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| 31.1 Introduction to abs_integrate | ||
| 31.2 Functions and Variables for abs_integrate |
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The package abs_integrate extends Maxima's integration code to
some integrands that involve the absolute value, max, min, signum, or
unit step functions. For integrands of the form p(x) |q(x)|,
where p is a polynomial and q is a polynomial that
factor is able to factor into a product of linear or constant
terms, the abs_integrate package determines an antiderivative
that is continuous on the entire real line. Additionally, for an
integrand that involves one or more parameters, the function
conditional_integrate
tries to determine an antiderivative that
is valid for all parameter values.
Examples:
To use the abs_integrate package, you'll first need to load it:
(%i1) load("abs_integrate.mac")$
(%i2) integrate(abs(x),x);
x abs(x)
(%o2) --------
2
To convert (%o2) into an expression involving the absolute value function,
apply signum_to_abs
; thus
(%i3) signum_to_abs(%);
x abs(x)
(%o3) --------
2
When the integrand has the form p(x) |x - c1| |x - c2| ... |x - cn|,
where p(x) is a polynomial and c1, c2, ..., cn are constants,
the abs_integrate package returns an antiderivative that is valid on the
entire real line; thus without making assumptions on a and b;
for example
(%i4) factor(convert_to_signum(integrate(abs((x-a)*(x-b)),x,a,b)));
3 2
(b - a) signum (b - a)
(%o4) -----------------------
6
Additionally, abs_integrate is able to find antiderivatives of some
integrands involving max,
min,
signum,
and
unit_step,
examples:
(%i5) integrate(max(x,x^2),x);
3 2 3 2
2 x - 3 x 1 1 x x
(%o5) ((----------- + --) signum(x - 1) + --) signum(x) + -- + --
12 12 12 6 4
(%i6) integrate(signum(x) - signum(1-x),x);
(%o6) abs(x) + abs(x - 1)
A plot indicates that indeed (%o5) and (%o6) are continuous at zero and at one.
For definite integrals with numerical integration limits (including
both minus and plus infinity), the abs_integrate package
converts the integrand to signum form and then it tries to subdivide
the integration region so that the integrand simplifies to a
non-signum expression on each subinterval; for example
(%i1) load(abs_integrate)$ (%i2) integrate(1 / (1 + abs(x-5)),x,-5,6); (%o2) log(11) + log(2)
Finally, abs_integrate is able to determine antiderivatives of
some functions of the form F(x, |x - a|); examples
(%i3) integrate(1/(1 + abs(x)),x);
signum(x) (log(x + 1) + log(1 - x))
(%o3) -----------------------------------
2
log(x + 1) - log(1 - x)
+ -----------------------
2
(%i4) integrate(cos(x + abs(x)),x);
(signum(x) + 1) sin(2 x) - 2 x signum(x) + 2 x
(%o4) ----------------------------------------------
4
Barton Willis (Professor of Mathematics, University of Nebraska at
Kearney) wrote the abs_integrate package and its English
language user documentation. This documentation also describes the
partition package for integration. Richard Fateman wrote
partition. Additional documentation for partition is
located at
http://www.cs.berkeley.edu/~fateman/papers/partition.pdf
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Default value: ['signum_int, 'abs_integrate_use_if]
The list extra_integration_methods is a list of functions for
integration. When integrate
is unable to find an
antiderivative, Maxima uses the methods in
extra_integration_methods to attempt to determine an
antiderivative.
Each function f in extra_integration_methods should have
the form f(integrand, variable). The function f may
either return false to indicate failure, or it may return an
expression involving an integration noun form. The integration methods
are tried from the first to the last member of
extra_integration_methods; when no method returns an expression
that does not involve an integration noun form, the value of the
integral is the last value that does not fail (or a pure noun form if
all methods fail).
When the function abs_integrate_use_if is successful, it returns
a conditional expression; for example
(%i1) load(abs_integrate)$
(%i2) integrate(1/(1 + abs(x+1) + abs(x-1)),x);
log(1 - 2 x) 2
(%o2) %if(- (x + 1) > 0, - ------------ + log(3) - -,
2 3
x log(3) 1 log(2 x + 1)
%if(- (x - 1) > 0, - + ------ - -, ------------))
3 2 3 2
(%i3) integrate(exp(-abs(x-1) - abs(x)),x);
2 x - 1
%e - 1
(%o3) %if(- x > 0, --------- - 2 %e ,
2
- 1 1 - 2 x
- 1 3 %e %e
%if(- (x - 1) > 0, %e x - -------, - ---------))
2 2
For definite integration, these conditional expressions can cause trouble:
(%i4) integrate(exp(-abs(x-1) - abs(x)),x, minf,inf);
- 1 2 x
%e (%e - 4)
(%o4) limit %if(- x > 0, -----------------,
x -> inf- 2
- 1 1 - 2 x
%e (2 x - 3) %e
%if(- (x - 1) > 0, ---------------, - ---------))
2 2
- 1 2 x
%e (%e - 4)
- limit %if(- x > 0, -----------------,
x -> minf+ 2
- 1 1 - 2 x
%e (2 x - 3) %e
%if(- (x - 1) > 0, ---------------, - ---------))
2 2
For such definite integrals, try disallowing the method
abs_integrate_use_if:
(%i5) integrate(exp(-abs(x-1) - abs(x)),x, minf,inf),
extra_integration_methods : ['signum_int];
- 1
(%o5) 2 %e
Related options extra_definite_integration_methods.
To use load(abs_integrate)
Default value: ['abs_defint]
The list extra_definite_integration_methods is a list of extra
functions for definite integration. When integrate
is
unable to find a definite integral, Maxima uses the methods in
extra_definite_integration_methods to attempt to determine an
antiderivative.
Each function f in extra_definite_integration_methods
should have the form f(integrand, variable, lo, hi), where
lo and hi are the lower and upper limits of integration,
respectively. The function f may either return false to
indicate failure, or it may return an expression involving an
integration noun form. The integration methods are tried from the
first to the last member of extra_definite_integration_methods;
when no method returns an expression that does not involve an
integration noun form, the value of the integral is the last value
that does not fail (or a pure noun form if all methods fail).
Related options extra_integration_methods.
To use load(abs_integrate).
This function uses the derivative divides rule for integrands of the
form f(w(x)) * diff(w(x),x). When infudu is unable to find
an antiderivative, it returns false.
(%i1) load(abs_integrate)$
(%i2) intfudu(cos(x^2) * x,x);
2
sin(x )
(%o2) -------
2
(%i3) intfudu(x * sqrt(1+x^2),x);
2 3/2
(x + 1)
(%o3) -----------
3
(%i4) intfudu(x * sqrt(1 + x^4),x);
(%o4) false
For the last example, the derivative divides rule fails, so
intfudu returns false.
A hashed array intable contains the antiderivative data. To append a
fact to the hash table, say integrate(f) = g, do this:
(%i5) intable[f] : lambda([u], [g(u),diff(u,%voi)]); (%o5) lambda([u], [g(u), diff(u, %voi)]) (%i6) intfudu(f(z),z); (%o6) g(z) (%i7) intfudu(f(w(x)) * diff(w(x),x),x); (%o7) g(w(x))
An alternative to calling intfudu directly is to use the
extra_integration_methods
mechanism; an example:
(%i1) load(abs_integrate)$
(%i2) load(basic)$
(%i3) load("partition.mac")$
(%i4) integrate(bessel_j(1,x^2) * x,x);
2
bessel_j(0, x )
(%o4) - ---------------
2
(%i5) push('intfudu, extra_integration_methods)$
(%i6) integrate(bessel_j(1,x^2) * x,x);
2
bessel_j(0, x )
(%o6) - ---------------
2
To use load(partition).
Additional documentation
http://www.cs.berkeley.edu/~fateman/papers/partition.pdf.
Related functions intfugudu.
This function uses the derivative divides rule for integrands of the
form f(w(x)) * g(w(x)) * diff(w(x),x). When infugudu is
unable to find an antiderivative, it returns false.
(%i1) load(abs_integrate)$
(%i2) diff(jacobi_sn(x,2/3),x);
2 2
(%o2) jacobi_cn(x, -) jacobi_dn(x, -)
3 3
(%i3) intfugudu(%,x);
2
(%o3) jacobi_sn(x, -)
3
(%i4) diff(jacobi_dn(x^2,a),x);
2 2
(%o4) - 2 a x jacobi_cn(x , a) jacobi_sn(x , a)
(%i5) intfugudu(%,x);
2
(%o5) jacobi_dn(x , a)
For a method for automatically calling infugudu from integrate,
see the documentation for intfudu.
To use load(partition).
Additional documentation
http://www.cs.berkeley.edu/~fateman/papers/partition.pdf
Related functions intfudu.
This function replaces subexpressions of the form q signum(q) by abs(q). Before it does these substitutions, it replaces subexpressions of the form signum(p) * signum(q) by signum(p * q); examples:
(%i1) load(abs_integrate)$
(%i2) map('signum_to_abs, [x * signum(x),
x * y * signum(x)* signum(y)/2]);
abs(x) abs(y)
(%o2) [abs(x), -------------]
2
To use load(abs_integrate).
Appended the facts f_1, f_2, …, f_n to the current context and simplify e. The facts are removed before returning the simplified expression e.
(%i1) load(abs_integrate)$ (%i2) simp_assuming(x + abs(x), x < 0); (%o2) 0
The facts in the current context aren't ignored:
(%i3) assume(x > 0)$ (%i4) simp_assuming(x + abs(x),x < 0); (%o4) 2 x
Since simp_assuming is a macro, effectively simp_assuming quotes
is arguments; this allows
(%i5) simp_assuming(asksign(p), p < 0); (%o5) neg
To use load(abs_integrate).
For an integrand with one or more parameters, this function tries to determine an antiderivative that is valid for all parameter values. When successful, this function returns a conditional expression for the antiderivative.
(%i1) load(abs_integrate)$
(%i2) conditional_integrate(cos(m*x),x);
sin(m x)
(%o2) %if(m # 0, --------, x)
m
(%i3) conditional_integrate(cos(m*x)*cos(x),x);
(%o3) %if((m - 1 # 0) %and (m + 1 # 0),
(m - 1) sin((m + 1) x) + (- m - 1) sin((1 - m) x)
-------------------------------------------------,
2
2 m - 2
sin(2 x) + 2 x
--------------)
4
(%i4) sublis([m=6],%);
5 sin(7 x) + 7 sin(5 x)
(%o4) -----------------------
70
(%i5) conditional_integrate(exp(a*x^2+b*x),x);
2
b
- ---
4 a 2 a x + b
sqrt(%pi) %e erf(-----------)
2 sqrt(- a)
(%o5) %if(a # 0, - ----------------------------------,
2 sqrt(- a)
b x
%e
%if(b # 0, -----, x))
b
This function replaces subexpressions of the form abs(q), unit_step(q),
min(q1, q2, ..., qn) and max(q1, q2, ..., qn) by equivalent
signum terms.
(%i1) load(abs_integrate)$
(%i2) map('convert_to_signum, [abs(x), unit_step(x),
max(a,2), min(a,2)]);
signum(x) (signum(x) + 1)
(%o2) [x signum(x), -------------------------,
2
(a - 2) signum(a - 2) + a + 2 - (a - 2) signum(a - 2) + a + 2
-----------------------------, -------------------------------]
2 2
To convert unit_step
to signum form, the function
convert_to_signum uses unit_step(x) = (1 + signum(x))/2.
To use load(abs_integrate).
Related functions signum_to_abs.
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