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13.3 Glossary
--------------------------------------------------------------------
* abs
abs(x) is the absolute value function of x. x can be complex numbers.
Absolute value of the complex number x is defined by
sqrt(re(x)^2+im(x)^2). The results are in the range 0 to inf.
e.g. abs(-1) gives 1, abs(-i) gives -1.
See also: sgn, re, im.
* acos
acos(x) is the arc cosine function of x. The inverse function of
cos(x). The result is given in radians.
See also: cos.
* acosh
acosh(x) is the inverse hyerbolic cosine function of x.
The inverse function of cosh(x).
See also: cosh.
* acot
acot(x) is the arc cotangent function of x. The inverse function
of cot(x). The result is given in radians. acot(x)=atan(1/x).
See also: cot, atan.
* acoth
acoth(x) is the inverse hyperbolic cotangent function of x. The
inverse function of coth(x). acoth(x)=atanh(1/x).
See also: coth, atanh.
* acsc
acsc(x) is the arc cosecant function of x. The inverse function of
csc(x). The result is in radians. acsc(x)=asin(1/x).
See also: csc, asin.
* acsch
acsch(x) is the inveres hyperbolic cosecant function of x. The
inverse function of csch(x). acsch(x)=asinh(1/x).
See also: csch, asinh.
* arg
arg(x) gives the argument of x. It gives the phase angle of x in
radians. Its result is in the range > -pi to <= pi. arg(x)=
atan2(re(x),im(x)). x can be complex number. For real x, arg(x)=0.
See also: abs, sgn, atan2.
* asec
asec(x) is the arc secant function of x. The inverse function of
sec(x). The result is given in radians. asec(x)=acos(1/x).
See also: sec, acos.
* asech
asech(x) is the inverse hyperbolic secant function of x. The inverse
function of sech(x). asech(x)=acosh(1/x).
See also: sech, acosh.
* asin
asin(x) is the arc sine function of x. The inverse function of
sin(x). The result is given in radians.
See also: sin.
* asinh
asinh(x) is the inverse hyperbolic sine function of x. The inverse
function of sinh(x).
See also: sinh.
* assume
assume(x > 1), or assume(x < 1), assumes a variable x > 1, or x < 1.
All variables are assumed as complex by default.
e.g. assume(x>0), assume(x<0), or isreal(x) := 1.
See aslo: sgn, isodd, iseven, isinteger, isreal, isnumber.
* assumed
assumed points out that the variable has been assumed.
See also: assume.
* atan
atan(x) is the arc tangent function of x. The inverse function of
tan(x). The result is given in radians.
See also: tan.
* atan2
atan2(x,y) returns the radian angle of (x,y).
atan2(x,y) = sgn(y)*pi/2 if x=0
= atan(y/x) if x>0
= atan(y/x)+pi if x<0, y>=0
= atan(y/x)-pi if x<0, y<0 .
Thus atan2(x,y) takes its value larger than -pi less or equal to pi.
* atanh
atanh(x) is the inverse hyperbolic tangent function of x. The inverse
function of tanh(x).
See also: tanh.
* basic
basic is a value of the switch output. It sets output in BASIC format.
e.g. output=basic.
See also: output, on, off, fortran, prolog.
* block
block(a,b,c) return only the last argument.
e.g. block(a,b,c) gives c.
* clear
clear(expr) clears values and definitions for the variable, function or
expression expr from memory.
e.g. clear(p), clear(f(x)).
* cleared
It says that the variable, function or expression has been cleared
from memory.
See also: clear.
* closefile
closefile('filename') closes the file 'filename' and return the output
to screen. The filename is any MS-DOS file name.
See also: openfile, include.
* coef
coef(expr,form) gives the coefficient of form in the polynomial expr.
It picks only terms that contain the particular form specified. x is
not considered part of x^6.
e.g. coef(2*x^6+x+4, x^6) gives 2, coef(2*x^6+x+4, x) gives 1.
* complex_inf
The complex infinite, both real and imaginary parts of complex numbers
are infinity, as the built-in constant. complex_inf=inf+inf*i,
inf-inf*i, -inf+inf*i, or -inf-inf*i.
See also: inf.
* constant
The indefinite integral constant.
* cos
cos(x) is the cosine function of x. The angle x is measured in radians
(multiply by degree to convert from degrees). x can be complex numbers.
See also: acos, sec.
* cosh
cosh(x) is the hyerbolic cosine function of x.
cosh(x)=(exp(x)+exp(-x))/2. x can be complex numbers.
See also: acosh.
* cot
cot(x) is the cotangent function of x. The angle x is measured in radians.
(multiply by degree to convert from degrees). cot(x)=1/tan(x).
See also: acot, tan.
* coth
coth(x) is the hyerbolic cotangent function of x. coth(x)=1/tanh(x).
See also: acoth, tanh.
* csc
csc (x) is the cosecant function of x. The angle x is measured in
radians (multiply by degree to convert from degrees). csc(x)=1/sin(x)
x can be complex numbers.
See also: acsc, sin.
* csch
csch(x) is the hyperbolic cosecant function of x. csch(x)=1/sinh(x).
x can be complex numbers.
See also: acsch, sinh.
* d
d() gives the partial derivative.
d(f, x) differentiate y with respect to x.
e.g. d(x^2/d(x)) gives 2*x.
d(f, x, n) gives the n-th order derivative of f with respect
to an undefined variable x.
d(f, x=c) gives the derivative of f with respect to an
undefined variable x at x=c.
d(f, x=c, n) gives the n-th order derivative of f with respect
to an undefined variable x at x=c.
d(y) implicit differentiation, used in differential
equations, e.g. x*d(x)+y*d(y) === 0.
* degree
degree gives the number of radians in one degree. degree=pi/180.
You can multipy by degree to convert from degree to radians.
e.g. 45*degree, sin(45*degree).
See also: pi.
* discont
The discontinuity. If the function value is discont, the function has
a discontinuity and only has the one-sided limit at x=c. Users should
evaluate its left-sided limit or right-sided limit by x=c-zero or
x=c+zero.
* do
do(expr, x,xmin,xmax,dx) evaluates expr with the x looping from xmin
to xmax on step dx.
e.g. x=1, do(x=x+1, j,1,5,1) gives x=5.
* done
It indicates that the action has been done.
* dsolve
dsolve(d(y)/d(x)===f(x,y), y) solves the first order variables
separable and linear differential equations. The d(y)/d(x) must be
alone on the left hand side of the equations.
e.g. dsolve(d(y)/d(x) === x*y + 1, y).
See also: solve, nsolve.
* E
E is the exponential part of a floating point number. e.g. 1.1E2 is the
same as 1.1*10^2.
See also: e, exp.
* e
(1) e is the exponential constant (baes of natural logarithms),
e=2.718..., the built-in constant, e is converted to 2.718... when the
switch Numerical=On. e^x is the same as exp(x).
e.g. e^2, e^x.
(2) e is the exponential part of a floating point number, the same as E.
e.g. 1.1e2 is the same as 1.1E2.
See also: E, exp.
* ei
ei(n,x) is the exponential integral function En(x),
ei(n,x)=inte(t^n*e^t, t,-inf,x), d(ei(n,x),x)=x^n*e^x, ei(-1, x)=ei(x),
ei(0,x)=e^x.
ei(x) is the exponential integral function Ei(x),
ei(x) = inte(e^t/t, t,-inf,x), d(ei(x),x)=e^x/x,
See also: gamma.
* erf
erf(x) is the error function of x. It is the probability integral
function or the integral of the Gaussian distribution. erf(x)=
2/sqrt(pi)*inte(exp(-t^2),t,0,z).
* exp
exp(x) is the exponential function of x (base of e). The same as e^z,
e=2.718... It is the inverse to ln(x). x can be complex numbers.
See also: ^.
* expand
expand(expr) expands out products and positive powers in expr.
expand works only on positive integer powers.
e.g. expand((a+b)^2) gives a^2 + 2*a*b + b^2.
See also: factor.
the switch of expansion.
expand=on e.g. c*(a+b) to c*a+c*b.
expand=off disable expansion, this is default.
* expexpand
The switch of exponential expansion.
expexpand=on e.g. c^(a+b) to c^a*c^b.
expexpand=off disable exponential expansion, this is default.
* fac
fac(n) is the factorial function of n. The same as n!.
e.g. fac(3) gives 6.
See also: n!.
* factor
factor(expr) factorises from expr.
e.g. factor(a^2 + 2*a*b + b^2) gives (a+b)^2.
See also: expand, Expand, ExpExpand, LnExpand.
* float
float(x) converts x to the floating-point number.
e.g. float(1/2) gives 0.5.
See also: ratio.
* fortran
fortran is the value of the switch Output. It forces the output in
Fortran format.
e.g. output=fortran.
See also: output, basic, twodim, prolog, on, off.
* from
The separator, the same as the comma (,).
* gamma
gamma(n,x) is the incomplete gamma function, gamma(n,x)=
inte(t^n*e^(-t), t,0,x), d(gamma(n,x),x)=x^n*e^(-x).
gamma(n,0)=0, gamma(n,inf)=gamma(n+1)=n!.
gamma(n) is the gamma function Γ(n), gamma(n)=inte(t^(n-1)*e^(-t), t,0,inf).
gamma(n,x) is similar to gamma(n), but its power term is t^n, instead
of t^(n-1). gamma(n)=(n-1)!.
See also: ei.
* i
i represents the imaginative unit of the complex numbers, i=sqrt(-1),
as the built-in constant.
e.g. 1+2*i.
See also: re, im.
* if
if(condition, x, y) gives x if condition evaluates to 1, y if it
evaluates to 0, or no output if it evaluates
to neither 1 or 0.
if(condition, x) gives x if condition evaluates to 1, or no
output otherwise.
It is useful in definition of the use-defined function to left the
function unevaluted if the argument of the function is not number.
e.g. define f(x_):=if(isnumber(x), 1), then call f(x), f(10) gives 1,
and f(a) gives f(a).
See also: :=, =.
* im
im(x) gives the imaginative part of the complex number x.
e.g. im(1+2*i) gives 2.
See also: re, abs, sgn, arg.
*include
include('filename') includes (or runs) the file 'filename'. The filename
is any MS-DOS file name.
See also: openfile, closefile.
* inf
inf is a positive infinity, as the built-in constant.
e.g. inf+2*inf gives inf, 1/inf gives 0.
See also: complex_inf.
* inte
The integral function.
inte(f,x) find the indefinite integral of f with respect
to an undefined variable x.
inte(f,x,xmin,xmax) find the definite integral of f with respect to
an undefined variable x taken from x=a to x=b.
inte(y) implicit integration, used to integrate the
differential equations.
See also: ninte.
* iseven
iseven(x) gives 1 if x is an even integer, or 0 otherwise.
You can assume x is even by iseven(x) := 1.
e.g. iseven(2) gives 1, iseven(3) gives 0.
See also: isodd, isinteger, isreal, isnumber, isfree, sgn.
* isfree
isfree(y,x) gives 1 if y is free of x, or 0 otherwise.
You can assume y is free of x by iseven(y,x) := 1.
e.g. isfree(a*b,x) gives 1, isfree(x*y,x) gives 0.
See also: isodd, isinteger, isreal, isnumber, isfree, sgn.
* isinteger
isinteger(x) gives 1 if x is an integer, or 0 otherwise.
You can assume x is integer by isinteger(x) := 1.
e.g. isinteger(2) gives 1, isinteger(3.2) gives 0.
See also: iseven, isodd, isreal, isnumber, isfree, sgn.
* isodd
isodd(x) gives 1 if x is an odd integer, or 0 otherwise.
You can assume x is odd by isodd(x) := 1.
e.g. isodd(3) gives 1, isodd(2) gives 0.
See also: iseven, isinteger, isreal, isnumber, isfree, sgn.
* isreal
isreal(x) gives 1 if x is real, or 0 otherwise.
You can assume x is real by isreal(x) := 1.
e.g. isreal(2.2) gives 1, isreal(a) gives 0.
See also: iseven, isodd, isinteger, isnumber, isfree, sgn.
* isnumber
isnumber(x) gives 1 if x is a number, or 0 otherwise.
You can assume x is a number by isnumber(x) := 1.
e.g. isnumber(2.2) gives 1, isnumber(a) gives 0.
See also: iseven, isodd, isinteger, isreal, isfree, sgn.
* last
last represents the last output, as the built-in variable.
last[1] the first element of the last output list.
* left
left(x===a) gives the left hand side of an equation.
e.g. left(x+y===2) gives x+y.
See also: righ.
* lim
lim(expr, x=x0) finds the limiting value of expr when x
approaches x0.
lim(expr, x=x0+zero) finds the right-sided limit as x approches x0
from the positive (+inf) direction.
lim(expr, x=x0-zero) finds the left-sided limit as x approches x0
from the negative (-inf) direction.
e.g. lim(sin(x)/x, x=0) gives 1.
Note that the correct answers are only for the indeterminate forms:
0/0, inf/inf, 0*inf, 0^0, inf^0.
* list
list(f,x,xmin,xmax,dx) produces a list of f when x runs from xmin
to xmax on step dx.
list(a+b) transforms sum to list.
e.g. list(x^2, x,1,3,1) gives [1,4,9], list(a+b) gives [a,b].
See also: table, sum, prod.
* ln
ln(x) is the natural logarithmic function of x. Its base is e. It is
the inverse to exp(x). Warming that if it has multi-values, the ln(x)
only gives a principle value (P.V.) and other values are
P.V.+2*k*pi*i (where k=0, 1, 2,..., -1, -2, ...).
If x is complex number (x=A+B*i) then
ln(x)=ln(abs(x))+i*atan2(A,B).
See also: exp.
* lnexpand
The switch of the logarithmic expansion.
lnexpand=on log expansion, e.g. ln(a*b) is expanded into
ln(a)+ln(b).
lnexpand=off disable log expansion, this is default.
See also: expexpand, expand.
* lowercase
The swicth of the case conversion.
lowercase=on converts the letters to lower-case letters, e.g.
SIN(x) is converted to sin(x).
lowercase=off disables the case convertion, this is default.
It only effects the input.
* mod
mod(m,n) gives the remainder on division of m by n.
See also: div.
* n!
n! gives the factorial of n. The same as fac(n).
e.g. 3! gives 6.
See also: fac.
* num
num(expr) gives the numerical value of expr. It converts all numbers
to real form.
e.g. num(pi) gives 3.1416.
See also: numerical.
* numerical
The switch of numerical calculation.
numerical=on numerical computation.
numerical=off disable numerical computation, this is default.
See also: num.
* null
null is a symbol used to indicate the absence of an expression or a
result. When it appreas as an output expression, no output is printed.
e.g. block(output=on, null).
* off
When the switch is set to off, it is inactive.
e.g. numerical=off, output=off, expand=off.
* on
When the switch is set to on, it is active.
e.g. numerical=on, expand=on, expexpand=on, lnexpand=on, lowercase=on,
output=on.
* openfile
openfile('filename') opens the disk file 'filename' for writing. The
filename is any MS-DOS file name. After doing something, the file must
be closed by closefile('filename').
See also: closefile, include.
* output
The switch of the output format,
e.g. output=basic, output=fortran, output=twodim, output=on, output=off.
* pi
pi=3.1416..., as the built-in constant, pi is converted to 3.1416...
when the switch numerical=on.
* plot
plot(f,x,xmin,xmax,dx) generates a plot of f as a function of x from
xmin to xmax with step dx.
plot(f,x,xmin,xmax) plots a function with default step (xmax-xmin)*
0.05 (20 plot points).
plot(f,x) plots a function with the default range (-6 to 6)
and the default step (20 plot points).
* prod
prod(f,x,xmin,xmax,dx) evaluates the product of f when x runs from
xmin to xmax with step dx.
prod(f,x,xmin,xmax) with the default step dx = 1.
* prolog
prolog is the value of the switch output. It forces the output in the
Prolog format.
See also: output, basic, fortran.
* random
random(0) gives a uniformly distributed pseudorandom real in
the range 0 to 1.
random(n) gives a uniformly distributed pseudorandom integer
in the range 0 to n. (n < 36000).
e.g. random(0) gives 0.11111, random(5) gives 2.
* ratio
ratio(x) converts x to a rational number.
e.g. ratio(0.2) gives 1/5.
See also: float, num.
* re
re(x) gives the real part of the complex number x.
e.g. re(1+2*i) gives 1.
See also: im(x), abs(x), sgn(x).
* repeat
repeat(expr,test) repeats expr until test gives 1.
e.g. x=1, repeat(x=x+1, x>5) gives x=6.
See also: do.
* reverse
reverse(list) reverses the order of the elements in the list.
e.g. reverse([1,2,3] gives [3,2,1].
* right
right(x===a) gives the right hand side of an equation.
e.g. right(x+y === 3) gives 3.
See also: left, solve.
* round
round(x) converts x to the rounded integer closest to x.
e.g. round(2.4) gives 2, round(2.5) gives 3.
See also: trunc.
* sec
sec(x) is the secant function of x. The angle x is measured in
radians (multiply by degree to convert from degrees). sec(x)=1/cos(x).
See also: asec, cos.
* sech
sech(x) is the hyperbolic secant function of x. sech(x)=1/cosh(x).
See also: asech, cosh.
* sgn
sgn(x) is the sign function of x. Its value is 1, 0 or -1.
/ 1 if re(x) > 0, or re(x) = 0 and im(x) > 0;
sgn(x) = 0 if x=0;
\ -1 otherwise.
You can assume x is positive or negative by sgn(x) := 1 or
sgn(x) := -1.
e.g. sgn(2) gives 1, sgn(1+i) gives 1.
See also: abs, re, im.
* sin
sin(x) is the sine function of x. The angle x is measured in radians.
(multiply by degree to convert from degrees).
See also: asin, csc.
* sinh
sinh(x) is the hyperbolic sine function of x.
sinh(x)=(exp(x)-exp(-z))/2.
See also: asinh, acsch.
* solve
solve(x^2===0, x) solves a polynomial ordered up to 4.
solve([expr1===expr2,expr3===expr4], [x,y]) solves systems of linear
equations.
It gives all symbolic solutions.
e.g. solve(x^2+5*x+6===0, x), solve([x+y===3, x-y===1], [x,y]).
See also: nsolve.
* sqrt
sqrt(x) is the square root function of x. It is the same as x^0.5.
It only gives the principal value (P.V.) (sgn(sqrt(x)) >= 0).
e.g. sqrt(4) gives 2, sqrt(2*i) gives 1+i.
* subs
subs(expr, x=x0) substitutes x in expr by x0.
e.g. subs(x^2, x=a) gives a^2.
* sum
sum(f,x,xmin,xmax,dx) evaluates the sum of f when x runs from xmin
to xmax with step dx.
sum(f,x,xmin,xmax) with the default step dx = 1.
sum([a,b]) transforms list to sum.
e.g. sum([a,b]) gives a+b, sum(2^n,n,1,5,1.1), sum(x^n,n,1,5).
See also: prod, list, table
* step
step is the separator, the same as the comma (,).
See also: from, to, ,
* system
system(DOS) executes the operating system (DOS) command.
e.g. system(dir).
* table
table(f,x,xmin,xmax,dx) produces a table of the function values when
x runs from xmin to xmax with step dx.
table(f,x,xmin,xmax) with the default step dx = 1.
table(l[x],x,xmin,xmax,dx) transforms a list l into a table.
e.g. table(x^2,x,1,4,1).
* tan
tan(x) is the tangent function of x. The angle x is measured in
radians (multiply by degree to convert from degrees).
See also: atan, cot.
* tanh
tanh(x) is the hyperbolic tangent function of x.
See also: atanh, coth.
* to
to is the separator, the same as the comma (,).
See also: from, step, ,.
* trunc
trunc(x) converts x to the truncated integer.
e.g. trunc(2.9) gives 2.
See also: round.
* twodim
towdim is a value of the switch output. It forces output in two
dimension format.
e.g. output=twodim.
See also: output, off, on, basic, fortran, prolog.
* undefined
The built-in constant. It indicates that the value of the expression
is undefined.
e.g. the indeterminate forms: 0/0, inf/inf, 0*inf, 0^0.
Users should try again by lim(f, x=x0).
* zero
The right-hand sided value at x=0, as the built-in constant. -zero
is the left-sided limit from the negative direction. e.g. 1+zero is
to approach to 1 from the positive (+infinity) direction (the right-
hand sided value), and 1-zero is to approach to 1 from the negative
(-infinity) direction (the left-hand sided value), i.e. limit as
zero -> 0. e.g. exp(1/(0+zero)) gives inf, exp(1/(0-zero)) gives 0.
+ add or positive sign.
- subtract or negative sign.
* multiply.
/ divide.
^ power in BASIC, the same as ** in FORTRAN, e.g. 2^3 gives 8.
** power in FORTRAN, the same as ^ in BASIC.
! factorial, the same as fac(x), e.g. 3! or fac(3) gives 6.
< less than.
<= less than.
> greater than.
>= greater than.
<> unequal
a <> b gives 1 if a is not equal to b, 0 if a is equal to b, or left
unevaluated otherwise. It only test two real numbers.
e.g. 2 <> 3 gives 1, 2 <> 2 gives 0.
:= delayed assigment.
= immediate assignment.
== equal
a==b gives 1 if a is equal to b, 0 if a is not equal to b, or left
unevaluated otherwise. It can test two complex numbers or expressions.
It gives 1 if a and b are the same expression or left unevaluated otherwise.
e.g. 2==2 gives 1, a==a gives 1.
=== equation sign
e.g. x^6===0
, comma
The words from, to, step and comma are the same.
# comment statement
e.g. # this is demo.
-------------------------------------------------------------------------