Vorige: Introduction to abs_integrate, Nach oben: abs_integrate [Inhalt][Index]
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.
Vorige: Introduction to abs_integrate, Nach oben: abs_integrate [Inhalt][Index]