Next: Introduction to QUADPACK, Previous: Introduction to Integration, Up: Integration [Contents][Index]
Makes the change of variable given by f(x,y) = 0 in all integrals
occurring in expr with integration with respect to x.
The new variable is y.
The change of variable can also be written f(x) = g(y).
(%i1) assume(a > 0)$
(%i2) 'integrate (%e**sqrt(a*y), y, 0, 4);
4
/
[ sqrt(a) sqrt(y)
(%o2) I %e dy
]
/
0
(%i3) changevar (%, y-z^2/a, z, y);
0
/
[ abs(z)
2 I z %e dz
]
/
- 2 sqrt(a)
(%o3) - ----------------------------
a
An expression containing a noun form, such as the instances of 'integrate
above, may be evaluated by ev with the nouns flag.
For example, the expression returned by changevar above may be evaluated
by ev (%o3, nouns).
changevar may also be used to make changes in the indices of a sum or
product. However, it must be realized that when a change is made in a
sum or product, this change must be a shift, i.e., i = j+ ..., not a
higher degree function. E.g.,
(%i4) sum (a[i]*x^(i-2), i, 0, inf);
inf
====
\ i - 2
(%o4) > a x
/ i
====
i = 0
(%i5) changevar (%, i-2-n, n, i);
inf
====
\ n
(%o5) > a x
/ n + 2
====
n = - 2
A double-integral routine which was written in
top-level Maxima and then translated and compiled to machine code.
Use load ("dblint") to access this package. It uses the Simpson’s rule
method in both the x and y directions to calculate
/b /s(x) | | | | f(x,y) dy dx | | /a /r(x)
The function f must be a translated or compiled function of two variables,
and r and s must each be a translated or compiled function of one
variable, while a and b must be floating point numbers. The routine
has two global variables which determine the number of divisions of the x and y
intervals: dblint_x and dblint_y, both of which are initially 10,
and can be changed independently to other integer values (there are
2*dblint_x+1 points computed in the x direction, and 2*dblint_y+1
in the y direction). The routine subdivides the X axis and then for each value
of X it first computes r(x) and s(x); then the Y axis
between r(x) and s(x) is subdivided and the integral
along the Y axis is performed using Simpson’s rule; then the integral along the
X axis is done using Simpson’s rule with the function values being the
Y-integrals. This procedure may be numerically unstable for a great variety of
reasons, but is reasonably fast: avoid using it on highly oscillatory functions
and functions with singularities (poles or branch points in the region). The Y
integrals depend on how far apart r(x) and s(x) are,
so if the distance s(x) - r(x) varies rapidly with X, there
may be substantial errors arising from truncation with different step-sizes in
the various Y integrals. One can increase dblint_x and dblint_y
in an effort to improve the coverage of the region, at the expense of
computation time. The function values are not saved, so if the function is very
time-consuming, you will have to wait for re-computation if you change anything
(sorry). It is required that the functions f, r, and s be
either translated or compiled prior to calling dblint. This will result
in orders of magnitude speed improvement over interpreted code in many cases!
demo ("dblint") executes a demonstration of dblint applied to an
example problem.
Attempts to compute a definite integral. defint is called by
integrate when limits of integration are specified, i.e., when
integrate is called as
integrate (expr, x, a, b).
Thus from the user’s point of view, it is sufficient to call integrate.
defint returns a symbolic expression, either the computed integral or the
noun form of the integral. See quad_qag and related functions for
numerical approximation of definite integrals.
Default value: true
When erfflag is false, prevents risch from introducing the
erf function in the answer if there were none in the integrand to
begin with.
Computes the inverse Laplace transform of expr with
respect to s and parameter t. expr must be a ratio of
polynomials whose denominator has only linear and quadratic factors;
there is an extension of ilt, called pwilt (Piece-Wise
Inverse Laplace Transform) that handles several other cases where
ilt fails.
By using the functions laplace and ilt together with the
solve or linsolve functions the user can solve a single
differential or convolution integral equation or a set of them.
(%i1) 'integrate (sinh(a*x)*f(t-x), x, 0, t) + b*f(t) = t**2;
t
/
[ 2
(%o1) I f(t - x) sinh(a x) dx + b f(t) = t
]
/
0
(%i2) laplace (%, t, s);
a laplace(f(t), t, s) 2
(%o2) b laplace(f(t), t, s) + --------------------- = --
2 2 3
s - a s
(%i3) linsolve ([%], ['laplace(f(t), t, s)]);
2 2
2 s - 2 a
(%o3) [laplace(f(t), t, s) = --------------------]
5 2 3
b s + (a - a b) s
(%i4) ilt (rhs (first (%)), s, t);
Is a b (a b - 1) positive, negative, or zero?
pos;
sqrt(a b (a b - 1)) t
2 cosh(---------------------) 2
b a t
(%o4) - ----------------------------- + -------
3 2 2 a b - 1
a b - 2 a b + a
2
+ ------------------
3 2 2
a b - 2 a b + a
Default value: true
When true, definite integration tries to find poles in the integrand in
the interval of integration. If there are, then the integral is evaluated
appropriately as a principal value integral. If intanalysis is false,
this check is not performed and integration is done assuming there are no poles.
See also ldefint.
Examples:
Maxima can solve the following integrals, when intanalysis is set to
false:
(%i1) integrate(1/(sqrt(x)+1),x,0,1);
1
/
[ 1
(%o1) I ----------- dx
] sqrt(x) + 1
/
0
(%i2) integrate(1/(sqrt(x)+1),x,0,1),intanalysis:false;
(%o2) 2 - 2 log(2)
(%i3) integrate(cos(a)/sqrt((tan(a))^2 +1),a,-%pi/2,%pi/2);
The number 1 isn't in the domain of atanh
-- an error. To debug this try: debugmode(true);
(%i4) intanalysis:false$
(%i5) integrate(cos(a)/sqrt((tan(a))^2+1),a,-%pi/2,%pi/2);
%pi
(%o5) ---
2
Attempts to symbolically compute the integral of expr with respect to
x. integrate (expr, x) is an indefinite integral,
while integrate (expr, x, a, b) is a definite
integral, with limits of integration a and b. The limits should
not contain x, although integrate does not enforce this
restriction. a need not be less than b.
If b is equal to a, integrate returns zero.
See quad_qag and related functions for numerical approximation of
definite integrals. See residue for computation of residues
(complex integration). See antid for an alternative means of computing
indefinite integrals.
The integral (an expression free of integrate) is returned if
integrate succeeds. Otherwise the return value is
the noun form of the integral (the quoted operator 'integrate)
or an expression containing one or more noun forms.
The noun form of integrate is displayed with an integral sign.
In some circumstances it is useful to construct a noun form by hand, by quoting
integrate with a single quote, e.g.,
'integrate (expr, x). For example, the integral may depend
on some parameters which are not yet computed.
The noun may be applied to its arguments by ev (i, nouns)
where i is the noun form of interest.
integrate handles definite integrals separately from indefinite, and
employs a range of heuristics to handle each case. Special cases of definite
integrals include limits of integration equal to zero or infinity (inf or
minf), trigonometric functions with limits of integration equal to zero
and %pi or 2 %pi, rational functions, integrals related to the
definitions of the beta and psi functions, and some logarithmic
and trigonometric integrals. Processing rational functions may include
computation of residues. If an applicable special case is not found, an attempt
will be made to compute the indefinite integral and evaluate it at the limits of
integration. This may include taking a limit as a limit of integration goes to
infinity or negative infinity; see also ldefint.
Special cases of indefinite integrals include trigonometric functions,
exponential and logarithmic functions,
and rational functions.
integrate may also make use of a short table of elementary integrals.
integrate may carry out a change of variable
if the integrand has the form f(g(x)) * diff(g(x), x).
integrate attempts to find a subexpression g(x) such that
the derivative of g(x) divides the integrand.
This search may make use of derivatives defined by the gradef function.
See also changevar and antid.
If none of the preceding heuristics find the indefinite integral, the Risch
algorithm is executed. The flag risch may be set as an evflag,
in a call to ev or on the command line, e.g.,
ev (integrate (expr, x), risch) or
integrate (expr, x), risch. If risch is present,
integrate calls the risch function without attempting heuristics
first. See also risch.
integrate works only with functional relations represented explicitly
with the f(x) notation. integrate does not respect implicit
dependencies established by the depends function.
integrate may need to know some property of a parameter in the integrand.
integrate will first consult the assume database,
and, if the variable of interest is not there,
integrate will ask the user.
Depending on the question,
suitable responses are yes; or no;,
or pos;, zero;, or neg;.
integrate is not, by default, declared to be linear. See declare
and linear.
integrate attempts integration by parts only in a few special cases.
Examples:
(%i1) integrate (sin(x)^3, x);
3
cos (x)
(%o1) ------- - cos(x)
3
(%i2) integrate (x/ sqrt (b^2 - x^2), x);
2 2
(%o2) - sqrt(b - x )
(%i3) integrate (cos(x)^2 * exp(x), x, 0, %pi);
%pi
3 %e 3
(%o3) ------- - -
5 5
(%i4) integrate (x^2 * exp(-x^2), x, minf, inf);
sqrt(%pi)
(%o4) ---------
2
assume and interactive query.
(%i1) assume (a > 1)$
(%i2) integrate (x**a/(x+1)**(5/2), x, 0, inf);
2 a + 2
Is ------- an integer?
5
no;
Is 2 a - 3 positive, negative, or zero?
neg;
3
(%o2) beta(a + 1, - - a)
2
gradef, and one using the
derivation diff(r(x)) of an unspecified function r(x).
(%i3) gradef (q(x), sin(x**2)); (%o3) q(x)
(%i4) diff (log (q (r (x))), x);
d 2
(-- (r(x))) sin(r (x))
dx
(%o4) ----------------------
q(r(x))
(%i5) integrate (%, x); (%o5) log(q(r(x)))
'integrate noun form. In this example, Maxima
can extract one factor of the denominator of a rational function, but cannot
factor the remainder or otherwise find its integral. grind shows the
noun form 'integrate in the result. See also
integrate_use_rootsof for more on integrals of rational functions.
(%i1) expand ((x-4) * (x^3+2*x+1));
4 3 2
(%o1) x - 4 x + 2 x - 7 x - 4
(%i2) integrate (1/%, x);
/ 2
[ x + 4 x + 18
I ------------- dx
] 3
log(x - 4) / x + 2 x + 1
(%o2) ---------- - ------------------
73 73
(%i3) grind (%);
log(x-4)/73-('integrate((x^2+4*x+18)/(x^3+2*x+1),x))/73$
f_1 in this
example contains the noun form of integrate. The quote-quote operator
'' causes the integral to be evaluated, and the result becomes the
body of f_2.
(%i1) f_1 (a) := integrate (x^3, x, 1, a);
3
(%o1) f_1(a) := integrate(x , x, 1, a)
(%i2) ev (f_1 (7), nouns); (%o2) 600
(%i3) /* Note parentheses around integrate(...) here */
f_2 (a) := ''(integrate (x^3, x, 1, a));
4
a 1
(%o3) f_2(a) := -- - -
4 4
(%i4) f_2 (7); (%o4) 600
Default value: %c
When a constant of integration is introduced by indefinite integration of an
equation, the name of the constant is constructed by concatenating
integration_constant and integration_constant_counter.
integration_constant may be assigned any symbol.
Examples:
(%i1) integrate (x^2 = 1, x);
3
x
(%o1) -- = x + %c1
3
(%i2) integration_constant : 'k; (%o2) k
(%i3) integrate (x^2 = 1, x);
3
x
(%o3) -- = x + k2
3
Default value: 0
When a constant of integration is introduced by indefinite integration of an
equation, the name of the constant is constructed by concatenating
integration_constant and integration_constant_counter.
integration_constant_counter is incremented before constructing the next
integration constant.
Examples:
(%i1) integrate (x^2 = 1, x);
3
x
(%o1) -- = x + %c1
3
(%i2) integrate (x^2 = 1, x);
3
x
(%o2) -- = x + %c2
3
(%i3) integrate (x^2 = 1, x);
3
x
(%o3) -- = x + %c3
3
(%i4) reset (integration_constant_counter); (%o4) [integration_constant_counter]
(%i5) integrate (x^2 = 1, x);
3
x
(%o5) -- = x + %c1
3
Default value: false
When integrate_use_rootsof is true and the denominator of
a rational function cannot be factored, integrate returns the integral
in a form which is a sum over the roots (not yet known) of the denominator.
For example, with integrate_use_rootsof set to false,
integrate returns an unsolved integral of a rational function in noun
form:
(%i1) integrate_use_rootsof: false$
(%i2) integrate (1/(1+x+x^5), x);
/ 2
[ x - 4 x + 5
I ------------ dx 2 x + 1
] 3 2 2 5 atan(-------)
/ x - x + 1 log(x + x + 1) sqrt(3)
(%o2) ----------------- - --------------- + ---------------
7 14 7 sqrt(3)
Now we set the flag to be true and the unsolved part of the integral will be expressed as a summation over the roots of the denominator of the rational function:
(%i3) integrate_use_rootsof: true$
(%i4) integrate (1/(1+x+x^5), x);
==== 2
\ (%r4 - 4 %r4 + 5) log(x - %r4)
> -------------------------------
/ 2
==== 3 %r4 - 2 %r4
3 2
%r4 in rootsof(%r4 - %r4 + 1, %r4)
(%o4) ----------------------------------------------------------
7
2 x + 1
2 5 atan(-------)
log(x + x + 1) sqrt(3)
- --------------- + ---------------
14 7 sqrt(3)
Alternatively the user may compute the roots of the denominator separately,
and then express the integrand in terms of these roots, e.g.,
1/((x - a)*(x - b)*(x - c)) or 1/((x^2 - (a+b)*x + a*b)*(x - c))
if the denominator is a cubic polynomial.
Sometimes this will help Maxima obtain a more useful result.
Attempts to compute the Laplace transform of expr with respect to the
variable t and transform parameter s. The Laplace
transform of the function f(t) is the one-sided transform defined by
where \(F(s)\) is the transform of \(f(t)\), represented by expr.
laplace recognizes in expr the functions delta, exp,
log, sin, cos, sinh, cosh, and erf,
as well as derivative, integrate, sum, and ilt. If
laplace fails to find a transform the function specint is called.
specint can find the laplace transform for expressions with special
functions like the bessel functions bessel_j, bessel_i, …
and can handle the unit_step function. See also specint.
If specint cannot find a solution too, a noun laplace is returned.
expr may also be a linear, constant coefficient differential equation in
which case atvalue of the dependent variable is used.
The required atvalue may be supplied either before or after the transform is
computed. Since the initial conditions must be specified at zero, if one has
boundary conditions imposed elsewhere he can impose these on the general
solution and eliminate the constants by solving the general solution
for them and substituting their values back.
laplace recognizes convolution integrals of the form
Other kinds of convolutions are not recognized.
Functional relations must be explicitly represented in expr;
implicit relations, established by depends, are not recognized.
That is, if \(f\) depends on \(x\) and \(y\),
\(f (x, y)\) must appear in expr.
See also ilt, the inverse Laplace transform.
Examples:
(%i1) laplace (exp (2*t + a) * sin(t) * t, t, s);
a
%e (2 s - 4)
(%o1) ---------------
2 2
(s - 4 s + 5)
(%i2) laplace ('diff (f (x), x), x, s);
(%o2) s laplace(f(x), x, s) - f(0)
(%i3) diff (diff (delta (t), t), t);
2
d
(%o3) --- (delta(t))
2
dt
(%i4) laplace (%, t, s);
!
d ! 2
(%o4) - -- (delta(t))! + s - delta(0) s
dt !
!t = 0
(%i5) assume(a>0)$
(%i6) laplace(gamma_incomplete(a,t),t,s),gamma_expand:true;
- a - 1
gamma(a) gamma(a) s
(%o6) -------- - -----------------
s 1 a
(- + 1)
s
(%i7) factor(laplace(gamma_incomplete(1/2,t),t,s));
s + 1
sqrt(%pi) (sqrt(s) sqrt(-----) - 1)
s
(%o7) -----------------------------------
3/2 s + 1
s sqrt(-----)
s
(%i8) assume(exp(%pi*s)>1)$
(%i9) laplace(sum((-1)^n*unit_step(t-n*%pi)*sin(t),n,0,inf),t,s),
simpsum;
%i %i
------------------------ - ------------------------
- %pi s - %pi s
(s + %i) (1 - %e ) (s - %i) (1 - %e )
(%o9) ---------------------------------------------------
2
(%i9) factor(%);
%pi s
%e
(%o9) -------------------------------
%pi s
(s - %i) (s + %i) (%e - 1)
Attempts to compute the definite integral of expr by using limit
to evaluate the indefinite integral of expr with respect to x
at the upper limit b and at the lower limit a.
If it fails to compute the definite integral,
ldefint returns an expression containing limits as noun forms.
ldefint is not called from integrate, so executing
ldefint (expr, x, a, b) may yield a different
result than integrate (expr, x, a, b).
ldefint always uses the same method to evaluate the definite integral,
while integrate may employ various heuristics and may recognize some
special cases.
Computes the inverse Laplace transform of expr with
respect to s and parameter t. Unlike ilt,
pwilt is able to return piece-wise and periodic functions
and can also handle some cases with polynomials of degree greater than 3
in the denominator.
Two examples where ilt fails:
(%i1) pwilt (exp(-s)*s/(s^3-2*s-s+2), s, t);
t - 1 - 2 (t - 1)
(t - 1) %e 2 %e
(%o1) hstep(t - 1) (--------------- - ---------------)
3 9
(%i2) pwilt ((s^2+2)/(s^2-1), s, t);
t - t
3 %e 3 %e
(%o2) delta(t) + ----- - -------
2 2
The calculation makes use of the global variable potentialzeroloc[0]
which must be nonlist or of the form
[indeterminatej=expressionj, indeterminatek=expressionk, ...]
the former being equivalent to the nonlist expression for all right-hand
sides in the latter. The indicated right-hand sides are used as the
lower limit of integration. The success of the integrations may
depend upon their values and order. potentialzeroloc is initially set
to 0.
Default value: false
When prefer_d is true, specint will prefer to
express solutions using parabolic_cylinder_d rather than
hypergeometric functions.
In the example below, the solution contains parabolic_cylinder_d
when prefer_d is true.
(%i1) assume(s>0); (%o1) [s > 0]
(%i2) factor(specint(ex:%e^-(t^2/8)*exp(-s*t),t));
2
2 s
(%o2) - sqrt(2) sqrt(%pi) %e (erf(sqrt(2) s) - 1)
(%i3) specint(ex,t),prefer_d=true;
2
s
--
s 8
parabolic_cylinder_d(- 1, -------) %e
sqrt(2)
(%o3) ---------------------------------------
sqrt(2)
Computes the residue in the complex plane of the expression expr when the
variable z assumes the value z_0. The residue is the coefficient of
(z - z_0)^(-1) in the Laurent series for expr.
(%i1) residue (s/(s**2+a**2), s, a*%i);
1
(%o1) -
2
(%i2) residue (sin(a*x)/x**4, x, 0);
3
a
(%o2) - --
6
Integrates expr with respect to x using the
transcendental case of the Risch algorithm. (The algebraic case of
the Risch algorithm has not been implemented.) This currently
handles the cases of nested exponentials and logarithms which the main
part of integrate can’t do. integrate will automatically apply
risch if given these cases.
erfflag, if false, prevents risch from introducing the
erf function in the answer if there were none in the integrand to begin
with.
(%i1) risch (x^2*erf(x), x);
2
3 2 - x
%pi x erf(x) + (sqrt(%pi) x + sqrt(%pi)) %e
(%o1) -------------------------------------------------
3 %pi
(%i2) diff(%, x), ratsimp;
2
(%o2) x erf(x)
Compute the Laplace transform of expr with respect to the variable t. The integrand expr may contain special functions. The parameter s maybe be named something else; it is determined automatically, as the examples below show where p is used in some places.
The following special functions are handled by specint: incomplete gamma
function, error functions (but not the error function erfi, it is easy to
transform erfi e.g. to the error function erf), exponential
integrals, bessel functions (including products of bessel functions), hankel
functions, hermite and the laguerre polynomials.
Furthermore, specint can handle the hypergeometric function
%f[p,q]([],[],z), the Whittaker function of the first kind
%m[u,k](z) and of the second kind %w[u,k](z).
The result may be in terms of special functions and can include unsimplified
hypergeometric functions. If variable prefer_d is true
then the parabolic_cylinder_d function may be used in the result
in preference to hypergeometric functions.
When laplace fails to find a Laplace transform, specint is called.
Because laplace knows more general rules for Laplace transforms, it is
preferable to use laplace and not specint.
demo("hypgeo") displays several examples of Laplace transforms computed by
specint.
Examples:
(%i1) assume (p > 0, a > 0)$
(%i2) specint (t^(1/2) * exp(-a*t/4) * exp(-p*t), t);
sqrt(%pi)
(%o2) ------------
a 3/2
2 (p + -)
4
(%i3) specint (t^(1/2) * bessel_j(1, 2 * a^(1/2) * t^(1/2))
* exp(-p*t), t);
- a/p
sqrt(a) %e
(%o3) ---------------
2
p
Examples for exponential integrals:
(%i4) assume(s>0,a>0,s-a>0)$
(%i5) ratsimp(specint(%e^(a*t)
*(log(a)+expintegral_e1(a*t))*%e^(-s*t),t));
log(s)
(%o5) ------
s - a
(%i6) logarc:true$
(%i7) gamma_expand:true$
radcan(specint((cos(t)*expintegral_si(t)
-sin(t)*expintegral_ci(t))*%e^(-s*t),t));
log(s)
(%o8) ------
2
s + 1
ratsimp(specint((2*t*log(a)+2/a*sin(a*t)
-2*t*expintegral_ci(a*t))*%e^(-s*t),t));
2 2
log(s + a )
(%o9) ------------
2
s
Results when using the expansion of gamma_incomplete and when changing
the representation to expintegral_e1:
(%i10) assume(s>0)$
(%i11) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
1
gamma_incomplete(-, k s)
2
(%o11) ------------------------
sqrt(%pi) sqrt(s)
(%i12) gamma_expand:true$
(%i13) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
erfc(sqrt(k) sqrt(s))
(%o13) ---------------------
sqrt(s)
(%i14) expintrep:expintegral_e1$
(%i15) ratsimp(specint(1/(t+a)^2*%e^(-s*t),t));
a s
a s %e expintegral_e1(a s) - 1
(%o15) - ---------------------------------
a
Equivalent to ldefint with tlimswitch set to true.
Next: Introduction to QUADPACK, Previous: Introduction to Integration, Up: Integration [Contents][Index]