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    SymbMath 2.2: A Symbolic Calculator with Learning
        (Version 2.2)

        by Dr. Weiguang HUANG
    Dept. Analytical Chemsitry, University of New South Wales,
    Kensington, Sydney, NSW 2033, Australia
    Phone: [phone]
    Fax:   [phone]
    E-mail: [e-mail]

        Copyright (C) 1990-1993
        April 15, 1993

    This is Part Two of SymbMath manual. The SymbMath manual has
three parts in the files SymbMath.DOC, SymbMath.DO2 and SymbMath.DO3.


            7. Examples

    In the following examples, a line of "Input:" means that
typing the program in the Editor, then leaving the Editor by <Esc>,
finally running the program by the command "Run", while a line of
"Output:" means outputs in the "Output" window. # is a comment
statement. All examples should be terminated by the end statement,
but the end statement may omit in the output of the following examples.


        7.1 Simplification and Assumption

    SymbMath automatically simplifies the output expression.
Users can further simplify it by using the built-in variable "last"
in a singe line again and again until they are happy.

    Example 7.1.1. Reduce sqrt(x^2).
    Input:
sqrt(x^2)
end
    Output:
x*sgn(x)
end

    A first way to reduce this output expression is to substitute
sgn(x) by 1 if x is positive.
    Input:
sqrt(x^2)
subs(last, sgn(x)=1)
end
    Output:
x*sgn(x)
x
end
    A second way to reduce the output expression is to assume
x>0 before evaluation. On this way, all x is affected. The first
method is local simplification, but the second method is global
simplification. If users declare the variable x is positive or
negative by

        assume(x > 0)
        assume(x < 0)

the output expression is simpler than that if users do not declare it.

    Example 7.1.2. Assume x>0 before reduce sqrt(x^2).
    Input:
assume(x >0)
sqrt(x^2)
end
    Output:
assumed
x
end

    Users can assume the variable b is even, odd, integer, real
number, positive or negative  by

        iseven(b) := 1          # assume b is even
        isodd(b) := 1           # assume b is odd
        isinteger(b) := 1       # assume b is integer
        isreal(b) := 1          # assume b is real
        isnumber(b) := 1        # assume b is number
        islist(b) := 1        # assume b is a list
        isfree(y,x) := 1    # assume y is free of x
        sgn(b) := 1             # assume b is positive
        sgn(b) := -1            # assume b is negative

    All variables are complex by default.

    Example 7.1.3.
    Input:
isreal(b) := 1
sqrt(b^2)
end
    Output:
isreal(b) := 1
abs(b)
end

    Example 7.1.4. Set f1=x^2+y^3, then put f1 into sin(f1)+
cos(2*f1) and assign the last output to f2.
    Input:
f1=x^2+y^3
sin(f1)+cos(2*f1)
f2=last
end
    Output:
f1 = x^2 + y^3
sin(x^2 + y^3) + cos(2 (x^2 + y^3))
f2 = sin(x^2 + y^3) + cos(2*(x^2 + y^3))
end

where a special word "last" stands for the last output, e.g. here
"last" is sin(x^2+y^3)+cos(2*(x^2+y^3)).


            7.2 Calculation

    SymbMath gives the exact value of calculation when the switch
numerical=off (default), or the approximate value of calculation when
the switch numerical=on or by the function num().
    SymbMath can manipulate units as well as numbers, be used as
a symbolic calculator, and do exact computation. The range of real
numbers is from -infinity to infinity, e.g. ln(-inf), exp(inf+pi*i),
etc. SymbMath contains many built-in algorithms for performing
numerical calculations when the switch numerical=on, e.g. ln(-9), i^i,
(-2.3)^(-3.2), 2^3^4^5^6^7^8^9, etc.
    Warming that SymbMath only gives a principle value if there
are multi-values, except for the function solve().

    Example 7.2.1. Find the values of sin(x) when x=pi/2
and x=90 degree.
    Input:
num(sin(pi/2))
num(sin(90*degree))
end
    Output:
1
1
end

    Example 7.2.2. Set the units converter from the minute to
the second, then calculate numbers with different units.
    Input:
minute=60*second
v=2*meter/second
t=2*minute
d0=10*meter
v*t+d0
end
    Output:
250 meter

    Example 7.2.3
    Input:
1/2+1/3
end
    Output:
5/6

    Example 7.2.4. Assign sqrt(x) to z, then evaluate z
when x=3 and y=4.
    Input:
z=sqrt(x)
subs(z, x=3)            # evaluate z by substituting x=3
x=4                     # assign 4 to x
z                       # evaluate z
end
    Output:
z = sqrt(x)
sqrt(3)
x=4
2

    Note that after assignment by x=4, x should be cleared from
memory by clear(x) before differentiation of the new function.
Otherwise the values of x still is 4 until new values assigned. If
evaluating z by the function subs(z, x=3), the variable x is
automatically cleared after evaluation, i.e. the variable x in subs()
is local variable. This rule applies to other functions.


    7.3  Defining Your Own Functions, Procedures and Rules

    Anytime you find yourself using the same expression over and
over, you should turn it into a function.
    You can define your own functions for evaluation by

        f(x_) := x^2
        f(x_) := if(isnumber(x), x^2)

On the first definition, when f() is called, it gives x^2, regardless
what x is. On the second definition, when f() is called it gives x^2
if x is a number, or left unevaluated otherwise.
    You can define the function by the immediate assignment =
or the delayed assignment :=, but you cannot define a conditional
function by the immediate assiment =. It is recommanded to define
the function by the delayed assigment :=.
    The pattern x_ should be only on the left side of the
assignment.
    Here are some sample function definitions:
           f(x_) := cos(x + pi/3)
           g(x_, y_) := x^2 - y^2
Once defined, functions can be used in expressions or in other
function definitions:
           y = f(3.2)
           z = g(4.1, -5.3)

    Example 7.3.1.
Define a new function f(x)=x^2, then evaluate it.
    Input:
f(x_) := x^2
f(-2)
f(3)
f(a)
end
    Output:
f(x_) := x^2
4
9
a^2

    Input:
f(x_) := if(isnumber(x), x^2)
f(-2)
f(3)
f(a)
end
    Output:
f(x_) := if(isnumber(x), x^2)
4
9
f(a)

    To define a conditional function by

        f(x_) := if(x>0, x^2)
        f(x_) := if(x>0, x^2, x)
        f(x_) := x*(x<0) + x^2*(x>0)

On the first definition, when f() is called it gives x^2 if x>0, or
left unevaluated otherwise. On the second definition, when f() is
called it gives x^2 if x>0, x if x<=0, or left unevaluated otherwise.
On the last definition, when f() is called, it is evaluated regardless
what x is.
    You cannot differentiate nor integrate the conditional function
if you define it by if(). But you can do so if you define it by relative
operators (e.g. the last definition).

    Input:
f(x_) := if(x>0, x^2)
f(2)
f(a)
end
    Output:
f(x_) := if(x > 0, x^2)
4
f(-2)
f(a)

    Input:
f(x_) := if(x>0, x^2, x)
f(2)
f(-2)
f(a)
end
    Output:
f(x_) := if(x > 0, x^2, x)
4
2
f(a)


    Example 7.3.2. Define a conditional function

      / x       if x < 0
f(x) =  0       if x = 0
      \ x^2     if x > 0

then evaluate f(-2), f(0), f(3).
    Input:
f(x_) := x*(x<0)+x^2*(x>0)
f(-2)
f(0)
f(3)
f(a)
d(f(t), t=3)
end
    Output:
f(x_) := x*(x < 0) + x^2*(x > 0)
-2
0
9
a*(a < 0) + a^2*(a > 0)
6

    To define a recursion function.
    Input:
factorial(n_) := if(n > 1, (n-1)*factorial(n-1))
factorial(1) := 1
end

    To define a function as a procedure.
    e.g. define a numerical integration procedure ninte() and
calculate integral of x^2 from x=1 to x=2 by call ninte().
    Input:
ninte(y_,x_,a_,b_) := block( num( dd=(b-a)/50,
    aa=a+dd,
    bb=b-dd,
    y0=subs(y, x=a),
    yn=subs(y, x=b),
    ff=(sum(y,x,aa,bb,dd)+(y0+yn)/2)*dd),
    ff )
ninte(x^2,x,1,2)
end

    Note that all variable within procedure are global. The mult-
statement should be grouped by block(). The block() output only result
of the last statement. The mult-line can be teminated by a comma (,).

    You can define transform rules. Defining rules is similar to
defining functions. In defining functions, all arguments must be simple
variables, but in defining rules, the first argument can be a
complicated expression. In this version of SymbMath the rules only have
two arguments and one pattern.
    e.g. define Laplace transform rules.
    Input:
laplace(sqrt(t_), t_) := sqrt(pi)/2/t^(3/2)
laplace(1/sqrt(t_), t_) := sqrt(pi/t)
laplace(sin(t_), t_) := 1/(t^2+1)
laplace(sin(s), s)
end
    Output:
laplace(sqrt(t_), t_) := sqrt(pi)/2/t^(3/2)
laplace(1/sqrt(t_), t_) := sqrt(pi/t)
laplace(sin(t_), t_) := 1/(t^2+1)
1/(s^2+1)
end

            7.4 Limits

    SymbMath finds real or complex limits, and discontinuity
when x approaches to x=x0 by functions

        subs(f, x=x0)
        lim(f, x=x0)

    First use subs() to find limits, if the result is undefined
(indeterminate forms, e.g. 0/0, inf/inf, 0*inf, and 0^0), then use
lim() to try again; if the result is discont, then use the one-side
limit by c+zero or c-zero.

    Example 7.4.1. Find limits of types 0/0 and inf/inf.
    Input:
p=(x^2-4)/(2*x-4)
subs(p, x=2)
lim(p, x=2)
subs(p, x=inf)
lim(p, x=inf)
end
    Output:
p = (x^2 - 4)/(2 x - 4)
undefined
2
undefined
inf

    The "discont" (discontinuity) means that the expression has
a discontinuity and only has the one-sided limit value at x=x0. Users
should use x0+zero or x0-zero to find the one-sided limit. The
f(x0+zero) or f(x0-zero) is the right-sided limit or left-sided limit
as approaching x0 from positive (+inf) or negative (-inf) direction,
respectively, i.e. limit as zero -> 0.
    SymbMath find a left-sided or right-sided limit when x
approaches to x0 from positive (+inf) or negative (-inf) direction at
discontinuity by functions

        subs(f, x=x0+zero)
        subs(f, x=x0-zero)
        lim(f, x=x0+zero)
        lim(f, x=x0-zero)

    Example 7.4.2. Find the left-sided and right-sided limits of
y=exp(1/x), (i.e. when x approaches 0 from positive and negative
directions).
    Input:
y=exp(1/x)
subs(y, x=0)
subs(y, x=0+zero)
subs(y, x=0-zero)
end
    Output:
y = exp(1/x)
discont
inf
0

    The built-in constants of inf or -inf, zero or -zero, and
discont or undefined can be used as numbers in calculation of
expressions or functions.

    Example 7.4.3.
    Input:
1/sgn(0)
1/sgn(zero)
end
    Output:
discont
1


            7.5 Differentiation

    SymbMath differentiates an expression expr by functions

        d(expr, x)
        d(expr, x, order)
        d(expr, x=x0)
        d(expr, x=x0, order)

    Example 7.5.1. Differentiate x^(x^x).
    Input:
d(x^(x^x), x)
end
    Output:
x^(x^x)*(x^(-1 + x) + x^x*ln(x)*(1 + ln(x)))

    Example 7.5.2. Differentiate the expression f=sin(x^2+y^3)+
cos(2*(x^2+y^3)) with respect to x, and with respect to both x and y.
    Input:
f=sin(x^2+y^3)+cos(2*(x^2+y^3))
d(f, x)
d(d(f, x), y)
end
    Output:
f = sin(x^2 + y^3) + cos(2*(x^2 + y^3))
2*x*cos(x^2 + y^3) - 4*x*sin(2*(x^2 + y^3))
-6*x*y^2*sin(x^2 + y^3) - 12*x*y^2*cos(2*(x^2 + y^3))

    To define a derivative.
    Input:
d(si(x_),x_) := sin(x)/x
d(si(t),t)
end
    Output:
d(si(x_),x_) := sin(x)/x
sin(t)/t

    Or the package d.sm is included before finding the derviatives.
    Example:
    Input:
include 'd.sm'
d(si(t),t)
end
    Output:
done
sin(t)/t

    If SymbMath cannot find some derivatives, you should include
the package 'd.sm' in your program or in the initial package 'init.sm'.


            7.6 Integration

    SymbMath system itself can find integrals of x^m*e^(x^n),
x^m*e^(-x^n), e^((a*x+b)^n), e^(-(a*x+b)^n), x^m*ln(x)^n, ln(a*x+b)^n,
etc., where m and n are any real number.
    The package 'inte.sm' and/or 'd.sm' should be included before
symbolic integration so that it become more powerful on integration. It
is recommended that to expand the integrand by the function expand()
and/or by setting the switch expand=on before symbolic integration.
    If symbolic integration fails, the user can define the simple
integral or derivative, then evaluates the integration again (see
7.13 Learning from User).
    If the user wants numerical integration by ninte(), the
package 'NInte.sm' should be included before doing numerical
integration (see 8.5 Numeric Integration Package).


        7.6.1 Indefinite Integration

    SymbMath finds indefinite integrals by functions

        inte(expr, x)
Note that the arbitrary constant is not represented.

    Example 7.6.1. Find indefinite integrals.
    Input:
inte(sinh(x)*e^sinh(x)*cosh(x), x)
inte(sinh(x)^2*cosh(x), x)
inte(x^1.5*exp(x), x)
end
    Output:
-e^sinh(x) + sinh(x)*e^sinh(x)
 (1/3)*sinh(x)^3
ei(1.5, x)

    Example 7.6.2. Find indefinite double integrals.
    Input:
inte(inte(x*y, x), y)
end
    Output:
 (1/4)*x^2*y^2

    Example 7.6.3. Find the line integral.
    Input:
x=2*t
y=3*t
z=5*t
u=x+y
v=x-y
w=x+y+z
inte((u*d(u,t)+v*d(v,t)+w*d(w,t), t)
end
    Output:
63*t^2

    Example 7.6.4. Find the integral of sin(x)/x by the mean of
the 'inte.sm' package.
    Input:
include('inte.sm')
inte(sin(x)/x, x)
end
    Output:
done
si(x)

    Defining an integral is similar to defining a rule.
    Example 7.6.5
    Input:
inte(sin(x_)/x_, x_) := si(x)
inte(sin(t)/t, t)
end
    Output:
inte(sin(x_)/x_, x_) := si(x)
si(t)

        7.6.2 Definite Integration

    SymbMath finds definite integrals by functions
        inte(expr, x, xmin, xmax)
    Example 7.6.6. Find the definite integral of y=exp(1-x) with
respect to x taken from x=0 to x=infinity.
    Input:
inte(exp(1-x), x from 0 to inf)
end
    Output:
e

    Example 7.6.7. Discontinuous integration of 1/x^2 and 1/x^3
with discontinuity at x=0.
    Input:
inte(1/x^2, x from -1 to 2)
inte(1/x^3, x from -1 to 1)
end
    Output:
inf
0


            7.7 Equations
            7.7.1 Algebraic Equations

    The equations can be operated (e.g. +, -, *, /, ^, expand(),
diff(), inte()). The operation is done on both sides of the equation,
as by hand. SymbMath is able to find roots of a polynomial, algebraic
equations, systems of equations, differential and integral equations.

    Example 7.7.1. Solve an equation
sqrt(x+2*k) - sqrt(x-k) === sqrt(k),
then check the solution by substituting the root into the equation.
    Input:
eq1=(sqrt(x + 2*k) - sqrt(x - k) === sqrt(k))
eq1^2
expand(last)
last-k-2*x
last/(-2)
last^2
expand(last)
last-x^2+2*k^2
last/k
subs(eq1, x=right(last))
end
    Output:
eq1=(sqrt(x + 2*k) - sqrt(x - k) === sqrt(k))
((2*k + x)^0.5 - ((-k) + x)^0.5)^2 === k
2*x + k + (-2)*(2*k + x)^0.5*((-k) + x)^0.5 === k
(-2)*(2*k + x)^0.5*((-k) + x)^0.5 === (-2)*x
(2*k + x)^0.5*((-k) + x)^0.5 === x
(2*k + x)*((-k) + x) === x^2
(-2)*k^2 + k*x + x^2 === x^2
k*x === 2*k^2
x === 2*k
k^0.5 === k^0.5

    SymbMath can solve many algebraic equations step by step, as
by hand. This method is useful on teaching, e.g. to show students how
to solve equations.
    SymbMath has the built-in function

        solve(expr1 === expr2, x)
        solve([expr1 === expr2, expr3 === expr4], [x, y])

to solve a polynomial and systems of linear equations on one step.
It is recommended to set the switch expand=on when solve the
complicated equations. All of the real and complex roots of the
equation will be found by solve(). The function solve() outputs a
list of roots when there are multi-roots. Users can get one of roots
from the list, (see 7.9 List, Array, Vector and Matrices).
    Users can get the left side of the equation by

        left(left_side === right_side)

or the right side by

        right(left_side === right_side)

    Example 7.7.2. Solve a+b*x+x^2 === 0, save the root to x.
    Input:
eq1 = (a+b*x+x^2===0)
solve(eq1, x)
x=right(last)
x[1]
x[2]
end
    Output:
eq1 = (a+b*x+x^2 === 0)
x === [-b/2 + sqrt((b/2)^2 - a),  -b/2 - sqrt((b/2)^2 - a)]
x = [-b/2 + sqrt((b/2)^2 - a),  -b/2 - sqrt((b/2)^2 - a)]
-b/2 + sqrt((b/2)^2 - a)
-b/2 - sqrt((b/2)^2 - a)


    Example 7.7.3. Solve x^3+x^2+x+5 === 2*x+6.
    Input:
num(solve(x^3+x^2+x+5 === 2*x+6, x))
end
    Output:
x === [1, -1, -1]

    Function solve() not only solve for a simple variable x
but also for an unknown function, e.g. ln(x).

    Example 7.7.4. Solve the equation for ln(x).
    Input:
solve(ln(x)^2+5*ln(x) === -6, ln(x))
exp(last)
end
    Output:
ln(x) === [-2, -3]
x === [exp(-2), exp(-3)]

    Example 7.7.5. Rearrange the equations.
    Input:
f = [x+y === 3+a+b, x-y === 1+a-b]
solve(f, [x,y])
solve(f, [a,b])
solve(f, [a,y])
solve(f, [x,b])
end
    Output:
f = [x + y === 3 + a + b, x - y === 1 + a - b]
[x === -1/2*(-4 - 2 a), y === -1/2*(-2 - 2 b)]
[a === -1/2*(4 - 2 x), b === -1/2*(2 - 2 y)]
[b === -1/2*(2 - 2 y), x === -1/2*(-4 - 2 a)]
[a === 1/2*(-4 + 2 x), y === 1/2*(2 + 2 b)]



        7.7.2 Differential Equations

    SymbMath can solve the variables separable differential
equations:
        y(x)*d(y(x), x) === f(x)
by integration inte().
    Example: 7.7.2.1
    Input:
y(x)*d(y(x), x) === sin(x)
inte(last, x)
end
    Output:
y(x)*d(y(x), x) === sin(x)
1/2*y(x)^2 === constant - cos(x)

    The function

        dsolve(d(y)/d(x) === f(x,y), y)

can solve the first order variables separable and linear differential
equations

        d(y)/d(x) === h(x)
        d(y)/d(x) === g(y)
        d(y)/d(x) === g(y)*x
        d(y)/d(x) === g(x)*y
        d(y)/d(x) === g(x)*y+h(x)

on one step. Notice that d(y)/d(x) must be alone on the left hand
side of the equation. It is recommended to set the switch
expand=on and/or to include 'inte.sm' when solving the complicated
differential equations.

    Example 7.7.2.2 Solve d(y)/d(x)=== -sinh(x)*y+sinh(x)*cosh(x).
    Input:
dsolve(d(y)/d(x) === -sinh(x)*y+sinh(x)*cosh(x), y)
end
    Output:
y === (constant + exp(cosh(x)) - cosh(x)*exp(cosh(x)))*exp(-cosh(x))

    Example 7.7.2.3 Solve differential equations. If the result is
a polynomial, then rearrange the equation by solve().
    Input:
dsolve(d(y)/d(x) === x/(2+y), y)
solve(last, y)
end
    Output:
2*y + 1/2*y^2 === constant + x^2
y === [-2 + sqrt(4 - 2*(-constant - x^2)), -2 - sqrt(4 - 2*(-constant - x^2))]

    Example 7.7.2.4. Solve differential equations.
    Input:
dsolve(d(y)/d(x) === x*exp(y), y)
dsolve(d(y)/d(x) === y^2+5*y+6, y)
dsolve(d(y)/d(x) === y/x, y)
dsolve(d(y)/d(x) === y*x+1, y)
end
    Output:
-exp(-y) === constant + x^2
ln((4 + 2*y)/(6 + 2*y)) === constant + x
y === constant*x*sgn(x)
y === exp(1/2*x^2)*(constant + sqrt(1/2)*sqrt(pi)*erf(sqrt(1/2)*x))


          7.8 Sums and Products

    Users can compute partial, finite or infinite sums and
products. Sums and products can be differentiated and integrated. You
construct functions like Taylor polynomials or finite Fourier series.
The procedure is the same for sums as products so all examples will
be restricted to sums.  The general formats for these functions are:

        sum(expr, x from xmin to xmax step dx)
        prod(expr, x from xmin to xmax step dx)

The expression expr is evaluated at xmin, xmin+dx, ...  up to the last
entry in the series not greater than xmax, and the resulting values
are added or multiplied.  The part "step dx" is optional and defaults
to 1.  The values of xmin, xmax and dx can be any real number.

Here are some examples:
     sum(j, j from 1 to 10)
for 1 + 2 + .. + 10.
      sum(3^j, j from 0 to 10 step 2)
for 1 + 3^2 + ... + 3^10.
      Here are some sample Taylor polynomials:
    sum(x^j/j!, j from 0 to n)
for exp(x).
    sum((-1)^j*x^(2*j+1)/(2*j+1)!, j from 0 to n)
for sin(x) of degree 2*n+2.

    The package SUM.sm should be included before computing
partial and infinite sums, and the package PRODUCT.sm should be
included before computing partial and infinite products (see 8.8
Infinite Sum Package).
    Remember, the keywords "from", "to" and "step" can be
replaced by the comma (,).


        7.9 Lists, Arrays, Vectors and Matrices

    SymbMath can construct lists of arbitrary length, and the
entries in the lists can be of any type of value whatsoever:
constants, expressions with undefined variables, or even other lists.
Lists are another kind of value in SymbMath, and they can be assigned
to variables just like simple values.  (Since variables in SymbMath
language are untyped, you can assign any value to any variable.).


          7.9.1     Entering Lists

    To define a list, put its elements between square brackets:

         a = [1,2,3]
         b = [f(2), g(1), h(1)]      # assumes f,g,h defined
         c = [[1,2],3,[4,5]]

    A function can have a list for its value:
         f(x) = [1,x,x^2]
    You can define lists another way, with the list command:

         list(f(x), x from xmin to xmax step dx)

    This is similar to the sum command,  but the result is a list:
           [f(xmin), f(xmin+dx), ..., f(xmin+x*dx), ...]
which continues until the last value of xmin + x*dx  <= xmax.
    Try

         a = list(j^2, j from 0 to 10 step 1)
         f(x) = list(x^j, j from 0 to 6 step 1)
         b = f(-2)

      The third way to construct a list is to transform the sum to
the list.

    Input:
y1=[a,b,c]
sum(y1)+d
list(last)
end
    Output:
y1 = [a, b, c]
a + b + c + d
[a, b, c, d]

      This is how you extend an existing list to include a new
element.


          7.9.2     Accessing Lists

    To find the value of the j-th element in a list x, use the
formula  x[j]. The first element of x is always  x[1].  If the x[j]
is itself a list, then its k-th element is accessed by repeating the
similar step. Try:

    Input:
x = [[1,2],3,[4,5]]
x[1]
x[2]
end
    Output:
x = [[1, 2], 3, [4, 5]]
[1, 2]
3

    An entire sub-list of a list x  can be accessed with the
command x[j], which is the list:
          [x[j], x[j+1], ... ]


          7.9.3     Modifying Lists

    The function subs() substitutes the value of the element in
the list, as in the variables. e.g.

    Input:
l=[a,b,c]
subs(l, a=a0)
end
    Output:
l = [a, b, c]
[a0, b, c]

    But you can't use this form unless the element of the list is
already defined.


          7.9.4     List Operations

    Lists can be added, subtracted, multiplied, and divided by
other lists or by constants.  When two lists are combined, they are
combined term-by-term, and the combination stops when the shortest
list is exhausted.  When a scalar is combined with a list, it is
combined with each element of the list.  Try:

     a = [1,2,3]
     b = [4,5,6]
     a + b
     a / b
     3 * a
     b - 4

    Example 7.9.1. Two lists are added.
    Input:
list1=[a1,a2,a3]
list2=[b1,b2,b3]
list1+list2
last[1]
end
    Output:
list1 = [a1,a2,a3]
list2 = [b1,b2,b3]
[a1 + b1, a2 + b2, a3 + b3]
a1 + b1

    If L is a list, then  f(L) results in a list of the values,
even though f() is the differentiation or integration function d() or
inte(). Try list with:
    Input:
sqrt([a, b, c])
d([x, x^2, x^3], x)
end

    You can find the number of elements in a list with:
        length(a)
    If you use a list as the value of a variable in a user-
defined function, SymbMath will try to use the list in the
calculation.
    You can sum all the elements in a list x with the user-
defined function:

               listsum(x)
in the statistics package 'stat.sm'.
    Example:

            x=[1,2,3]
            listsum(x^2)

    This functions takes the sum of the squares of the elements of
the list x.
    See the statistics package 'stat.sm' for other statistics
operations (e.g. average, max, min).
    See the list plot package 'listplot.sm' for plotting a list.

          7.9.5     Vector Operations

    SymbMath uses lists to represent vectors, and lists of lists to
represent matrices.
    Vectors and matrices can be operated by "+" and "-" with vectors
and matrixes, by "*" and "/" with a scalar, and by subs(), diff() and
inte(). These operations are on each element, as in lists and arrays.
    You can use lists as vectors, adding them and multiplying them
by scalars. For example, the dot product of two vectors of a and b is:

               dot = listsum(a*b)

    You can even make this into a function:

               Dot(x_, y_) = listsum(x*y)
               a = [2,3,5]
               b = [4,3,2]
               Dot(a,b)

    How about the cross product:

Cross(a,b) = [a[2]*b[3]-b[2]*a[3],a[3]*b[1]-b[3]*a[1],a[1]*b[2]-b[1]*a[2]]


        7.10    Complex Analysis

    The complex numbers, complex infinity and some complex
functions can be calculated, and the complex expressions can be
differentiated and integrated.
    Sign of complex numbers is defined as

     /  1    if re(z)>0, or re(z)=0 and im(z)>0, (i.e. z>0),
sgn(z) =    0    if z=0,
     \ -1    otherwise, (i.e. z<0).

    Example 7.10.1.
    Input:
sgn(1+i)
sgn(1-i)
sgn(-1-i)
end
    Output:
1
1
-1

    Input:
exp(inf+pi*i)
ln(last)
end
    Output:
-inf
inf + pi*i

    Input:
inte(1/x, x from i to 2*i)
end
    Output:
ln(2)

        10.11     Tables of Function Values

    If you want to look at a table of values for a formula, you
can use the table command:

      table(f(x), x)
      table(f(x), x from xmin to xmax)
      table(f(x), x from xmin to xmax step dx)

It causes a table of values for f(x) to be displayed with x=xmin,
xmin+dx, ..., xmax.  If xmin, xmax, and step omit, then xmin=-5, xmax=5,
and dx=1 for default. You can specify a function to be in table(),
    Example. Make a table of x^2.
    Input:
table(x^2, x)
end
    Output:
-5,    25
-4,    16
-3,    9
-2,    4
:       :
:       :

    Its output can be written into a disk file for interfacing
with other software (e.g. the numeric computation software).


        7.12 Transformation and Piece of Expressions

    Different types of data may be transformed each other.
    The complex number z is transformed to the real number x by

        re(z), im(z), abs(z), sgn(z).

    The functions

        left(x===a) and right(x===a)

pick up one side of the equation.
    A list can be transformed to a table with the table command if
the elements of the list are the numbers. e.g.
    Input:
x=[5,4,3,2,1]
table(x[j], j from 1 to 4 step 1)
end
    Output:
1,      5
2,      4
3,      3
4,      2

    A piece of an expression can be picked up. e.g.
    Input:
y=a+b*x+c*x^2+d*x^3
coef(y, x)
coef(y, x^2)*x^2
list(y)
last[3]
end
    Output:
y = a + b*x + c*x^2 + d*x^3
b
c*x^2
[a, b*x, c*x^2, d*x^3]
c*x^2


    Table 7.2      Expand and Factor
---------------------------------------------------------------------
Expand                  Factor

(a+b)^2                 a^2+2*a*b+b^2
(a+b)^n                 a^n+ ...... +b^n        n is positive integer
a*(b+c)                 a*b + a*c
---------------------------------------------------------------------
    a+b can be many terms or a-b.



            7.13 Learning from Users

    One of the most important feature of SymbMath is its ability
to deduce and expand its knowledge. If the user provides it with the
necessary facts, SymbMath can solve many problems that it was unable
to solve before. The followings are several ways in which SymbMath is
able to learn from the user.


7.13.1 Learning indefinite and definite integrals from a derivative

    Finding derivatives is much easier than finding integrals.
Therefore, you can find the integrals of a function from the
derivative of that function.
    If the user provides the derivative of a known or unknown
function, SymbMath can deduce the indefinite and definite integrals of
that function. If the function is not a simple function, the user only
need to provide the derivative of its simple function. For example,
the user wants to evaluate the integral of f(a*x+b), the user only need
to provide d(f(x), x).

Example 7.13.1.1 :

    If a user know a derivative of an function f(x) (f(x) is a
known or unknown function), SymbMath can learn the integrals of that
function from its derivative.
    First check SymbMath wether or not it had already known
indefinite and definite integrals of an unknown function f(x).

In the input window type :

inte(f(x), x)
inte(f(x), x, 1, 2)
end

In the output window you'll see :

inte(f(x), x)
inte(f(x), x, 1, 2)
end

As the output windows displayed only what was typed in the input
windows without any computed results, imply that SymbMath has no
knowlege of the indefinite and definite integrals of the functions
in question. Now teach SymbMath the derivative of f(x) on the first
line, and then run the program again.

    Input:
d(f(x_), x_) = exp(x)/x
inte(f(x), x)
inte(f(x), x, 1, 2)
end
    Output:
d(f(x_), x_) = e^x/x
x*f(x) - e^x
e - f(1) + 2*f(2) - e^2

As demonstrated, the user only supplied the derivative of the function
and in exchange SymbMath logically deduced its integral.
    An other example is
    Input:
d(f(x_) ,x_) := 1/sqrt(1-x^2)
inte(f(x), x)
inte(k*f(a*x+b), x)
inte(x*f(a*x^2+b), x)
end
    Output:
d(f(x_), x_) := 1/sqrt(1 - x^2)
sqrt(1 - x^2) + x*f(x)
k*(sqrt(1 - (b + a*x)^2) + (b + a*x)*f(b + a*x))/a
sqrt(1-(a*x^2 + b)^2) + (a*x^2 + b)*f(a*x^2 + b)

    The derivative of the function that user supplied can be another
derivative or integral.

Example 7.13.1.2 :

    Input window :
d(f(x_), x_) = inte(sin(x), x)
inte(f(x), x)
end
    Output window :
d(f(x_), x_) =  - cos(x)
cos(x)


7.13.2 Learning complicated indefinite integrals from a simple
    indefinite integral

The user supplies a simple indefinite integral, and in return, SymbMath
will perform the related complicated integrals.

Example 7.13.2.1 :

Check whether SymbMath already knowns the following integrals or not.

    Input window :
inte(tan(x)^2, x)
inte((2*tan(x)^2+x), x)
inte(inte(tan(x)^2+y), x), y)
end
    Output window :
inte(tan(x)^2, x)
inte((2*tan(x)^2+x), x)
inte(inte(tan(x)^2+y), x), y)

Supply like in the previous examples the following in information
integral of tan (x) is tan (x) - x; then ask the indefinite integral
of 2*tan(x)^2+x, and a double indefinite integral of 2*tan (x)^2 + x,
and a double indefinite integral of respect to both x and y. Change
the first line, and then the program again.

    Input window :
inte(tan(x)^2, x) = tan(x) - x
inte((2*tan(x)^2+x), x)
inte(inte(tan(x)^2+y), x), y)
end
    Output window :
inte(tan(x)^2, x) = tan(x) - x
2*(tan(x) - x) + 1/2*x^2
tan(x)*y - x*y + x*y^2

    The user can also ask SymbMath to perform the following
integrals:
inte(inte(tan(x)^2+y^2), x), y),
inte(inte(tan(x)^2*y), x), y),
inte(x*tan(x)^2, x),
triple integral of tan(x)^2-y+z, or others.


    7.13.3 Learning definite integral from indefinite integral

    The user continues to ask indefinite integral.

    Input window :
inte(inte(tan(x)^2+y, x from 0 to 1), y from 0 to 2)
end
    Output:
2 tan(1)


    7.13.4 Learning complicated derivative from simple derivative

SymbMath can learn complicated derivatives from a simple derivative
even thought the function to be differentiated is any function, not
standard function.

Example 7.13.4.1 :

Differentiate f(x^2)^6, where f(x) is an known function, instead of
a standard function.

    Input:
d(f(x^2)^6, x)
end
    Output:
12*x*f(x^2)^5*d(f(x^2), x^2)

Output is only the part derivative. d(f(x^2), x^2) in the output
suggest that the user should teach SymbMath d(f(x_), x_). e.g. the
derivative of f(x) is another unknown function df(x), i.e.
d(f(x_), x_) = df(x), do so and run it again.

    Input:
d(f(x_), x_) = df(x)
d(f(x^2)^6, x)
end
    Output:
d(f(x_), x_) = df(x)
12*x*f(x^2)^5*df(x^2)

This time we get complete derivative.

    7.13.5 Learning integration from algebra

If the user shows SymbMath algebra, SymbMath can learn integrals.


Example 7.13.5.1 :

Input sin(x)^2=1/2-1/2*cos(2*x), then ask for the integral of sin(x)^2.

    Input window :
sin(x)^2=1/2-1/2*cos(2*x)
inte(sin(x)^2, x)
end
    Output window :
sin(x)^2 = 1/2 - 1/2*cos(2*x)
1/2*x - 1/4*sin(2*x)

SymbMath is very flexible, It learned to solve these problems, even though
the types of problems are different, e.g. learning integrals from
derivatives or algebra.


    7.13.6 Learning complicated algebra from simple algebra

SymbMath has the ability to learn complicated algebra from simple algebra.

Example F1 :

Transform sin(x)/cos(x) into tan(x) in an expression.

input window:

sin(x)/cos(x) = tan(x)
x+sin(x)/cos(x)+a

Output window :
sin(x)/cos(x) = tan(x)
a + x + tan(x)


    The difference between learning and programming is as follows :
the learning process of SymbMath is very similar to the way human
beings learn, and that is accomplished by knowing certain rule that
can be applied to several problems. Programming is diffrent in the way
that the programmer have to accomplish many tasks before he can begin
to solve a problem. First, the programmer defines many subroutines for
the individual integrands (e.g. tan(x)^2, tan(x)^2+y^2, 2*tan(x)^2+x,
x*tan(x)^2, etc.), and for individual integrals (e.g. the indefinite
integral, definite integral, the indefinite double integrals,
indefinite triple integrals, definite double integrals, definite
triple integrals, etc.), second, write many lines of program for the
individual subroutines, (i.e. to tell the computer how to calculate
these integrals), third, load these subroutines, finally, call these
subroutines. That is precisely what SymbMath do not ask the user to do.
    In one word, programming means that programmers must
provide step-by-step procedures telling the computer how to solve
each problems. By contrast, learning means that users need only supply
the necessary facts, SymbMath will determine how to go about
solutions.
    If the learning is saved into the initial file init.sm, the
learning  will become the knowledge of the SymbMath system, users need
not to teach SymbMath again when users run SymbMath next time.


            8. Packages
    A package is a SymbMath program file for special use.
    In order to expand the special ability of SymbMath, some
packages should be included in the user program by
        include('filename')
    The filename is any MS-DOS filename.
    It is recommended that the filename is the same as the function
name in the program file, and the extension of the filename is .sm in
order to remember easily the name of the package. e.g. inte.sm is the
filename of the integral package as the name of integral function is
inte().
    After including the package, users can call the functions in
the package from their program.
    The include command must be in a single line anywhere.
    Many packages can be included at a time, however, all names of
the variables are public and name conflicts must be avoided.
    Alternately, a part of the package file rather than the whole
package file can be copied into the Edit window by pressing <F7>
(external copy) in order to save memory space.
    There are many packages. The following are some of them.

        Table 8.1      Packages
------------------------------------------------------------------------
File Name               Package Function

init.sm                 initial package when running the program in
            the Edit window. It contains switches on the
            default status.
plot.sm            plotting functions.
listplot.sm        plotting a list of data.
d.sm                    derivatives.
inte.sm                 integrals.

--------------- shareware version only has above packages --------------

units.sm                an units conversion.
trig.sm                 reduction of trig functions.
hyperbol.sm             reduction of hyerbolic functions.
dt.sm                   total derivatives.
NInte.sm                numeric integration.
NSolve.sm               numeric solver of equation.
ODE.sm            ordinary differential equation.
gamma.sm                gamma function.
ei.sm            exponential integral function.
series.sm        Taylor series.
sum.sm            symbolic sum.
InfSum.sm        infinite sum.
chemical.sm             the atomic weight of chemical elements.
inorgani.sm             inorganic chemical reactions.
-----------------------------------------------------------------------


        8.1 Initial Package
---------------------------------------------------------------------
#    The initial package init.sm
output=off
numerical=off
expand=off
sum(y_,x_,a_,b_) := sum(y,x,a,b,1)
prod(y_,x_,a_,b_) := prod(y,x,a,b,1)
list(y_,x_,a_,b_) := list(y,x,a,b,1)
table(y_,x_,a_,b_) := table(y,x,a,b,1)
table(y_,x_) := table(y,x,-5,5,1)
degree=pi/180
0!=1
inf!=inf
sgn(0)=0
sgn(zero)=1
sgn(e)=1
sgn(pi)=1
sgn(inf)=1
abs(zero)=zero
abs(e)=e
abs(pi)=pi
abs(inf)=inf
ln(0)=-inf
ln(inf)=inf
 include('plot.sm')
# include('d.sm')
block(output=basic,null)
end
---------------------------------------------------------------------

    When running the program in the Edit window, SymbMath
automatically first includes (or runs) the initial package 'init.sm'.
The commands in the 'init.sm' package seems the SymbMath system
commands. You can include other packages (e.g. d.sm) in the initial
package 'init.sm' so the derivatives table in the package 'd.sm' seems
to be in SymbMath system. Doing this is by changing the comment
statement in the last third line
        # include('d.sm')
to include statement
        include('d.sm')


        8.2 Derivative Package
       The package 'd.sm' is extention of the d(f(x),x) function.

        8.3 Total Derivative Package
       The package 'dt.sm' is for the total derivative function
dt(f(x,y),x,y).
    Example:
    Input:
include('dt.sm')
dt(x*y, x, y)
end
    Output:
y*d(x) + x*d(y)

        8.4 Integral Package
       The package 'inte.sm' is the extention of the function inte(f(x),x).

        8.5 Numeric Integration Package
    The function
        ninte(y, x from x0 to x1)
in the package 'NInte.sm' can do numerical integration.
    Example 8.5.1. Compare numerical and symbolic integrations
for 4/(x^2+1) with respect to x taken from x=0 to x=1.
    Input:
include('NInte.sm')
ninte(4/(x^2+1), x from 0 to 1)
num(inte(4/(x^2+1), x from 0 to 1))
end
    Output:
done
3.1415
3.1416

        8.6 Numerical Solving
    The function
        nsolve( f(x) === x, x, x0)
in the package 'NSolve.sm' can numerically solve an algebraic equation
with an initial value x0.
    Example 8.6.1. solve the equation sin(x) === 1 with x0=1.
    Input:
include('NSolve.sm')
nsolve( sin(x) === 1, x, 1)
end
    Output:
done
1.5364150214

        8.7 Series Package
    The series package 'Series.sm' has a function series() for the
Taylor series at x=0:
    You can call the function
        series(f(x), x, order)
        series(f(x), x)
to find the Taylor series. The arguement (order) is optional and
defaults to 6.
    Example 8.7.1. Find the power series expansion for cos(x)
about the point x=0 to order x^4.
    Input:
include('series.sm')
series(cos(x), x, 4)
end
    Output:
1 - 1/2*x^2 + 1/24*x^4

        8.8 Infinite Sum Package
    The infinite sum package 'InfSum.sm' has a function
        infsum(f(x),x)
to find the infinite sum, sum(f(x), x from 0 to inf).
    Example:
    Input:
include('infsum.sm')
infsum(1/n!, n)
end
    Output:
e

        8.9 Chemical Calculation Package
    SymbMath recognizes 100 symbols of chemical elements and
converts them into their atomic weights after the chemical package of
'Chemical.sm' is included.

    Example 8.9.1. Calculate the weight percentage of the
element C in the molecule CH4.
    Input:
include('chemical.sm')
numerical=on
C/(C+H*4)*100*%
end
    Output:
done
numerical = on
74.868 %

    Example 8.9.2. Calculate the molar concentration of CuO when
3 gram of CuO is in 0.5 litre of a solution.
    Input:
include('chemical.sm')
numerical=on
g=1/(Cu+O)*mol
3*g/(0.5*l)
end
    Output:
0.07543*mol/l

        8.10 Chemical Reactions Package
    SymbMath can provide the answers for inorganic chemical
reactions after the inorganic reaction package "Inorgani.sm" is included.

    Example 8.10.1.  What are the products when the reactors are
HCl + NaOH ?
    Input:
include('inorgani.sm')
HCl+NaOH
end
    Output:
H2O + NaCl


        8.11 Plot Function Package
---------------------------------------------------------------------
# The plot function package plot.sm
# to plot a function.
# plot(x^2, x)        # plot the function of x^2
# plot(x^2, x,-6,6)    # plot the function of x^2 from x=-6 to x=6
# end

output=off
plot(y_,x_,x1_,x2_,dx_) := block(openfile('symbmath.out'),
    table(y,x,x1,x2,dx),
    closefile('symbmath.out'),
    system(plotdata))
plot(y_,x_,x1_,x2_) := plot(y,x,x1,x2,(x2-x1)*0.5)
plot(y_,x_) := plot(y,x,-5,5,0.5)
output=on
end
----------------------------------------------------------------------

    If SymbMath is interfaced with the software PlotData, SymbMath
produces the data table of functions, and PlotData plots from the
table. So SymbMath seems to plot the function. This interface can be
used to solve equations graphically.
    In the plot package 'plot.sm' the funcion
        plot(y, x, x1, x2, dx)
first open a file 'SymbMath.Out' for writing, then write the data table
of the function y into the file 'SymbMath.out', then close the file,
and finally automatically call the software PlotData to plot. After it
exits from PlotData, it automatically return to SymbMath.
    The functions
        plot(y, x)
        plot(y, x from xmin to xmax)
call the function plot(y,x,x1,x2,dx)
    e.g. plot x^2.
    Input:
plot(x^2, x)
end

in the software PlotData, just select the option to read the file
'SymbMath.out' and to plot. PlotData read the data in the SymbMath
format without any modification (and in many data format).
    In PlotData,
in the main menu:
1 <Enter>
in the read menu:
2 <Enter>
<Enter>
in the main menu:
2 <Enter>
in the graph menu:
1 <Enter>

where <Enter> is the <Enter> key.
Refer to PlotData for detail.
    Note that if your monitor is Hercules, you must load the
MSHERC.COM program as a TRS program before you run PlotData.
Otherwise you will get Error when you plot.

        8.12 List Plot Package
----------------------------------------------------------------------
# the list plot package 'listplot.sm'
# to plot a list of data.
# listplot([1,2,3,4,5,4,3,2,1])
# end

output=off
listplot(y_) := if(islist(y), block(yy=y, length=length(yy),
    openfile('symbmath.out'),
    table(yy[xx],xx,1,length),
    closefile('symbmath.out'),
    system(plotdata) ))
output=on
end
---------------------------------------------------------------------

        15. Interface with Other Software
    Interface with other software, (e.g. CurFit, Lotus 123) is
similar to interface with the software PlotData in the plot package
'plot.sm'.

                    (continued on the file SymbMath.DO3)