fact that
+--# the signs of the parameters 'b' and 'm' were declared PRIOR to the second
+--# attempt at computation.
+--
+--	SymbMath:
+--	Input:
+--inte(x^n*exp(-a*x), x from 0 to inf)
+--	Output:
+--inte(x^n*exp(-a*x), x, 0, inf)
+--
+--
+--************************ Example 8 *********************************
+--	Mathematica:
+--In[1]:= Integrate[1/x,{x,-1,1}]
+--Out[1]= -Log[-1]
+--
+--	Maple:
+--has the same problem.
+--
+--	SymbMath:
+--	Input:
+--inte(1/x, x from -1 to 1)
+--inte(1/x, x from -1 to 2)
+--	Output:
+--0
+--ln(2)
+--
+--
+--*************************** Example 9 *******************************
+--	Maple:
+--has a problem for int(tan(x), x=0..pi).
+--
+--	SymbMath:
+--	Input:
+--inte(tan(x), x from 0 to pi)
+--	Output:
+--0
+--
+--**************************** Example 10 ****************************
+--	Mathematica:
+--cannot evaluate integral of sgn(x).
+--
+--	Maple:
+--# cannot evaluate inegral of signum(x) by int(). Help with a procedure:
+--# Load the procedure into Maple:
+--
+--> read `pvint.txt`;
+--
+--pvint := proc(f,x,a,b,s)
+--         local i1,i2,eps;
+--             signum(eps) := 1;
+--             i1 := int(f,x = a .. s-eps);
+--             i2 := int(f,x = s+eps .. b);
+--             simplify(i1+i2);
+--             limit(",eps = 0,right)
+--         end
+--
+--
+--> pvint(signum(x), x, -1, 1, 0);
+--
+--                          - eps                   1
+--                            /                     /
+--                           |                     |
+--               limit       |    signum(x) dx +   |  signum(x) dx
+--               eps -> 0+   |                     |
+--                          /                     /
+--                          -1                   eps
+--
+--# Maple refuses to evaluate this P.V. integral:
+--
+--> simplify(");
+--
+--                          - eps                   1
+--                            /                     /
+--                           |                     |
+--               limit       |    signum(x) dx +   |  signum(x) dx
+--               eps -> 0+   |                     |
+--                          /                     /
+--                          -1                   eps
+--
+--> eval(");
+--
+--                          - eps                   1
+--                            /                     /
+--                           |                     |
+--               limit       |    signum(x) dx +   |  signum(x) dx
+--               eps -> 0+   |                     |
+--                          /                     /
+--                          -1                   eps
+--
+--	SymbMath:
+--	Input:
+--inte(sgn(x), x from -1 to 1)
+--inte(sgn(x), x from -1 to 2)
+--	Output:
+--0
+--1
+--
+--************************* Example 11 ********************************
+--Implicit diff. gives 1+y'[x](1+1/y[x])==0; y'[x]==-y[x]/(y[x]+1).
+--
+--	Mathematica:
+-- given this eq. as input to DSolve says that built-in
+-- procedure can't solve it.
+--
+--	 MACSYMA:
+--(c1) depends(y,x)$
+--
+--(c2) ode2(diff(y,x) = -y/(y+1),y,x);
+--
+--(d2)                   - log(y) - y = x + %c
+--
+--(c3) method;
+--
+--(d3)                         separable
+--
+--
+--	SymbMath:
+--solve the differential equation by integration inte() or by dsolve().
+--	Input:
+--d(y)/d(x)*(1+1/y) === -1
+--inte(last*d(x))
+--Expand=On
+--dsolve(d(y)/d(x) === -y/(y+1), y)
+--	Output:
+--(1 + 1/y)*d(y)/d(x) === -1
+--y + ln(y*sgn(y)) === constant - x
+--Expand = On
+---y - ln(y*sgn(y)) === constant + x
+--
+--
+--************************* Example 12 ********************************
+--	Mathematica:
+--                 y'[x] = y[x]^(1/2)
+--                  y[0] = 0
+--
+--DSolve could not handle it. (It rarely solves anything!!), and the
+--RungeKutta package only gave me the solution
+--
+--                    y[x]=0
+--
+--Obviously, there is another solution viz.
+--
+--                   y[x] = (x/2)^2
+--
+--	SymbMath:
+--	Input:
+--dsolve(d(y)/d(x) === sqrt(y), y)
+--(last/2)^2
+--	Output:
+--2*sqrt(y) === constant + x
+--y === 1/4*(constant + x)^2
+--
+--************************* Example 13 ********************************
+--	Maple:	
+--> sqrt(x*x);
+--      x
+--
+--	Mathematica:
+-- Sqrt[a^2] evaluates to Sqrt[a^2].
+--
+--	SymbMath:
+--	Input:
+--sqrt(x^2)
+--assume(a > 0)
+--sqrt(a^2)
+--assume(b <0 )
+--sqrt(b^2)
+--	Output:
+--x*sgn(x)
+--assumed
+--a
+--assumed
+---b
+--
+--********************** Example 14 **********************************
+--	Maple and Mathematica cannot find the integrals of abs(x).
+--
+--	SymbMath:
+--	Input:
+--inte(abs(x), x from -1 to 1)
+--inte(abs(x)^5*d(x))
+--	Output:
+--1
+--constant + 1/6*abs(x)^6*sgn(x)
+--
+-----------------------------------------------------------------------
+--
+--	The following problems are taken from Swokowski's Calculus
+--book.  They cause Mathematica to fail because of a singularity in
+--the interior of the interval of integration:
+--
+--        Section 10.4 Problems 3, 12, 15, 16, 23, 29.
+--
+--The comments "INTEGRAL IS DIVERGENT" and "Principal Value" come from
+--Macsyma.  Mma gives no indication that anything is amiss.
+--
+--************************ Problem3 *********************************
+--	Mathematica:
+--In[4]:= Integrate[1/x^2,{x,-3,1}]
+--
+--          4
+--Out[4]= -(-)                            (* INTEGRAL IS DIVERGENT *)
+--          3
+--
+--	SymbMath:
+--	Input:
+--inte(1/x^2, x from -3 to 1)
+--	Output:
+--inf
+--
+--************************** Problem12 ***************************
+--	Mathemtica:
+--In[6]:= Integrate[x^(-4/3),{x,-1,1}]
+--
+--Out[6]= -6                              (* INTEGRAL IS DIVERGENT *)
+--
+--	SymbMath:
+--	Input:
+--inte(x^(-4/3), x from -1 to 1)
+--        Output:
+--inf
+--
+--**************************** Problem15  ************************
+--	Mathematica:
+--In[7]:= Integrate[1/x,{x,-1,2}]
+--
+--Out[7]= -Log[-1] + Log[2]
+--
+--	Maple:
+--has the same problem.
+--	
+--	Macsyma:
+--	(c7) integrate(1/x,x,-1,2);
+--	Principal Value
+--	(d7)                                log(2)
+--
+--	SymbMath:
+--	Input:
+--inte(1/x, x from -1 to 2)
+--	Output:
+--ln(2)
+--
+--
+--********************** Problem16 ************************************
+--	Mathematica:
+--In[8]:= Integrate[1/(x^2-x-2),{x,0,4}]
+--
+--        -Log[-2]   Log[2]   Log[5]
+--Out[8]= -------- + ------ - ------
+--           3         3        3
+--
+--	Macsyma:
+--	(c8) integrate(1/(x^2-x-2),x,0,4);
+--	Principal Value
+--	                                     log(5)
+--	(d8)                               - ------
+--	                                       3
+--
+--
+--	SymbMath:
+--	Input:
+--inte(1/(x^2-x-2), x from 0 to 4)
+--	Output:
+---1/3*ln(2) + 1/3*ln(2/5)
+--
+--***************************** Problem23 **********************
+--	Mathematica:
+--In[10]:= Integrate[(1/x^2)Cos[1/x],{x,-1,2}]
+--
+--                       1
+--Out[10]= Sin[-1] - Sin[-]              (* INTEGRAL IS DIVERGENT *)
+--		       2
+--
+--	SymbMath:
+--	Input:
+--y=1/x^2*cos(1/x)
+--inte(y, x from -1 to 0-zero) + inte(y, x from 0+zero to 2)
+--	Output:
+--y = x^(-2)*cos(1/x)
+--sin(-1) - sin(1/2) + 2*sin(inf)
+--
+--************************** Problem29 ********************************
+--	Mathematica:
+--In[12]:= Integrate[1/(x-4)^2,{x,0,Infinity}]
+--
+--           1
+--Out[12]= -(-)                          (* INTEGRAL IS DIVERGENT *)
+--           4
+--
+--	SymbMath:
+--	Input:
+--y=1/(x-4)^2
+--inte(y, x from 0 to 4-zero) + inte(y, x from 4+zero to inf)
+--	Output:
+--y = (-4 + x)^(-2)
+--inf
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}