# Integrals¶

The integrals module in Diofant implements methods to calculate definite and indefinite integrals of expressions.

Principal method in this module is integrate()

• integrate(f, x) returns the indefinite integral $$\int f\,dx$$

• integrate(f, (x, a, b)) returns the definite integral $$\int_{a}^{b} f\,dx$$

## Examples¶

Diofant can integrate a vast array of functions. It can integrate polynomial functions:

>>> init_printing(pretty_print=True, use_unicode=False, wrap_line=False, no_global=True)
>>> x = Symbol('x')
>>> integrate(x**2 + x + 1, x)
3    2
x    x
-- + -- + x
3    2


Rational functions:

>>> integrate(x/(x**2+2*x+1), x)
1
log(x + 1) + -----
x + 1


Exponential-polynomial functions. These multiplicative combinations of polynomials and the functions exp, cos and sin can be integrated by hand using repeated integration by parts, which is an extremely tedious process. Happily, Diofant will deal with these integrals.

>>> integrate(x**2 * exp(x) * cos(x), x)
x  2           x  2                         x           x
E *x *sin(x)   E *x *cos(x)    x            E *sin(x)   E *cos(x)
------------ + ------------ - E *x*sin(x) + --------- - ---------
2              2                           2           2


even a few nonelementary integrals (in particular, some integrals involving the error function) can be evaluated:

>>> integrate(exp(-x**2)*erf(x), x)
____    2
\/ pi *erf (x)
--------------
4


## Integral Transforms¶

Diofant has special support for definite integrals, and integral transforms.

diofant.integrals.transforms.mellin_transform(f, x, s, **hints)[source]

Compute the Mellin transform $$F(s)$$ of $$f(x)$$,

$F(s) = \int_0^\infty x^{s-1} f(x) \mathrm{d}x.$
For all “sensible” functions, this converges absolutely in a strip

$$a < \operatorname{Re}(s) < b$$.

The Mellin transform is related via change of variables to the Fourier transform, and also to the (bilateral) Laplace transform.

This function returns (F, (a, b), cond) where F is the Mellin transform of f, (a, b) is the fundamental strip (as above), and cond are auxiliary convergence conditions.

If the integral cannot be computed in closed form, this function returns an unevaluated MellinTransform object.

For a description of possible hints, refer to the docstring of diofant.integrals.transforms.IntegralTransform.doit(). If noconds=False, then only $$F$$ will be returned (i.e. not cond, and also not the strip (a, b)).

>>> from diofant.abc import s
>>> mellin_transform(exp(-x), x, s)
(gamma(s), (0, oo), True)

diofant.integrals.transforms.inverse_mellin_transform(F, s, x, strip, **hints)[source]

Compute the inverse Mellin transform of $$F(s)$$ over the fundamental strip given by strip=(a, b).

This can be defined as

$f(x) = \int_{c - i\infty}^{c + i\infty} x^{-s} F(s) \mathrm{d}s,$

for any $$c$$ in the fundamental strip. Under certain regularity conditions on $$F$$ and/or $$f$$, this recovers $$f$$ from its Mellin transform $$F$$ (and vice versa), for positive real $$x$$.

One of $$a$$ or $$b$$ may be passed as None; a suitable $$c$$ will be inferred.

If the integral cannot be computed in closed form, this function returns an unevaluated InverseMellinTransform object.

Note that this function will assume x to be positive and real, regardless of the diofant assumptions!

For a description of possible hints, refer to the docstring of diofant.integrals.transforms.IntegralTransform.doit().

>>> from diofant.abc import s
>>> inverse_mellin_transform(gamma(s), s, x, (0, oo))
E**(-x)


The fundamental strip matters:

>>> f = 1/(s**2 - 1)
>>> inverse_mellin_transform(f, s, x, (-oo, -1))
(x/2 - 1/(2*x))*Heaviside(x - 1)
>>> inverse_mellin_transform(f, s, x, (-1, 1))
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
>>> inverse_mellin_transform(f, s, x, (1, oo))
(-x/2 + 1/(2*x))*Heaviside(-x + 1)

diofant.integrals.transforms.laplace_transform(f, t, s, **hints)[source]

Compute the Laplace Transform $$F(s)$$ of $$f(t)$$,

$F(s) = \int_0^\infty e^{-st} f(t) \mathrm{d}t.$

For all “sensible” functions, this converges absolutely in a half plane $$a < \operatorname{Re}(s)$$.

This function returns (F, a, cond) where F is the Laplace transform of f, $$\operatorname{Re}(s) > a$$ is the half-plane of convergence, and cond are auxiliary convergence conditions.

If the integral cannot be computed in closed form, this function returns an unevaluated LaplaceTransform object.

For a description of possible hints, refer to the docstring of diofant.integrals.transforms.IntegralTransform.doit(). If noconds=True, only $$F$$ will be returned (i.e. not cond, and also not the plane a).

>>> from diofant.abc import s
>>> laplace_transform(t**a, t, s)
(s**(-a)*gamma(a + 1)/s, 0, -re(a) < 1)

diofant.integrals.transforms.inverse_laplace_transform(F, s, t, plane=None, **hints)[source]

Compute the inverse Laplace transform of $$F(s)$$, defined as

$f(t) = \int_{c-i\infty}^{c+i\infty} e^{st} F(s) \mathrm{d}s,$

for $$c$$ so large that $$F(s)$$ has no singularites in the half-plane $$\operatorname{Re}(s) > c-\epsilon$$.

The plane can be specified by argument plane, but will be inferred if passed as None.

Under certain regularity conditions, this recovers $$f(t)$$ from its Laplace Transform $$F(s)$$, for non-negative $$t$$, and vice versa.

If the integral cannot be computed in closed form, this function returns an unevaluated InverseLaplaceTransform object.

Note that this function will always assume $$t$$ to be real, regardless of the diofant assumption on $$t$$.

For a description of possible hints, refer to the docstring of diofant.integrals.transforms.IntegralTransform.doit().

>>> from diofant.abc import s
>>> a = Symbol('a', positive=True)
>>> inverse_laplace_transform(exp(-a*s)/s, s, t)
Heaviside(-a + t)

diofant.integrals.transforms.fourier_transform(f, x, k, **hints)[source]

Compute the unitary, ordinary-frequency Fourier transform of $$f$$, defined as

$F(k) = \int_{-\infty}^\infty f(x) e^{-2\pi i x k} \mathrm{d} x.$

If the transform cannot be computed in closed form, this function returns an unevaluated FourierTransform object.

For a description of possible hints, refer to the docstring of diofant.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> fourier_transform(exp(-x**2), x, k)
E**(-pi**2*k**2)*sqrt(pi)
>>> fourier_transform(exp(-x**2), x, k, noconds=False)
(E**(-pi**2*k**2)*sqrt(pi), True)

diofant.integrals.transforms.inverse_fourier_transform(F, k, x, **hints)[source]

Compute the unitary, ordinary-frequency inverse Fourier transform of $$F$$, defined as

$f(x) = \int_{-\infty}^\infty F(k) e^{2\pi i x k} \mathrm{d} k.$

If the transform cannot be computed in closed form, this function returns an unevaluated InverseFourierTransform object.

For a description of possible hints, refer to the docstring of diofant.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x)
E**(-x**2)
>>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x, noconds=False)
(E**(-x**2), True)

diofant.integrals.transforms.sine_transform(f, x, k, **hints)[source]

Compute the unitary, ordinary-frequency sine transform of $$f$$, defined as

$F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \sin(2\pi x k) \mathrm{d} x.$

If the transform cannot be computed in closed form, this function returns an unevaluated SineTransform object.

For a description of possible hints, refer to the docstring of diofant.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> sine_transform(x*exp(-a*x**2), x, k)
sqrt(2)*E**(-k**2/(4*a))*k/(4*a**(3/2))
>>> sine_transform(x**(-a), x, k)
2**(-a + 1/2)*k**(a - 1)*gamma(-a/2 + 1)/gamma(a/2 + 1/2)

diofant.integrals.transforms.inverse_sine_transform(F, k, x, **hints)[source]

Compute the unitary, ordinary-frequency inverse sine transform of $$F$$, defined as

$f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \sin(2\pi x k) \mathrm{d} k.$

If the transform cannot be computed in closed form, this function returns an unevaluated InverseSineTransform object.

For a description of possible hints, refer to the docstring of diofant.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> inverse_sine_transform(2**((1-2*a)/2)*k**(a - 1)*
...     gamma(-a/2 + 1)/gamma((a+1)/2), k, x)
x**(-a)
>>> inverse_sine_transform(sqrt(2)*k*exp(-k**2/(4*a))/(4*sqrt(a)**3), k, x)
E**(-a*x**2)*x

diofant.integrals.transforms.cosine_transform(f, x, k, **hints)[source]

Compute the unitary, ordinary-frequency cosine transform of $$f$$, defined as

$F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \cos(2\pi x k) \mathrm{d} x.$

If the transform cannot be computed in closed form, this function returns an unevaluated CosineTransform object.

For a description of possible hints, refer to the docstring of diofant.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> cosine_transform(exp(-a*x), x, k)
sqrt(2)*a/(sqrt(pi)*(a**2 + k**2))
>>> cosine_transform(exp(-a*sqrt(x))*cos(a*sqrt(x)), x, k)
E**(-a**2/(2*k))*a/(2*k**(3/2))

diofant.integrals.transforms.inverse_cosine_transform(F, k, x, **hints)[source]

Compute the unitary, ordinary-frequency inverse cosine transform of $$F$$, defined as

$f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \cos(2\pi x k) \mathrm{d} k.$

If the transform cannot be computed in closed form, this function returns an unevaluated InverseCosineTransform object.

For a description of possible hints, refer to the docstring of diofant.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> inverse_cosine_transform(sqrt(2)*a/(sqrt(pi)*(a**2 + k**2)), k, x)
E**(-a*x)
>>> inverse_cosine_transform(1/sqrt(k), k, x)
1/sqrt(x)

diofant.integrals.transforms.hankel_transform(f, r, k, nu, **hints)[source]

Compute the Hankel transform of $$f$$, defined as

$F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r.$

If the transform cannot be computed in closed form, this function returns an unevaluated HankelTransform object.

For a description of possible hints, refer to the docstring of diofant.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> from diofant.abc import r, nu, k

>>> ht = hankel_transform(1/r**m, r, k, nu)
>>> ht
2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2)

>>> inverse_hankel_transform(ht, k, r, nu)
r**(-m)

>>> ht = hankel_transform(exp(-a*r), r, k, 0)
>>> ht
a/(k**3*(a**2/k**2 + 1)**(3/2))

>>> inverse_hankel_transform(ht, k, r, 0)
E**(-a*r)

diofant.integrals.transforms.inverse_hankel_transform(F, k, r, nu, **hints)[source]

Compute the inverse Hankel transform of $$F$$ defined as

$f(r) = \int_{0}^\infty F_\nu(k) J_\nu(k r) k \mathrm{d} k.$

If the transform cannot be computed in closed form, this function returns an unevaluated InverseHankelTransform object.

For a description of possible hints, refer to the docstring of diofant.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> from diofant.abc import r, nu, k

>>> ht = hankel_transform(1/r**m, r, k, nu)
>>> ht
2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2)

>>> inverse_hankel_transform(ht, k, r, nu)
r**(-m)

>>> ht = hankel_transform(exp(-a*r), r, k, 0)
>>> ht
a/(k**3*(a**2/k**2 + 1)**(3/2))

>>> inverse_hankel_transform(ht, k, r, 0)
E**(-a*r)


## Internals¶

There is a general method for calculating antiderivatives of elementary functions, called the Risch algorithm. The Risch algorithm is a decision procedure that can determine whether an elementary solution exists, and in that case calculate it. It can be extended to handle many nonelementary functions in addition to the elementary ones.

Diofant currently uses a simplified version of the Risch algorithm, called the Risch-Norman algorithm. This algorithm is much faster, but may fail to find an antiderivative, although it is still very powerful. Diofant also uses pattern matching and heuristics to speed up evaluation of some types of integrals, e.g. polynomials.

For non-elementary definite integrals, Diofant uses so-called Meijer G-functions. Details are described here.

## API reference¶

diofant.integrals.integrals.integrate(f, var, ...)[source]

Compute definite or indefinite integral of one or more variables using Risch-Norman algorithm and table lookup. This procedure is able to handle elementary algebraic and transcendental functions and also a huge class of special functions, including Airy, Bessel, Whittaker and Lambert.

var can be:

• a symbol – indefinite integration

• a tuple (symbol, a) – indefinite integration with result

given with $$a$$ replacing $$symbol$$

• a tuple (symbol, a, b) – definite integration

Several variables can be specified, in which case the result is multiple integration. (If var is omitted and the integrand is univariate, the indefinite integral in that variable will be performed.)

Indefinite integrals are returned without terms that are independent of the integration variables. (see examples)

Definite improper integrals often entail delicate convergence conditions. Pass conds=’piecewise’, ‘separate’ or ‘none’ to have these returned, respectively, as a Piecewise function, as a separate result (i.e. result will be a tuple), or not at all (default is ‘piecewise’).

Strategy

Diofant uses various approaches to definite integration. One method is to find an antiderivative for the integrand, and then use the fundamental theorem of calculus. Various functions are implemented to integrate polynomial, rational and trigonometric functions, and integrands containing DiracDelta terms.

Diofant also implements the part of the Risch algorithm, which is a decision procedure for integrating elementary functions, i.e., the algorithm can either find an elementary antiderivative, or prove that one does not exist. There is also a (very successful, albeit somewhat slow) general implementation of the heuristic Risch algorithm. This algorithm will eventually be phased out as more of the full Risch algorithm is implemented. See the docstring of Integral._eval_integral() for more details on computing the antiderivative using algebraic methods.

The option risch=True can be used to use only the (full) Risch algorithm. This is useful if you want to know if an elementary function has an elementary antiderivative. If the indefinite Integral returned by this function is an instance of NonElementaryIntegral, that means that the Risch algorithm has proven that integral to be non-elementary. Note that by default, additional methods (such as the Meijer G method outlined below) are tried on these integrals, as they may be expressible in terms of special functions, so if you only care about elementary answers, use risch=True. Also note that an unevaluated Integral returned by this function is not necessarily a NonElementaryIntegral, even with risch=True, as it may just be an indication that the particular part of the Risch algorithm needed to integrate that function is not yet implemented.

Another family of strategies comes from re-writing the integrand in terms of so-called Meijer G-functions. Indefinite integrals of a single G-function can always be computed, and the definite integral of a product of two G-functions can be computed from zero to infinity. Various strategies are implemented to rewrite integrands as G-functions, and use this information to compute integrals (see the meijerint module).

In general, the algebraic methods work best for computing antiderivatives of (possibly complicated) combinations of elementary functions. The G-function methods work best for computing definite integrals from zero to infinity of moderately complicated combinations of special functions, or indefinite integrals of very simple combinations of special functions.

The strategy employed by the integration code is as follows:

• If computing a definite integral, and both limits are real, and at least one limit is +- oo, try the G-function method of definite integration first.

• Try to find an antiderivative, using all available methods, ordered by performance (that is try fastest method first, slowest last; in particular polynomial integration is tried first, Meijer G-functions second to last, and heuristic Risch last).

• If still not successful, try G-functions irrespective of the limits.

The option meijerg=True, False, None can be used to, respectively: always use G-function methods and no others, never use G-function methods, or use all available methods (in order as described above). It defaults to None.

Examples

>>> integrate(x*y, x)
x**2*y/2

>>> integrate(log(x), x)
x*log(x) - x

>>> integrate(log(x), (x, 1, a))
a*log(a) - a + 1

>>> integrate(x)
x**2/2


Terms that are independent of x are dropped by indefinite integration:

>>> integrate(sqrt(1 + x), (x, 0, x))
2*(x + 1)**(3/2)/3 - 2/3
>>> integrate(sqrt(1 + x), x)
2*(x + 1)**(3/2)/3

>>> integrate(x*y)
Traceback (most recent call last):
...
ValueError: specify integration variables to integrate x*y


Note that integrate(x) syntax is meant only for convenience in interactive sessions and should be avoided in library code.

>>> integrate(x**a*exp(-x), (x, 0, oo)) # same as conds='piecewise'
Piecewise((gamma(a + 1), -re(a) < 1),
(Integral(E**(-x)*x**a, (x, 0, oo)), true))

>>> integrate(x**a*exp(-x), (x, 0, oo), conds='none')
gamma(a + 1)

>>> integrate(x**a*exp(-x), (x, 0, oo), conds='separate')
(gamma(a + 1), -re(a) < 1)

diofant.integrals.integrals.line_integrate(field, Curve, variables)[source]

Compute the line integral.

Examples

>>> C = Curve([E**t + 1, E**t - 1], (t, 0, ln(2)))
>>> line_integrate(x + y, C, [x, y])
3*sqrt(2)

diofant.integrals.deltafunctions.deltaintegrate(f, x)[source]

The idea for integration is the following:

• If we are dealing with a DiracDelta expression, i.e. DiracDelta(g(x)), we try to simplify it.

If we could simplify it, then we integrate the resulting expression. We already know we can integrate a simplified expression, because only simple DiracDelta expressions are involved.

If we couldn’t simplify it, there are two cases:

1. The expression is a simple expression: we return the integral, taking care if we are dealing with a Derivative or with a proper DiracDelta.

2. The expression is not simple (i.e. DiracDelta(cos(x))): we can do nothing at all.

• If the node is a multiplication node having a DiracDelta term:

First we expand it.

If the expansion did work, then we try to integrate the expansion.

If not, we try to extract a simple DiracDelta term, then we have two cases:

1. We have a simple DiracDelta term, so we return the integral.

2. We didn’t have a simple term, but we do have an expression with simplified DiracDelta terms, so we integrate this expression.

Examples

>>> deltaintegrate(x*sin(x)*cos(x)*DiracDelta(x - 1), x)
sin(1)*cos(1)*Heaviside(x - 1)
>>> deltaintegrate(y**2*DiracDelta(x - z)*DiracDelta(y - z), y)
z**2*DiracDelta(x - z)*Heaviside(y - z)

diofant.integrals.rationaltools.ratint(f, x, **flags)[source]

Performs indefinite integration of rational functions.

Given a field $$K$$ and a rational function $$f = p/q$$, where $$p$$ and $$q$$ are polynomials in $$K[x]$$, returns a function $$g$$ such that $$f = g'$$.

>>> ratint(36/(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2), x)
(12*x + 6)/(x**2 - 1) + 4*log(x - 2) - 4*log(x + 1)


References

• M. Bronstein, Symbolic Integration I: Transcendental Functions, Second Edition, Springer-Verlag, 2005, pp. 35-70

diofant.integrals.rationaltools.ratint_logpart(f, g, x, t=None)[source]

Lazard-Rioboo-Trager algorithm.

Given a field K and polynomials f and g in K[x], such that f and g are coprime, deg(f) < deg(g) and g is square-free, returns a list of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i in K[t, x] and q_i in K[t], and:

          ___    ___
d  f   d  \     \  
-- - = --  )      )   a log(s_i(a, x))
dx g   dx /__,   /__,
i=1..n a | q_i(a) = 0


Examples

>>> ratint_logpart(Poly(1, x), Poly(x**2 + x + 1, x), x)
[(Poly(x + 3*_t/2 + 1/2, x, domain='QQ[_t]'),
Poly(3*_t**2 + 1, _t, domain='ZZ'))]
>>> ratint_logpart(Poly(12, x), Poly(x**2 - x - 2, x), x)
[(Poly(x - 3*_t/8 - 1/2, x, domain='QQ[_t]'),
Poly(_t**2 - 16, _t, domain='ZZ'))]

diofant.integrals.rationaltools.ratint_ratpart(f, g, x)[source]

Given a field K and polynomials f and g in K[x], such that f and g are coprime and deg(f) < deg(g), returns fractions A and B in K(x), such that f/g = A’ + B and B has square-free denominator.

Examples

>>> ratint_ratpart(Poly(1, x), Poly(x + 1, x), x)
(0, 1/(x + 1))
>>> ratint_ratpart(Poly(1, x, domain=EX),
...                Poly(x**2 + y**2, x, domain=EX), x)
(0, 1/(x**2 + y**2))
>>> ratint_ratpart(Poly(36, x),
...                Poly(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2, x), x)
((12*x + 6)/(x**2 - 1), 12/(x**2 - x - 2))

diofant.integrals.heurisch.components(f, x)[source]

Returns a set of all functional components of the given expression which includes symbols, function applications and compositions and non-integer powers. Fractional powers are collected with with minimal, positive exponents.

>>> components(sin(x)*cos(x)**2, x)
{x, sin(x), cos(x)}

diofant.integrals.heurisch.heurisch(f, x, rewrite=False, hints=None, mappings=None, retries=3, degree_offset=0, unnecessary_permutations=None)[source]

Compute indefinite integral using heuristic Risch algorithm.

This is a heuristic approach to indefinite integration in finite terms using the extended heuristic (parallel) Risch algorithm, based on Manuel Bronstein’s “Poor Man’s Integrator”.

The algorithm supports various classes of functions including transcendental elementary or special functions like Airy, Bessel, Whittaker and Lambert.

Note that this algorithm is not a decision procedure. If it isn’t able to compute the antiderivative for a given function, then this is not a proof that such a functions does not exist. One should use recursive Risch algorithm in such case. It’s an open question if this algorithm can be made a full decision procedure.

This is an internal integrator procedure. You should use toplevel ‘integrate’ function in most cases, as this procedure needs some preprocessing steps and otherwise may fail.

Parameters
• f (Expr) – expression

• x (Symbol) – variable

• rewrite (Boolean, optional) – force rewrite ‘f’ in terms of ‘tan’ and ‘tanh’, default False.

• hints (None or list) – a list of functions that may appear in anti-derivate. If None (default) - no suggestions at all, if empty list - try to figure out.

Examples

>>> heurisch(y*tan(x), x)
y*log(tan(x)**2 + 1)/2


References

diofant.integrals.heurisch.heurisch_wrapper(f, x, rewrite=False, hints=None, mappings=None, retries=3, degree_offset=0, unnecessary_permutations=None)[source]

A wrapper around the heurisch integration algorithm.

This method takes the result from heurisch and checks for poles in the denominator. For each of these poles, the integral is reevaluated, and the final integration result is given in terms of a Piecewise.

Examples

>>> heurisch(cos(n*x), x)
sin(n*x)/n
>>> heurisch_wrapper(cos(n*x), x)
Piecewise((x, Eq(n, 0)), (sin(n*x)/n, true))

diofant.integrals.trigonometry.trigintegrate(f, x, conds='piecewise')[source]

Integrate f = Mul(trig) over x

>>> trigintegrate(sin(x)*cos(x), x)
sin(x)**2/2

>>> trigintegrate(sin(x)**2, x)
x/2 - sin(x)*cos(x)/2

>>> trigintegrate(tan(x)*sec(x), x)
1/cos(x)

>>> trigintegrate(sin(x)*tan(x), x)
-log(sin(x) - 1)/2 + log(sin(x) + 1)/2 - sin(x)


References

The class $$Integral$$ represents an unevaluated integral and has some methods that help in the integration of an expression.

class diofant.integrals.integrals.Integral[source]

Represents unevaluated integral.

is_commutative

Returns whether all the free symbols in the integral are commutative.

as_sum(n, method='midpoint')[source]

Approximates the definite integral by a sum.

method … one of: left, right, midpoint, trapezoid

These are all basically the rectangle method , the only difference is where the function value is taken in each interval to define the rectangle.

References

Examples

>>> e = Integral(sin(x), (x, 3, 7))
>>> e
Integral(sin(x), (x, 3, 7))


For demonstration purposes, this interval will only be split into 2 regions, bounded by [3, 5] and [5, 7].

The left-hand rule uses function evaluations at the left of each interval:

>>> e.as_sum(2, 'left')
2*sin(5) + 2*sin(3)


The midpoint rule uses evaluations at the center of each interval:

>>> e.as_sum(2, 'midpoint')
2*sin(4) + 2*sin(6)


The right-hand rule uses function evaluations at the right of each interval:

>>> e.as_sum(2, 'right')
2*sin(5) + 2*sin(7)


The trapezoid rule uses function evaluations on both sides of the intervals. This is equivalent to taking the average of the left and right hand rule results:

>>> e.as_sum(2, 'trapezoid')
2*sin(5) + sin(3) + sin(7)
>>> (e.as_sum(2, 'left') + e.as_sum(2, 'right'))/2 == _
True


All but the trapexoid method may be used when dealing with a function with a discontinuity. Here, the discontinuity at x = 0 can be avoided by using the midpoint or right-hand method:

>>> e = Integral(1/sqrt(x), (x, 0, 1))
>>> e.as_sum(5).evalf(4)
1.730
>>> e.as_sum(10).evalf(4)
1.809
>>> e.doit().evalf(4)  # the actual value is 2
2.000


The left- or trapezoid method will encounter the discontinuity and return oo:

>>> e.as_sum(5, 'left')
oo
>>> e.as_sum(5, 'trapezoid')
oo


diofant.integrals.integrals.Integral.doit()

Perform the integration using any hints

doit(**hints)[source]

Perform the integration using any hints given.

Examples

>>> from diofant.abc import i
>>> Integral(x**i, (i, 1, 3)).doit()
Piecewise((2, Eq(log(x), 0)), (x**3/log(x) - x/log(x), true))

property free_symbols

This method returns the symbols that will exist when the integral is evaluated. This is useful if one is trying to determine whether an integral depends on a certain symbol or not.

Examples

>>> Integral(x, (x, y, 1)).free_symbols
{y}

transform(x, u)[source]

Performs a change of variables from $$x$$ to $$u$$ using the relationship given by $$x$$ and $$u$$ which will define the transformations $$f$$ and $$F$$ (which are inverses of each other) as follows:

1. If $$x$$ is a Symbol (which is a variable of integration) then $$u$$ will be interpreted as some function, f(u), with inverse F(u). This, in effect, just makes the substitution of x with f(x).

2. If $$u$$ is a Symbol then $$x$$ will be interpreted as some function, F(x), with inverse f(u). This is commonly referred to as u-substitution.

Once f and F have been identified, the transformation is made as follows:

$\int_a^b x \mathrm{d}x \rightarrow \int_{F(a)}^{F(b)} f(x) \frac{\mathrm{d}}{\mathrm{d}x}$

where $$F(x)$$ is the inverse of $$f(x)$$ and the limits and integrand have been corrected so as to retain the same value after integration.

Notes

The mappings, F(x) or f(u), must lead to a unique integral. Linear or rational linear expression, $$2*x$$, $$1/x$$ and $$sqrt(x)$$, will always work; quadratic expressions like $$x**2 - 1$$ are acceptable as long as the resulting integrand does not depend on the sign of the solutions (see examples).

The integral will be returned unchanged if $$x$$ is not a variable of integration.

$$x$$ must be (or contain) only one of of the integration variables. If $$u$$ has more than one free symbol then it should be sent as a tuple ($$u$$, $$uvar$$) where $$uvar$$ identifies which variable is replacing the integration variable. XXX can it contain another integration variable?

Examples

>>> from diofant.abc import u

>>> i = Integral(x*cos(x**2 - 1), (x, 0, 1))


transform can change the variable of integration

>>> i.transform(x, u)
Integral(u*cos(u**2 - 1), (u, 0, 1))


transform can perform u-substitution as long as a unique integrand is obtained:

>>> i.transform(x**2 - 1, u)
Integral(cos(u)/2, (u, -1, 0))


This attempt fails because x = +/-sqrt(u + 1) and the sign does not cancel out of the integrand:

>>> Integral(cos(x**2 - 1), (x, 0, 1)).transform(x**2 - 1, u)
Traceback (most recent call last):
...
ValueError:
The mapping between F(x) and f(u) did not give a unique integrand.


transform can do a substitution. Here, the previous result is transformed back into the original expression using “u-substitution”:

>>> ui = _
>>> _.transform(sqrt(u + 1), x) == i
True


We can accomplish the same with a regular substitution:

>>> ui.transform(u, x**2 - 1) == i
True


If the $$x$$ does not contain a symbol of integration then the integral will be returned unchanged. Integral $$i$$ does not have an integration variable $$a$$ so no change is made:

>>> i.transform(a, x) == i
True


When $$u$$ has more than one free symbol the symbol that is replacing $$x$$ must be identified by passing $$u$$ as a tuple:

>>> Integral(x, (x, 0, 1)).transform(x, (u + a, u))
Integral(a + u, (u, -a, -a + 1))
>>> Integral(x, (x, 0, 1)).transform(x, (u + a, a))
Integral(a + u, (a, -u, -u + 1))

class diofant.integrals.transforms.IntegralTransform[source]

Base class for integral transforms.

This class represents unevaluated transforms.

To implement a concrete transform, derive from this class and implement the _compute_transform(f, x, s, **hints) and _as_integral(f, x, s) functions. If the transform cannot be computed, raise IntegralTransformError.

Also set cls._name.

Implement self._collapse_extra if your function returns more than just a number and possibly a convergence condition.

doit(**hints)[source]

Try to evaluate the transform in closed form.

This general function handles linearity, but apart from that leaves pretty much everything to _compute_transform.

Standard hints are the following:

• simplify: whether or not to simplify the result

• noconds: if True, don’t return convergence conditions

• needeval: if True, raise IntegralTransformError instead of

returning IntegralTransform objects

The default values of these hints depend on the concrete transform, usually the default is (simplify, noconds, needeval) = (True, False, False).

property free_symbols

This method returns the symbols that will exist when the transform is evaluated.

property function

The function to be transformed.

property function_variable

The dependent variable of the function to be transformed.

property transform_variable

The independent transform variable.

class diofant.integrals.transforms.MellinTransform[source]

Class representing unevaluated Mellin transforms.

class diofant.integrals.transforms.InverseMellinTransform[source]

Class representing unevaluated inverse Mellin transforms.

class diofant.integrals.transforms.LaplaceTransform[source]

Class representing unevaluated Laplace transforms.

class diofant.integrals.transforms.InverseLaplaceTransform[source]

Class representing unevaluated inverse Laplace transforms.

class diofant.integrals.transforms.FourierTransform[source]

Class representing unevaluated Fourier transforms.

class diofant.integrals.transforms.InverseFourierTransform[source]

Class representing unevaluated inverse Fourier transforms.

class diofant.integrals.transforms.SineTransform[source]

Class representing unevaluated sine transforms.

class diofant.integrals.transforms.InverseSineTransform[source]

Class representing unevaluated inverse sine transforms.

class diofant.integrals.transforms.CosineTransform[source]

Class representing unevaluated cosine transforms.

class diofant.integrals.transforms.InverseCosineTransform[source]

Class representing unevaluated inverse cosine transforms.

class diofant.integrals.transforms.HankelTransform[source]

Class representing unevaluated Hankel transforms.

class diofant.integrals.transforms.InverseHankelTransform[source]

Class representing unevaluated inverse Hankel transforms.

## Numeric Integrals¶

Diofant has functions to calculate points and weights for Gaussian quadrature of any order and any precision:

diofant.integrals.quadrature.gauss_legendre(n, n_digits)[source]

Computes the Gauss-Legendre quadrature points and weights.

The Gauss-Legendre quadrature approximates the integral:

$\int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)$

The nodes $$x_i$$ of an order $$n$$ quadrature rule are the roots of $$P_n$$ and the weights $$w_i$$ are given by:

$w_i = \frac{2}{\left(1-x_i^2\right) \left(P'_n(x_i)\right)^2}$
Parameters
• n (the order of quadrature)

• n_digits (number of significant digits of the points and weights to return)

Returns

(x, w) (the x and w are lists of points and weights as Floats.) – The points $$x_i$$ and weights $$w_i$$ are returned as (x, w) tuple of lists.

Examples

>>> x, w = gauss_legendre(3, 5)
>>> x
[-0.7746, 0, 0.7746]
>>> w
[0.55556, 0.88889, 0.55556]

>>> x, w = gauss_legendre(4, 5)
>>> x
[-0.86114, -0.33998, 0.33998, 0.86114]
>>> w
[0.34785, 0.65215, 0.65215, 0.34785]


References

diofant.integrals.quadrature.gauss_laguerre(n, n_digits)[source]

Computes the Gauss-Laguerre quadrature points and weights.

The Gauss-Laguerre quadrature approximates the integral:

$\int_0^{\infty} e^{-x} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)$

The nodes $$x_i$$ of an order $$n$$ quadrature rule are the roots of $$L_n$$ and the weights $$w_i$$ are given by:

$w_i = \frac{x_i}{(n+1)^2 \left(L_{n+1}(x_i)\right)^2}$
Parameters
• n (the order of quadrature)

• n_digits (number of significant digits of the points and weights to return)

Returns

(x, w) (the x and w are lists of points and weights as Floats.) – The points $$x_i$$ and weights $$w_i$$ are returned as (x, w) tuple of lists.

Examples

>>> x, w = gauss_laguerre(3, 5)
>>> x
[0.41577, 2.2943, 6.2899]
>>> w
[0.71109, 0.27852, 0.010389]

>>> x, w = gauss_laguerre(6, 5)
>>> x
[0.22285, 1.1889, 2.9927, 5.7751, 9.8375, 15.983]
>>> w
[0.45896, 0.417, 0.11337, 0.010399, 0.00026102, 8.9855e-7]


References

diofant.integrals.quadrature.gauss_hermite(n, n_digits)[source]

Computes the Gauss-Hermite quadrature points and weights.

The Gauss-Hermite quadrature approximates the integral:

$\int_{-\infty}^{\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)$

The nodes $$x_i$$ of an order $$n$$ quadrature rule are the roots of $$H_n$$ and the weights $$w_i$$ are given by:

$w_i = \frac{2^{n-1} n! \sqrt{\pi}}{n^2 \left(H_{n-1}(x_i)\right)^2}$
Parameters
• n (the order of quadrature)

• n_digits (number of significant digits of the points and weights to return)

Returns

(x, w) (the x and w are lists of points and weights as Floats.) – The points $$x_i$$ and weights $$w_i$$ are returned as (x, w) tuple of lists.

Examples

>>> x, w = gauss_hermite(3, 5)
>>> x
[-1.2247, 0, 1.2247]
>>> w
[0.29541, 1.1816, 0.29541]

>>> x, w = gauss_hermite(6, 5)
>>> x
[-2.3506, -1.3358, -0.43608, 0.43608, 1.3358, 2.3506]
>>> w
[0.00453, 0.15707, 0.72463, 0.72463, 0.15707, 0.00453]


References

diofant.integrals.quadrature.gauss_gen_laguerre(n, alpha, n_digits)[source]

Computes the generalized Gauss-Laguerre quadrature points and weights.

The generalized Gauss-Laguerre quadrature approximates the integral:

$\int_{0}^\infty x^{\alpha} e^{-x} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)$

The nodes $$x_i$$ of an order $$n$$ quadrature rule are the roots of $$L^{\alpha}_n$$ and the weights $$w_i$$ are given by:

$w_i = \frac{\Gamma(\alpha+n)}{n \Gamma(n) L^{\alpha}_{n-1}(x_i) L^{\alpha+1}_{n-1}(x_i)}$
Parameters
• n (the order of quadrature)

• alpha (the exponent of the singularity, $$\alpha > -1$$)

• n_digits (number of significant digits of the points and weights to return)

Returns

(x, w) (the x and w are lists of points and weights as Floats.) – The points $$x_i$$ and weights $$w_i$$ are returned as (x, w) tuple of lists.

Examples

>>> x, w = gauss_gen_laguerre(3, -0.5, 5)
>>> x
[0.19016, 1.7845, 5.5253]
>>> w
[1.4493, 0.31413, 0.00906]

>>> x, w = gauss_gen_laguerre(4, 1.5, 5)
>>> x
[0.97851, 2.9904, 6.3193, 11.712]
>>> w
[0.53087, 0.67721, 0.11895, 0.0023152]


References

diofant.integrals.quadrature.gauss_chebyshev_t(n, n_digits)[source]

Computes the Gauss-Chebyshev quadrature points and weights of the first kind.

The Gauss-Chebyshev quadrature of the first kind approximates the integral:

$\int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)$

The nodes $$x_i$$ of an order $$n$$ quadrature rule are the roots of $$T_n$$ and the weights $$w_i$$ are given by:

$w_i = \frac{\pi}{n}$
Parameters
• n (the order of quadrature)

• n_digits (number of significant digits of the points and weights to return)

Returns

(x, w) (the x and w are lists of points and weights as Floats.) – The points $$x_i$$ and weights $$w_i$$ are returned as (x, w) tuple of lists.

Examples

>>> x, w = gauss_chebyshev_t(3, 5)
>>> x
[0.86602, 0, -0.86602]
>>> w
[1.0472, 1.0472, 1.0472]

>>> x, w = gauss_chebyshev_t(6, 5)
>>> x
[0.96593, 0.70711, 0.25882, -0.25882, -0.70711, -0.96593]
>>> w
[0.5236, 0.5236, 0.5236, 0.5236, 0.5236, 0.5236]


References

diofant.integrals.quadrature.gauss_chebyshev_u(n, n_digits)[source]

Computes the Gauss-Chebyshev quadrature points and weights of the second kind.

The Gauss-Chebyshev quadrature of the second kind approximates the integral:

$\int_{-1}^{1} \sqrt{1-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)$

The nodes $$x_i$$ of an order $$n$$ quadrature rule are the roots of $$U_n$$ and the weights $$w_i$$ are given by:

$w_i = \frac{\pi}{n+1} \sin^2 \left(\frac{i}{n+1}\pi\right)$
Parameters
• n (the order of quadrature)

• n_digits (number of significant digits of the points and weights to return)

Returns

(x, w) (the x and w are lists of points and weights as Floats.) – The points $$x_i$$ and weights $$w_i$$ are returned as (x, w) tuple of lists.

Examples

>>> x, w = gauss_chebyshev_u(3, 5)
>>> x
[0.70711, 0, -0.70711]
>>> w
[0.3927, 0.7854, 0.3927]

>>> x, w = gauss_chebyshev_u(6, 5)
>>> x
[0.90097, 0.62349, 0.22252, -0.22252, -0.62349, -0.90097]
>>> w
[0.084489, 0.27433, 0.42658, 0.42658, 0.27433, 0.084489]


References

diofant.integrals.quadrature.gauss_jacobi(n, alpha, beta, n_digits)[source]

Computes the Gauss-Jacobi quadrature points and weights.

The Gauss-Jacobi quadrature of the first kind approximates the integral:

$\int_{-1}^1 (1-x)^\alpha (1+x)^\beta f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)$

The nodes $$x_i$$ of an order $$n$$ quadrature rule are the roots of $$P^{(\alpha,\beta)}_n$$ and the weights $$w_i$$ are given by:

$w_i = -\frac{2n+\alpha+\beta+2}{n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)} {\Gamma(n+\alpha+\beta+1)(n+1)!} \frac{2^{\alpha+\beta}}{P'_n(x_i) P^{(\alpha,\beta)}_{n+1}(x_i)}$
Parameters
• n (the order of quadrature)

• alpha (the first parameter of the Jacobi Polynomial, $$\alpha > -1$$)

• beta (the second parameter of the Jacobi Polynomial, $$\beta > -1$$)

• n_digits (number of significant digits of the points and weights to return)

Returns

(x, w) (the x and w are lists of points and weights as Floats.) – The points $$x_i$$ and weights $$w_i$$ are returned as (x, w) tuple of lists.

Examples

>>> x, w = gauss_jacobi(3, 0.5, -0.5, 5)
>>> x
[-0.90097, -0.22252, 0.62349]
>>> w
[1.7063, 1.0973, 0.33795]

>>> x, w = gauss_jacobi(6, 1, 1, 5)
>>> x
[-0.87174, -0.5917, -0.2093, 0.2093, 0.5917, 0.87174]
>>> w
[0.050584, 0.22169, 0.39439, 0.39439, 0.22169, 0.050584]


References