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The abs function represents the mathematical absolute value function and
works for both numerical and symbolic values. If the argument, z, is a
real or complex number, abs returns the absolute value of z. If
possible, symbolic expressions using the absolute value function are
also simplified.
Maxima can differentiate, integrate and calculate limits for expressions
containing abs. The abs_integrate package further extends
Maxima’s ability to calculate integrals involving the abs function. See
(%i12) in the examples below.
When applied to a list or matrix, abs automatically distributes over
the terms. Similarly, it distributes over both sides of an
equation. To alter this behaviour, see the variable distribute_over.
See also cabs.
Examples:
Calculation of abs for real and complex numbers, including numerical
constants and various infinities. The first example shows how abs
distributes over the elements of a list.
(%i1) abs([-4, 0, 1, 1+%i]); (%o1) [4, 0, 1, sqrt(2)]
(%i2) abs((1+%i)*(1-%i)); (%o2) 2
(%i3) abs(%e+%i);
2
(%o3) sqrt(%e + 1)
(%i4) abs([inf, infinity, minf]); (%o4) [inf, inf, inf]
Simplification of expressions containing abs:
(%i1) abs(x^2);
2
(%o1) x
(%i2) abs(x^3);
2
(%o2) x abs(x)
(%i3) abs(abs(x)); (%o3) abs(x)
(%i4) abs(conjugate(x)); (%o4) abs(x)
Integrating and differentiating with the abs function. Note that more
integrals involving the abs function can be performed, if the
abs_integrate package is loaded. The last example shows the Laplace
transform of abs: see laplace.
(%i1) diff(x*abs(x),x),expand; (%o1) 2 abs(x)
(%i2) integrate(abs(x),x);
x abs(x)
(%o2) --------
2
(%i3) integrate(x*abs(x),x);
/
|
(%o3) | x abs(x) dx
|
/
(%i4) load("abs_integrate")$
(%i5) integrate(x*abs(x),x);
3
x signum(x)
(%o5) ------------
3
(%i6) integrate(abs(x),x,-2,%pi);
2
%pi
(%o6) ---- + 2
2
(%i7) laplace(abs(x),x,s);
1
(%o7) --
2
s
When x is a real number, return the least integer that is greater than or equal to x.
If x is a constant expression (10 * %pi, for example),
ceiling evaluates x using big floating point numbers, and
applies ceiling to the resulting big float. Because ceiling uses
floating point evaluation, it’s possible, although unlikely, that ceiling
could return an erroneous value for constant inputs. To guard against errors,
the floating point evaluation is done using three values for fpprec.
For non-constant inputs, ceiling tries to return a simplified value.
Here are examples of the simplifications that ceiling knows about:
(%i1) ceiling (ceiling (x)); (%o1) ceiling(x)
(%i2) ceiling (floor (x)); (%o2) floor(x)
(%i3) declare (n, integer)$
(%i4) [ceiling (n), ceiling (abs (n)), ceiling (max (n, 6))]; (%o4) [n, abs(n), max(6, n)]
(%i5) assume (x > 0, x < 1)$
(%i6) ceiling (x); (%o6) 1
(%i7) tex (ceiling (a)); $$\left \lceil a \right \rceil$$ (%o7) false
The ceiling function distributes over lists, matrices and equations.
See distribute_over.
Finally, for all inputs that are manifestly complex, ceiling returns
a noun form.
If the range of a function is a subset of the integers, it can be declared to
be integervalued. Both the ceiling and floor functions
can use this information; for example:
(%i1) declare (f, integervalued)$
(%i2) floor (f(x)); (%o2) f(x)
(%i3) ceiling (f(x) - 1); (%o3) f(x) - 1
Example use:
(%i1) unitfrac(r) := block([uf : [], q],
if not(ratnump(r)) then
error("unitfrac: argument must be a rational number"),
while r # 0 do (
uf : cons(q : 1/ceiling(1/r), uf),
r : r - q),
reverse(uf));
(%o1) unitfrac(r) := block([uf : [], q],
if not ratnump(r) then error("unitfrac: argument must be a rational number"
1
), while r # 0 do (uf : cons(q : ----------, uf), r : r - q),
1
ceiling(-)
r
reverse(uf))
(%i2) unitfrac (9/10);
1 1 1
(%o2) [-, -, --]
2 3 15
(%i3) apply ("+", %);
9
(%o3) --
10
(%i4) unitfrac (-9/10);
1
(%o4) [- 1, --]
10
(%i5) apply ("+", %);
9
(%o5) - --
10
(%i6) unitfrac (36/37);
1 1 1 1 1
(%o6) [-, -, -, --, ----]
2 3 8 69 6808
(%i7) apply ("+", %);
36
(%o7) --
37
Returns the largest integer less than or equal to x where x is
numeric. fix (as in fixnum) is a synonym for this, so
fix(x) is precisely the same.
When x is a real number, return the largest integer that is less than or equal to x.
If x is a constant expression (10 * %pi, for example), floor
evaluates x using big floating point numbers, and applies floor to
the resulting big float. Because floor uses floating point evaluation,
it’s possible, although unlikely, that floor could return an erroneous
value for constant inputs. To guard against errors, the floating point
evaluation is done using three values for fpprec.
For non-constant inputs, floor tries to return a simplified value. Here
are examples of the simplifications that floor knows about:
(%i1) floor (ceiling (x)); (%o1) ceiling(x)
(%i2) floor (floor (x)); (%o2) floor(x)
(%i3) declare (n, integer)$
(%i4) [floor (n), floor (abs (n)), floor (min (n, 6))]; (%o4) [n, abs(n), min(6, n)]
(%i5) assume (x > 0, x < 1)$
(%i6) floor (x); (%o6) 0
(%i7) tex (floor (a)); $$\left \lfloor a \right \rfloor$$ (%o7) false
The floor function distributes over lists, matrices and equations.
See distribute_over.
Finally, for all inputs that are manifestly complex, floor returns
a noun form.
If the range of a function is a subset of the integers, it can be declared to
be integervalued. Both the ceiling and floor functions
can use this information; for example:
(%i1) declare (f, integervalued)$
(%i2) floor (f(x)); (%o2) f(x)
(%i3) ceiling (f(x) - 1); (%o3) f(x) - 1
A synonym for entier (x).
The Heaviside unit step function, equal to 0 if x is negative, equal to 1 if x is positive and equal to 1/2 if x is equal to zero.
If you want a unit step function that takes on the value of 0 at x
equal to zero, use unit_step.
When L is a list or a set, return apply ('max, args (L)).
When L is not a list or a set, signal an error.
See also lmin and max.
When L is a list or a set, return apply ('min, args (L)).
When L is not a list or a set, signal an error.
See also lmax and min.
Return a simplified value for the numerical maximum of the expressions x_1
through x_n. For an empty argument list, max yields minf.
The option variable maxmin_effort controls which simplification methods are
applied. Using the default value of twelve for maxmin_effort,
max uses all available simplification methods. To to inhibit all
simplifications, set maxmin_effort to zero.
When maxmin_effort is one or more, for an explicit list of real numbers,
max returns a number.
Unless max needs to simplify a lengthy list of expressions, we suggest using
the default value of maxmin_effort. Setting maxmin_effort to zero
(no simplifications), will cause problems for some Maxima functions; accordingly,
generally maxmin_effort should be nonzero.
See also min, lmax., and lmin..
Examples:
In the first example, setting maxmin_effort to zero suppresses simplifications.
(%i1) block([maxmin_effort : 0], max(1,2,x,x, max(a,b))); (%o1) max(1,2,max(a,b),x,x) (%i2) block([maxmin_effort : 1], max(1,2,x,x, max(a,b))); (%o2) max(2,a,b,x)
When maxmin_effort is two or more, max compares pairs of members:
(%i1) block([maxmin_effort : 1], max(x,x+1,x+3)); (%o1) max(x,x+1,x+3) (%i2) block([maxmin_effort : 2], max(x,x+1,x+3)); (%o2) x+3
Finally, when maxmin_effort is three or more, max compares triples
members and excludes those that are in between; for example
(%i1) block([maxmin_effort : 4], max(x, 2*x, 3*x, 4*x)); (%o1) max(x,4*x)
Return a simplified value for the numerical minimum of the expressions x_1
through x_n. For an empty argument list, minf yields inf.
The option variable maxmin_effort controls which simplification methods are
applied. Using the default value of twelve for maxmin_effort,
max uses all available simplification methods. To to inhibit all
simplifications, set maxmin_effort to zero.
When maxmin_effort is one or more, for an explicit list of real numbers,
min returns a number.
Unless min needs to simplify a lengthy list of expressions, we suggest using
the default value of maxmin_effort. Setting maxmin_effort to zero
(no simplifications), will cause problems for some Maxima functions; accordingly,
generally maxmin_effort should be nonzero.
See also max, lmax., and lmin..
Examples:
In the first example, setting maxmin_effort to zero suppresses simplifications.
(%i1) block([maxmin_effort : 0], min(1,2,x,x, min(a,b))); (%o1) min(1,2,a,b,x,x) (%i2) block([maxmin_effort : 1], min(1,2,x,x, min(a,b))); (%o2) min(1,a,b,x)
When maxmin_effort is two or more, min compares pairs of members:
(%i1) block([maxmin_effort : 1], min(x,x+1,x+3)); (%o1) min(x,x+1,x+3) (%i2) block([maxmin_effort : 2], min(x,x+1,x+3)); (%o2) x
Finally, when maxmin_effort is three or more, min compares triples
members and excludes those that are in between; for example
(%i1) block([maxmin_effort : 4], min(x, 2*x, 3*x, 4*x)); (%o1) max(x,4*x)
When x is a real number, returns the closest integer to x.
Multiples of 1/2 are rounded to the nearest even integer. Evaluation of
x is similar to floor and ceiling.
The round function distributes over lists, matrices and equations.
See distribute_over.
For either real or complex numbers x, the signum function returns
0 if x is zero; for a nonzero numeric input x, the signum function
returns x/abs(x).
For non-numeric inputs, Maxima attempts to determine the sign of the input.
When the sign is negative, zero, or positive, signum returns -1,0, 1,
respectively. For all other values for the sign, signum a simplified but
equivalent form. The simplifications include reflection (signum(-x)
gives -signum(x)) and multiplicative identity (signum(x*y) gives
signum(x) * signum(y)).
The signum function distributes over a list, a matrix, or an
equation. See sign and distribute_over.
When x is a real number, return the closest integer to x not
greater in absolute value than x. Evaluation of x is similar
to floor and ceiling.
The truncate function distributes over lists, matrices and equations.
See distribute_over.
Next: Combinatorial Functions, Previous: Functions for Numbers, Up: Elementary Functions [Contents][Index]
Calculates the absolute value of an expression representing a complex
number. Unlike the function abs, the cabs function always
decomposes its argument into a real and an imaginary part. If x and
y represent real variables or expressions, the cabs function
calculates the absolute value of x + %i*y as
(%i1) cabs (1); (%o1) 1
(%i2) cabs (1 + %i); (%o2) sqrt(2)
(%i3) cabs (exp (%i)); (%o3) 1
(%i4) cabs (exp (%pi * %i)); (%o4) 1
(%i5) cabs (exp (3/2 * %pi * %i)); (%o5) 1
(%i6) cabs (17 * exp (2 * %i)); (%o6) 17
If cabs returns a noun form this most commonly is caused by
some properties of the variables involved not being known:
(%i1) cabs (a+%i*b);
2 2
(%o1) sqrt(b + a )
(%i2) declare(a,real,b,real); (%o2) done
(%i3) cabs (a+%i*b);
2 2
(%o3) sqrt(b + a )
(%i4) assume(a>0,b>0); (%o4) [a > 0, b > 0]
(%i5) cabs (a+%i*b);
2 2
(%o5) sqrt(b + a )
The cabs function can use known properties like symmetry properties of
complex functions to help it calculate the absolute value of an expression. If
such identities exist, they can be advertised to cabs using function
properties. The symmetries that cabs understands are: mirror symmetry,
conjugate function and complex characteristic.
cabs is a verb function and is not suitable for symbolic
calculations. For such calculations (including integration,
differentiation and taking limits of expressions containing absolute
values), use abs.
The result of cabs can include the absolute value function,
abs, and the arc tangent, atan2.
When applied to a list or matrix, cabs automatically distributes over
the terms. Similarly, it distributes over both sides of an equation.
For further ways to compute with complex numbers, see the functions
rectform, realpart, imagpart,
carg, conjugate and polarform.
Examples:
(%i1) cabs(sqrt(1+%i*x));
2 1/4
(%o1) (x + 1)
(%i2) cabs(sin(x+%i*y));
2 2 2 2
(%o2) sqrt(cos (x) sinh (y) + sin (x) cosh (y))
The error function, erf, has mirror symmetry, which is used here in
the calculation of the absolute value with a complex argument:
(%i1) cabs(erf(x+%i*y));
2
(erf(%i y + x) - erf(%i y - x))
(%o1) sqrt(--------------------------------
4
2
(- erf(%i y + x) - erf(%i y - x))
- ----------------------------------)
4
Maxima knows complex identities for the Bessel functions, which allow
it to compute the absolute value for complex arguments. Here is an
example for bessel_j.
(%i1) cabs(bessel_j(1,%i)); (%o1) bessel_i(1, 1)
Returns the complex argument of z. The complex argument is an angle
theta in (-%pi, %pi] such that r exp (theta %i) = z
where r is the magnitude of z.
carg is a computational function, not a simplifying function.
See also abs (complex magnitude), polarform,
rectform, realpart, and imagpart.
Examples:
(%i1) carg (1); (%o1) 0
(%i2) carg (1 + %i);
%pi
(%o2) ---
4
(%i3) carg (exp (%i));
sin(1)
(%o3) atan(------)
cos(1)
(%i4) carg (exp (%pi * %i)); (%o4) %pi
(%i5) carg (exp (3/2 * %pi * %i));
%pi
(%o5) - ---
2
(%i6) carg (17 * exp (2 * %i));
sin(2)
(%o6) atan(------) + %pi
cos(2)
If carg returns a noun form this most commonly is caused by
some properties of the variables involved not being known:
(%i1) carg (a+%i*b); (%o1) atan2(b, a)
(%i2) declare(a,real,b,real); (%o2) done
(%i3) carg (a+%i*b); (%o3) atan2(b, a)
(%i4) assume(a>0,b>0); (%o4) [a > 0, b > 0]
(%i5) carg (a+%i*b);
b
(%o5) atan(-)
a
Returns the complex conjugate of x.
(%i1) declare ([aa, bb], real, cc, complex, ii, imaginary); (%o1) done
(%i2) conjugate (aa + bb*%i); (%o2) aa - %i bb
(%i3) conjugate (cc); (%o3) conjugate(cc)
(%i4) conjugate (ii); (%o4) - ii
(%i5) conjugate (xx + yy); (%o5) yy + xx
Returns the imaginary part of the expression expr.
imagpart is a computational function, not a simplifying function.
See also abs, carg, polarform,
rectform, and realpart.
Example:
(%i1) imagpart (a+b*%i); (%o1) b
(%i2) imagpart (1+sqrt(2)*%i); (%o2) sqrt(2)
(%i3) imagpart (1); (%o3) 0
(%i4) imagpart (sqrt(2)*%i); (%o4) sqrt(2)
Returns an expression r %e^(%i theta) equivalent to expr,
such that r and theta are purely real.
Example:
(%i1) polarform(a+b*%i);
2 2 %i atan2(b, a)
(%o1) sqrt(b + a ) %e
(%i2) polarform(1+%i);
%i %pi
------
4
(%o2) sqrt(2) %e
(%i3) polarform(1+2*%i);
%i atan(2)
(%o3) sqrt(5) %e
Returns the real part of expr. realpart and imagpart will
work on expressions involving trigonometric and hyperbolic functions,
as well as square root, logarithm, and exponentiation.
Example:
(%i1) realpart (a+b*%i); (%o1) a
(%i2) realpart (1+sqrt(2)*%i); (%o2) 1
(%i3) realpart (sqrt(2)*%i); (%o3) 0
(%i4) realpart (1); (%o4) 1
Returns an expression a + b %i equivalent to expr,
such that a and b are purely real.
Example:
(%i1) rectform(sqrt(2)*%e^(%i*%pi/4)); (%o1) %i + 1
(%i2) rectform(sqrt(b^2+a^2)*%e^(%i*atan2(b, a))); (%o2) %i b + a
(%i3) rectform(sqrt(5)*%e^(%i*atan(2))); (%o3) 2 %i + 1
Next: Root, Exponential and Logarithmic Functions, Previous: Functions for Complex Numbers, Up: Elementary Functions [Contents][Index]
The double factorial operator.
For an integer, float, or rational number n, n!! evaluates to the
product n (n-2) (n-4) (n-6) ... (n - 2 (k-1)) where k is equal to
entier (n/2), that is, the largest integer less than or equal to
n/2. Note that this definition does not coincide with other published
definitions for arguments which are not integers.
For an even (or odd) integer n, n!! evaluates to the product of
all the consecutive even (or odd) integers from 2 (or 1) through n
inclusive.
For an argument n which is not an integer, float, or rational, n!!
yields a noun form genfact (n, n/2, 2).
The binomial coefficient x!/(y! (x - y)!).
If x and y are integers, then the numerical value of the binomial
coefficient is computed. If y, or x - y, is an integer, the
binomial coefficient is expressed as a polynomial.
Examples:
(%i1) binomial (11, 7); (%o1) 330
(%i2) 11! / 7! / (11 - 7)!; (%o2) 330
(%i3) binomial (x, 7);
(x - 6) (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) x
(%o3) -------------------------------------------------
5040
(%i4) binomial (x + 7, x);
(x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x + 6) (x + 7)
(%o4) -------------------------------------------------------
5040
(%i5) binomial (11, y); (%o5) binomial(11, y)
Tries to combine the coefficients of factorials in expr
with the factorials themselves by converting, for example, (n + 1)*n!
into (n + 1)!.
sumsplitfact if set to false will cause minfactorial to be
applied after a factcomb.
Example:
(%i1) sumsplitfact; (%o1) true
(%i2) (n + 1)*(n + 1)*n!;
2
(%o2) (n + 1) n!
(%i3) factcomb (%); (%o3) (n + 2)! - (n + 1)!
(%i4) sumsplitfact: not sumsplitfact; (%o4) false
(%i5) (n + 1)*(n + 1)*n!;
2
(%o5) (n + 1) n!
(%i6) factcomb (%); (%o6) n (n + 1)! + (n + 1)!
Represents the factorial function. Maxima treats factorial (x)
the same as x!.
For any complex number x, except for negative integers, x! is
defined as gamma(x+1).
For an integer x, x! simplifies to the product of the integers
from 1 to x inclusive. 0! simplifies to 1. For a real or complex
number in float or bigfloat precision x, x! simplifies to the
value of gamma (x+1). For x equal to n/2 where n is
an odd integer, x! simplifies to a rational factor times
sqrt (%pi) (since gamma (1/2) is equal to sqrt (%pi)).
The option variables factlim and gammalim control the numerical
evaluation of factorials for integer and rational arguments. The functions
minfactorial and factcomb simplifies expressions containing
factorials.
The functions gamma, bffac, and cbffac are
varieties of the gamma function. bffac and cbffac are called
internally by gamma to evaluate the gamma function for real and complex
numbers in bigfloat precision.
makegamma substitutes gamma for factorials and related functions.
Maxima knows the derivative of the factorial function and the limits for specific values like negative integers.
The option variable factorial_expand controls the simplification of
expressions like (n+x)!, where n is an integer.
See also binomial.
The factorial of an integer is simplified to an exact number unless the operand
is greater than factlim. The factorial for real and complex numbers is
evaluated in float or bigfloat precision.
(%i1) factlim : 10; (%o1) 10
(%i2) [0!, (7/2)!, 8!, 20!];
105 sqrt(%pi)
(%o2) [1, -------------, 40320, 20!]
16
(%i3) [4,77!, (1.0+%i)!]; (%o3) [4, 77!, 0.3430658398165453 %i + 0.6529654964201667]
(%i4) [2.86b0!, (1.0b0+%i)!];
(%o4) [5.046635586910012b0, 3.430658398165454b-1 %i
+ 6.529654964201667b-1]
The factorial of a known constant, or general expression is not simplified. Even so it may be possible to simplify the factorial after evaluating the operand.
(%i1) [(%i + 1)!, %pi!, %e!, (cos(1) + sin(1))!]; (%o1) [(%i + 1)!, %pi!, %e!, (sin(1) + cos(1))!]
(%i2) ev (%, numer, %enumer);
(%o2) [0.3430658398165453 %i + 0.6529654964201667,
7.188082728976031, 4.260820476357003, 1.227580202486819]
Factorials are simplified, not evaluated.
Thus x! may be replaced even in a quoted expression.
(%i1) '([0!, (7/2)!, 4.77!, 8!, 20!]);
105 sqrt(%pi)
(%o1) [1, -------------, 81.44668037931197, 40320,
16
2432902008176640000]
Maxima knows the derivative of the factorial function.
(%i1) diff(x!,x);
(%o1) x! psi (x + 1)
0
The option variable factorial_expand controls expansion and
simplification of expressions with the factorial function.
(%i1) (n+1)!/n!,factorial_expand:true; (%o1) n + 1
Default value: 100000
factlim specifies the highest factorial which is
automatically expanded. If it is -1 then all integers are expanded.
Default value: false
The option variable factorial_expand controls the simplification of
expressions like (x+n)!, where n is an integer.
See factorial for an example.
Returns the generalized factorial, defined as
x (x-z) (x - 2 z) ... (x - (y - 1) z). Thus, when x is an integer,
genfact (x, x, 1) = x! and genfact (x, x/2, 2) = x!!.
Examines expr for occurrences of two factorials
which differ by an integer.
minfactorial then turns one into a polynomial times the other.
(%i1) n!/(n+2)!;
n!
(%o1) --------
(n + 2)!
(%i2) minfactorial (%);
1
(%o2) ---------------
(n + 1) (n + 2)
Default value: true
When sumsplitfact is false,
minfactorial is applied after a factcomb.
(%i1) sumsplitfact; (%o1) true
(%i2) n!/(n+2)!;
n!
(%o2) --------
(n + 2)!
(%i3) factcomb(%);
n!
(%o3) --------
(n + 2)!
(%i4) sumsplitfact: not sumsplitfact ; (%o4) false
(%i5) n!/(n+2)!;
n!
(%o5) --------
(n + 2)!
(%i6) factcomb(%);
1
(%o6) ---------------
(n + 1) (n + 2)
Next: Trigonometric Functions, Previous: Combinatorial Functions, Up: Elementary Functions [Contents][Index]
Default value: false
When true, r some rational number, and x some expression,
%e^(r*log(x)) will be simplified into x^r . It should be noted
that the radcan command also does this transformation, and more
complicated transformations of this ilk as well. The logcontract
command "contracts" expressions containing log.
Default value: true
When %emode is true, %e^(%pi %i x) is simplified as
follows.
%e^(%pi %i x) simplifies to cos (%pi x) + %i sin (%pi x) if
x is a floating point number, an integer, or a multiple of 1/2, 1/3, 1/4,
or 1/6, and then further simplified.
For other numerical x, %e^(%pi %i x) simplifies to
%e^(%pi %i y) where y is x - 2 k for some integer k
such that abs(y) < 1.
When %emode is false, no special simplification of
%e^(%pi %i x) is carried out.
(%i1) %emode; (%o1) true
(%i2) %e^(%pi*%i*1); (%o2) - 1
(%i3) %e^(%pi*%i*216/144); (%o3) - %i
(%i4) %e^(%pi*%i*192/144);
sqrt(3) %i 1
(%o4) - ---------- - -
2 2
(%i5) %e^(%pi*%i*180/144);
%i 1
(%o5) - ------- - -------
sqrt(2) sqrt(2)
(%i6) %e^(%pi*%i*120/144);
%i sqrt(3)
(%o6) -- - -------
2 2
(%i7) %e^(%pi*%i*121/144);
121 %i %pi
----------
144
(%o7) %e
Default value: false
When %enumer is true, %e is replaced by its numeric value
2.718… whenever numer is true.
When %enumer is false, this substitution is carried out
only if the exponent in %e^x evaluates to a number.
(%i1) %enumer; (%o1) false
(%i2) numer; (%o2) false
(%i3) 2*%e; (%o3) 2 %e
(%i4) %enumer: not %enumer; (%o4) true
(%i5) 2*%e; (%o5) 2 %e
(%i6) numer: not numer; (%o6) true
(%i7) 2*%e; (%o7) 5.43656365691809
(%i8) 2*%e^1; (%o8) 5.43656365691809
(%i9) 2*%e^x;
x
(%o9) 2 2.718281828459045
Represents the exponential function. Instances of exp (x) in input
are simplified to %e^x; exp does not appear in simplified
expressions.
demoivre if true causes %e^(a + b %i) to simplify to
%e^(a (cos(b) + %i sin(b))) if b is free of %i.
See demoivre.
%emode, when true, causes %e^(%pi %i x) to be simplified.
See %emode.
%enumer, when true causes %e to be replaced by
2.718… whenever numer is true. See %enumer.
(%i1) demoivre; (%o1) false
(%i2) %e^(a + b*%i);
%i b + a
(%o2) %e
(%i3) demoivre: not demoivre; (%o3) true
(%i4) %e^(a + b*%i);
a
(%o4) %e (%i sin(b) + cos(b))
Represents the polylogarithm function of order s and argument z, defined by the infinite series
$$ {\rm Li}_s \left(z\right) = \sum_{k=1}^\infty {z^k \over k^s} $$li[1](z) is
\(-\log(1 - z).\)
li[2] and li[3] are the
dilogarithm and trilogarithm functions, respectively.
When the order is 1, the polylogarithm simplifies to - log (1 - z), which
in turn simplifies to a numerical value if z is a real or complex floating
point number or the numer evaluation flag is present.
When the order is 2 or 3,
the polylogarithm simplifies to a numerical value
if z is a real floating point number
or the numer evaluation flag is present.
Examples:
(%i1) assume (x > 0); (%o1) [x > 0]
(%i2) integrate ((log (1 - t)) / t, t, 0, x);
(%o2) - li (x)
2
(%i3) li[4](1);
4
%pi
(%o3) ----
90
(%i4) li[5](1); (%o4) zeta(5)
(%i5) li[2](1/2);
2 2
%pi log (2)
(%o5) ---- - -------
12 2
(%i6) li[2](%i);
2
%pi
(%o6) %catalan %i - ----
48
(%i7) li[2](1+%i);
2
%i %pi log(2) %pi
(%o7) ------------- + ---- + %catalan %i
4 16
(%i8) li [2] (7);
(%o8) li (7)
2
(%i9) li [2] (7), numer; (%o9) 1.2482731820994244 - 6.1132570288179915 %i
(%i10) li [3] (7);
(%o10) li (7)
3
(%i11) li [2] (7), numer; (%o11) 1.2482731820994244 - 6.1132570288179915 %i
(%i12) L : makelist (i / 4.0, i, 0, 8); (%o12) [0.0, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0]
(%i13) map (lambda ([x], li [2] (x)), L); (%o13) [0.0, 0.2676526390827326, 0.5822405264650125, 0.978469392930306, 1.6449340668482264, 2.1901770114416452 - 0.7010261415046585 %i, 2.3743952702724798 - 1.2738062049196004 %i, 2.448686765338203 - 1.7580848482107874 %i, 2.4674011002723395 - 2.177586090303602 %i]
(%i14) map (lambda ([x], li [3] (x)), L); (%o14) [0.0, 0.25846139579657335, 0.5372131936080402, 0.8444258088622044, 1.2020569031595942, 1.6428668813178295 - 0.07821473138972386 %i, 2.0608775073202805 - 0.258241985293288 %i, 2.433418898226189 - 0.49192601879440423 %i, 2.762071906228924 - 0.7546938294602477 %i]
Represents the natural (base e) logarithm of x.
Maxima does not have a built-in function for the base 10 logarithm or other
bases. log10(x) := log(x) / log(10) is a useful definition.
Simplification and evaluation of logarithms is governed by several global flags:
logexpandcauses log(a^b) to become b*log(a). If it is
set to all, log(a*b) will also simplify to log(a)+log(b).
If it is set to super, then log(a/b) will also simplify to
log(a)-log(b) for rational numbers a/b, a#1.
(log(1/b), for b integer, always simplifies.) If it is set to
false, all of these simplifications will be turned off.
logsimpif false then no simplification of %e to a power containing
log’s is done.
lognegintif true implements the rule log(-n) -> log(n)+%i*%pi for
n a positive integer.
%e_to_numlogwhen true, r some rational number, and x some expression,
the expression %e^(r*log(x)) will be simplified into x^r. It
should be noted that the radcan command also does this transformation,
and more complicated transformations of this as well. The logcontract
command "contracts" expressions containing log.
Default value: false
When doing indefinite integration where logs are generated, e.g.
integrate(1/x,x), the answer is given in terms of log(abs(...))
if logabs is true, but in terms of log(...) if
logabs is false. For definite integration, the logabs:true
setting is used, because here "evaluation" of the indefinite integral at the
endpoints is often needed.
The function logarc(expr) carries out the replacement of
inverse circular and hyperbolic functions with equivalent logarithmic
functions for an expression expr without setting the global
variable logarc.
When the global variable logarc is true,
inverse circular and hyperbolic functions are replaced by
equivalent logarithmic functions.
The default value of logarc is false.
Default value: false
Controls which coefficients are
contracted when using logcontract. It may be set to the name of a
predicate function of one argument. E.g. if you like to generate
SQRTs, you can do logconcoeffp:'logconfun$
logconfun(m):=featurep(m,integer) or ratnump(m)$ . Then
logcontract(1/2*log(x)); will give log(sqrt(x)).
Recursively scans the expression expr, transforming
subexpressions of the form a1*log(b1) + a2*log(b2) + c into
log(ratsimp(b1^a1 * b2^a2)) + c
(%i1) 2*(a*log(x) + 2*a*log(y))$
(%i2) logcontract(%);
2 4
(%o2) a log(x y )
The declaration declare(n,integer) causes
logcontract(2*a*n*log(x)) to simplify to a*log(x^(2*n)). The
coefficients that "contract" in this manner are those such as the 2 and the
n here which satisfy featurep(coeff,integer). The user can
control which coefficients are contracted by setting the option
logconcoeffp to the name of a predicate function of one argument.
E.g. if you like to generate SQRTs, you can do logconcoeffp:'logconfun$
logconfun(m):=featurep(m,integer) or ratnump(m)$ . Then
logcontract(1/2*log(x)); will give log(sqrt(x)).
Default value: true
If true, that is the default value, causes log(a^b) to become
b*log(a). If it is set to all, log(a*b) will also simplify
to log(a)+log(b). If it is set to super, then log(a/b)
will also simplify to log(a)-log(b) for rational numbers a/b,
a#1. (log(1/b), for integer b, always simplifies.) If it
is set to false, all of these simplifications will be turned off.
When logexpand is set to all or super,
the logarithm of a product expression simplifies to a summation of logarithms.
Examples:
When logexpand is true,
log(a^b) simplifies to b*log(a).
(%i1) log(n^2), logexpand=true; (%o1) 2 log(n)
When logexpand is all,
log(a*b) simplifies to log(a)+log(b).
(%i1) log(10*x), logexpand=all; (%o1) log(x) + log(10)
When logexpand is super,
log(a/b) simplifies to log(a)-log(b)
for rational numbers a/b with a#1.
(%i1) log(a/(n + 1)), logexpand=super; (%o1) log(a) - log(n + 1)
When logexpand is set to all or super,
the logarithm of a product expression simplifies to a summation of logarithms.
(%i1) my_product : product (X(i), i, 1, n);
n
_____
| |
(%o1) | | X(i)
| |
i = 1
(%i2) log(my_product), logexpand=all;
n
____
\
(%o2) > log(X(i))
/
----
i = 1
(%i3) log(my_product), logexpand=super;
n
____
\
(%o3) > log(X(i))
/
----
i = 1
When logexpand is false,
these simplifications are disabled.
(%i1) logexpand : false $
(%i2) log(n^2);
2
(%o2) log(n )
(%i3) log(10*x); (%o3) log(10 x)
(%i4) log(a/(n + 1));
a
(%o4) log(-----)
n + 1
(%i5) log ('product (X(i), i, 1, n));
n
_____
| |
(%o5) log(| | X(i))
| |
i = 1
Default value: false
If true implements the rule
log(-n) -> log(n)+%i*%pi for n a positive integer.
Default value: true
If false then no simplification of %e to a
power containing log’s is done.
Represents the principal branch of the complex-valued natural
logarithm with -%pi < carg(x) <= +%pi .
The square root of x. It is represented internally by
x^(1/2). See also rootscontract and radexpand.
Next: Random Numbers, Previous: Root, Exponential and Logarithmic Functions, Up: Elementary Functions [Contents][Index]
Next: Functions and Variables for Trigonometric, Previous: Trigonometric Functions, Up: Trigonometric Functions [Contents][Index]
Maxima has many trigonometric functions defined. Not all trigonometric
identities are programmed, but it is possible for the user to add many
of them using the pattern matching capabilities of the system. The
trigonometric functions defined in Maxima are: acos,
acosh, acot, acoth, acsc,
acsch, asec, asech, asin,
asinh, atan, atanh, cos,
cosh, cot, coth, csc, csch,
sec, sech, sin, sinh, tan,
and tanh. There are a number of commands especially for
handling trigonometric functions, see trigexpand,
trigreduce, and the switch trigsign. Two share
packages extend the simplification rules built into Maxima,
ntrig and atrig1. Do describe(command)
for details.
Previous: Introduction to Trigonometric, Up: Trigonometric Functions [Contents][Index]
Next: Options Controlling Simplification, Previous: Functions and Variables for Trigonometric, Up: Functions and Variables for Trigonometric [Contents][Index]
– Arc Cosine.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Hyperbolic Arc Cosine.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Arc Cotangent.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Hyperbolic Arc Cotangent.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Arc Cosecant.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Hyperbolic Arc Cosecant.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Arc Secant.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Hyperbolic Arc Secant.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Arc Sine.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Hyperbolic Arc Sine.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Arc Tangent.
See also atan2.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– yields the value of \(\tan^{-1}(y/x)\) in the interval \(-\pi\) to \(\pi\) taking into consideration the quadrant of the point \((x,y).\)
Along the branch cut with y = 0 and x < 0, atan2
is continuous with the second quadrant.
See also atan.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Hyperbolic Arc Tangent.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Cosine.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Hyperbolic Cosine.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Cotangent.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Hyperbolic Cotangent.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Cosecant.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Hyperbolic Cosecant.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Secant.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Hyperbolic Secant.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Sine.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Hyperbolic Sine.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Tangent.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
– Hyperbolic Tangent.
For variables that control simplification see %piargs, %iargs, halfangles, triginverses, and trigsign.
Next: Explicit Simplifications Using Identities, Previous: Trigonometric and Hyperbolic Functions, Up: Functions and Variables for Trigonometric [Contents][Index]
Default value: true
When %piargs is true,
trigonometric functions are simplified to algebraic constants
when the argument is an integer multiple
of
\(\pi,\)
\(\pi/2,\)
\(\pi/4,\)
or
\(\pi/6.\)
Maxima knows some identities which can be applied when \(\pi,\) etc., are multiplied by an integer variable (that is, a symbol declared to be integer).
Examples:
(%i1) %piargs : false$
(%i2) [sin (%pi), sin (%pi/2), sin (%pi/3)];
%pi %pi
(%o2) [sin(%pi), sin(---), sin(---)]
2 3
(%i3) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
%pi %pi %pi
(%o3) [sin(---), sin(---), sin(---)]
4 5 6
(%i4) %piargs : true$
(%i5) [sin (%pi), sin (%pi/2), sin (%pi/3)];
sqrt(3)
(%o5) [0, 1, -------]
2
(%i6) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
1 %pi 1
(%o6) [-------, sin(---), -]
sqrt(2) 5 2
(%i7) [cos (%pi/3), cos (10*%pi/3), tan (10*%pi/3),
cos (sqrt(2)*%pi/3)];
1 1 sqrt(2) %pi
(%o7) [-, - -, sqrt(3), cos(-----------)]
2 2 3
Some identities are applied when \(\pi\) and \(\pi/2\) are multiplied by an integer variable.
(%i1) declare (n, integer, m, even)$
(%i2) [sin (%pi * n), cos (%pi * m), sin (%pi/2 * m),
cos (%pi/2 * m)];
m/2
(%o2) [0, 1, 0, (- 1) ]
Default value: true
When %iargs is true,
trigonometric functions are simplified to hyperbolic functions
when the argument is apparently a multiple of the imaginary
unit
\(i.\)
Even when the argument is demonstrably real, the simplification is applied; Maxima considers only whether the argument is a literal multiple of \(i.\)
Examples:
(%i1) %iargs : false$
(%i2) [sin (%i * x), cos (%i * x), tan (%i * x)]; (%o2) [sin(%i x), cos(%i x), tan(%i x)]
(%i3) %iargs : true$
(%i4) [sin (%i * x), cos (%i * x), tan (%i * x)]; (%o4) [%i sinh(x), cosh(x), %i tanh(x)]
Even when the argument is demonstrably real, the simplification is applied.
(%i1) declare (x, imaginary)$
(%i2) [featurep (x, imaginary), featurep (x, real)]; (%o2) [true, false]
(%i3) sin (%i * x); (%o3) %i sinh(x)
Default value: false
When halfangles is true, trigonometric functions of arguments
expr/2 are simplified to functions of expr.
For a real argument x in the interval \(0 \le x < 2\pi,\) \(\sin{x\over 2}\) simplifies to a simple formula: $$ {\sqrt{1-\cos x}\over\sqrt{2}} $$
A complicated factor is needed to make this formula correct for all complex arguments z = x+iy: $$ (-1)^{\lfloor{x/(2\pi)}\rfloor} \left[1-\rm{unit\_step}(-y) \left(1+(-1)^{\lfloor{x/(2\pi)}\rfloor - \lceil{x/(2\pi)}\rceil}\right)\right] $$
Maxima knows this factor and similar factors for the functions sin,
cos, sinh, and cosh. For special values of the argument
z these factors simplify accordingly.
Examples:
(%i1) halfangles : false$
(%i2) sin (x / 2);
x
(%o2) sin(-)
2
(%i3) halfangles : true$
(%i4) sin (x / 2);
x
floor(-----)
2 %pi
(- 1) sqrt(1 - cos(x))
(%o4) ----------------------------------
sqrt(2)
(%i5) assume(x>0, x<2*%pi)$
(%i6) sin(x / 2);
sqrt(1 - cos(x))
(%o6) ----------------
sqrt(2)
Default value: true
When trigsign is true, it permits simplification of negative
arguments to trigonometric functions. E.g.,
\(\sin(-x)\)
will
become
\(-\sin x\)
only if trigsign is true.
Next: Additional Functions, Previous: Options Controlling Simplification, Up: Functions and Variables for Trigonometric [Contents][Index]
Expands trigonometric and hyperbolic functions of
sums of angles and of multiple angles occurring in expr. For best
results, expr should be expanded. To enhance user control of
simplification, this function expands only one level at a time,
expanding sums of angles or multiple angles. To obtain full expansion
into sines and cosines immediately, set the switch trigexpand: true.
trigexpand is governed by the following global flags:
trigexpandIf true causes expansion of all
expressions containing sin’s and cos’s occurring subsequently.
halfanglesIf true causes half-angles to be simplified
away.
trigexpandplusControls the "sum" rule for trigexpand,
expansion of sums (e.g. sin(x + y)) will take place only if
trigexpandplus is true.
trigexpandtimesControls the "product" rule for trigexpand,
expansion of products (e.g. sin(2 x)) will take place only if
trigexpandtimes is true.
Examples:
(%i1) x+sin(3*x)/sin(x),trigexpand=true,expand;
2 2
(%o1) - sin (x) + 3 cos (x) + x
(%i2) trigexpand(sin(10*x+y)); (%o2) cos(10 x) sin(y) + sin(10 x) cos(y)
Default value: true
trigexpandplus controls the "sum" rule for
trigexpand. Thus, when the trigexpand command is used or the
trigexpand switch set to true, expansion of sums
(e.g. sin(x+y)) will take place only if trigexpandplus is
true.
Default value: true
trigexpandtimes controls the "product" rule for trigexpand.
Thus, when the trigexpand command is used or the trigexpand
switch set to true, expansion of products (e.g. sin(2*x))
will take place only if trigexpandtimes is true.
Default value: true
triginverses controls the simplification of the
composition of trigonometric and hyperbolic functions with their inverse
functions.
If all, both e.g. atan(tan(x))
and tan(atan(x)) simplify to x.
If true, the arcfun(fun(x))
simplification is turned off.
If false, both the
arcfun(fun(x)) and
fun(arcfun(x))
simplifications are turned off.
Combines products and powers of trigonometric and hyperbolic sin’s and cos’s of x into those of multiples of x. It also tries to eliminate these functions when they occur in denominators. If x is omitted then all variables in expr are used.
See also poissimp.
(%i1) trigreduce(-sin(x)^2+3*cos(x)^2+x);
cos(2 x) cos(2 x) 1 1
(%o1) -------- + 3 (-------- + -) + x - -
2 2 2 2
Employs the identities
\(\sin\left(x\right)^2 + \cos\left(x\right)^2 = 1\)
and
\(\cosh\left(x\right)^2 - \sinh\left(x\right)^2 = 1\)
to
simplify expressions containing tan, sec,
etc., to sin, cos, sinh, cosh.
trigreduce, ratsimp, and radcan may be
able to further simplify the result.
demo ("trgsmp.dem") displays some examples of trigsimp.
Gives a canonical simplified quasilinear form of a trigonometrical expression;
expr is a rational fraction of several sin, cos or
tan, the arguments of them are linear forms in some variables (or
kernels) and %pi/n (n integer) with integer coefficients.
The result is a simplified fraction with numerator and denominator linear in
sin and cos. Thus trigrat linearize always when it is
possible.
(%i1) trigrat(sin(3*a)/sin(a+%pi/3)); (%o1) sqrt(3) sin(2 a) + cos(2 a) - 1
The following example is taken from Davenport, Siret, and Tournier, Calcul Formel, Masson (or in English, Addison-Wesley), section 1.5.5, Morley theorem.
(%i1) c : %pi/3 - a - b$
(%i2) bc : sin(a)*sin(3*c)/sin(a+b);
%pi
sin(a) sin(3 (- b - a + ---))
3
(%o2) -----------------------------
sin(b + a)
(%i3) ba : bc, c=a, a=c;
%pi
sin(3 a) sin(b + a - ---)
3
(%o3) -------------------------
%pi
sin(a - ---)
3
(%i4) ac2 : ba^2 + bc^2 - 2*bc*ba*cos(b);
2 2 %pi
sin (3 a) sin (b + a - ---)
3
(%o4) ---------------------------
2 %pi
sin (a - ---)
3
%pi
- (2 sin(a) sin(3 a) sin(3 (- b - a + ---)) cos(b)
3
%pi %pi
sin(b + a - ---))/(sin(a - ---) sin(b + a))
3 3
2 2 %pi
sin (a) sin (3 (- b - a + ---))
3
+ -------------------------------
2
sin (b + a)
(%i5) trigrat (ac2); (%o5) - (sqrt(3) sin(4 b + 4 a) - cos(4 b + 4 a) - 2 sqrt(3) sin(4 b + 2 a) + 2 cos(4 b + 2 a) - 2 sqrt(3) sin(2 b + 4 a) + 2 cos(2 b + 4 a) + 4 sqrt(3) sin(2 b + 2 a) - 8 cos(2 b + 2 a) - 4 cos(2 b - 2 a) + sqrt(3) sin(4 b) - cos(4 b) - 2 sqrt(3) sin(2 b) + 10 cos(2 b) + sqrt(3) sin(4 a) - cos(4 a) - 2 sqrt(3) sin(2 a) + 10 cos(2 a) - 9)/4
Previous: Explicit Simplifications Using Identities, Up: Functions and Variables for Trigonometric [Contents][Index]
The atrig1 package contains several additional simplification rules
for inverse trigonometric functions. Together with rules
already known to Maxima, the following angles are fully implemented:
0,
\(\pi/6,\)
\(\pi/4,\)
\(\pi/3,\)
and
\(\pi/2.\)
Corresponding angles in the other three quadrants are also available.
Do load("atrig1"); to use them.
The ntrig package contains a set of simplification rules that are
used to simplify trigonometric function whose arguments are of the form
f(n %pi/10) where f is any of the functions
sin, cos, tan, csc, sec and cot.
Previous: Trigonometric Functions, Up: Elementary Functions [Contents][Index]
A random state object represents the state of the random number generator. The state comprises 627 32-bit words.
make_random_state (n) returns a new random state object
created from an integer seed value equal to n modulo 2^32.
n may be negative.
make_random_state (s) returns a copy of the random state s.
make_random_state (true) returns a new random state object,
using the current computer clock time as the seed.
make_random_state (false) returns a copy of the current state
of the random number generator.
Copies s to the random number generator state.
set_random_state always returns done.
Returns a pseudorandom number. If x is an integer,
random (x) returns an integer from 0 through x - 1
inclusive. If x is a floating point number, random (x)
returns a nonnegative floating point number less than x. random
complains with an error if x is neither an integer nor a float, or if
x is not positive.
The functions make_random_state and set_random_state
maintain the state of the random number generator.
The Maxima random number generator is an implementation of the Mersenne twister MT 19937.
Examples:
(%i1) s1: make_random_state (654321)$
(%i2) set_random_state (s1); (%o2) done
(%i3) random (1000); (%o3) 768
(%i4) random (9573684); (%o4) 7657880
(%i5) random (2^75); (%o5) 11804491615036831636390
(%i6) s2: make_random_state (false)$
(%i7) random (1.0); (%o7) 0.2310127244107132
(%i8) random (10.0); (%o8) 4.3945536458708245
(%i9) random (100.0); (%o9) 32.28666704056853
(%i10) set_random_state (s2); (%o10) done
(%i11) random (1.0); (%o11) 0.2310127244107132
(%i12) random (10.0); (%o12) 4.3945536458708245
(%i13) random (100.0); (%o13) 32.28666704056853
Next: Maxima’s Database, Previous: Simplification [Contents][Index]