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In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or indefinite integral, of a given function f(x), i.e. to find a formula for a differentiable function F(x) such that
This is also denoted
The term symbolic is used to distinguish this problem from that of numerical integration, where the value of F is sought at a particular input or set of inputs, rather than a general formula for F.
Both problems were held to be of practical and theoretical importance long before the time of digital computers, but they are now generally considered the domain of computer science, as computers are most often used currently to tackle individual instances.
Finding the derivative of an expression is a straightforward process for which it is easy to construct an algorithm. The reverse question of finding the integral is much more difficult. Many expressions that are relatively simple do not have integrals that can be expressed in closed form. See antiderivative and nonelementary integral for more details.
A procedure called the Risch algorithm exists that is capable of determining whether the integral of an elementary function (function built from a finite number of exponentials, logarithms, constants, and nth roots through composition and combinations using the four elementary operations) is elementary and returning it if it is. In its original form, the Risch algorithm was not suitable for a direct implementation, and its complete implementation took a long time. It was first implemented in Reduce in the case of purely transcendental functions; the case of purely algebraic functions was solved and implemented in Reduce by James H. Davenport; the general case was solved by Manuel Bronstein, who implemented almost all of it in Axiom, though to date there is no implementation of the Risch algorithm that can deal with all of the special cases and branches in it. [1] [2]
However, the Risch algorithm applies only to indefinite integrals, while most of the integrals of interest to physicists, theoretical chemists, and engineers are definite integrals often related to Laplace transforms, Fourier transforms, and Mellin transforms. Lacking a general algorithm, the developers of computer algebra systems have implemented heuristics based on pattern-matching and the exploitation of special functions, in particular the incomplete gamma function. [3] Although this approach is heuristic rather than algorithmic, it is nonetheless an effective method for solving many definite integrals encountered by practical engineering applications. Earlier systems such as Macsyma had a few definite integrals related to special functions within a look-up table. However this particular method, involving differentiation of special functions with respect to its parameters, variable transformation, pattern matching and other manipulations, was pioneered by developers of the Maple [4] system and then later emulated by Mathematica, Axiom, MuPAD and other systems.
The main problem in the classical approach of symbolic integration is that, if a function is represented in closed form, then, in general, its antiderivative has not a similar representation. In other words, the class of functions that can be represented in closed form is not closed under antiderivation.
Holonomic functions are a large class of functions, which is closed under antiderivation and allows algorithmic implementation in computers of integration and many other operations of calculus.
More precisely, a holonomic function is a solution of a homogeneous linear differential equation with polynomial coefficients. Holonomic functions are closed under addition and multiplication, derivation, and antiderivation. They include algebraic functions, exponential function, logarithm, sine, cosine, inverse trigonometric functions, inverse hyperbolic functions. They include also most common special functions such as Airy function, error function, Bessel functions and all hypergeometric functions.
A fundamental property of holonomic functions is that the coefficients of their Taylor series at any point satisfy a linear recurrence relation with polynomial coefficients, and that this recurrence relation may be computed from the differential equation defining the function. Conversely given such a recurrence relation between the coefficients of a power series, this power series defines a holonomic function whose differential equation may be computed algorithmically. This recurrence relation allows a fast computation of the Taylor series, and thus of the value of the function at any point, with an arbitrary small certified error.
This makes algorithmic most operations of calculus, when restricted to holonomic functions, represented by their differential equation and initial conditions. This includes the computation of antiderivatives and definite integrals (this amounts to evaluating the antiderivative at the endpoints of the interval of integration). This includes also the computation of the asymptotic behavior of the function at infinity, and thus the definite integrals on unbounded intervals.
All these operations are implemented in the algolib library for Maple. [5] See also the Dynamic Dictionary of Mathematical Functions. [6]
For example:
is a symbolic result for an indefinite integral (here C is a constant of integration),
is a symbolic result for a definite integral, and
is a numerical result for the same definite integral.
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. The process of solving for antiderivatives is called antidifferentiation, and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as F and G.
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
In mathematics, an elementary function is a function of a single variable that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions.
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter.
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "computer algebra" or "symbolic computation", which has spurred work in algorithms over mathematical objects such as polynomials.
In calculus, the constant of integration, often denoted by , is a constant term added to an antiderivative of a function to indicate that the indefinite integral of , on a connected domain, is only defined up to an additive constant. This constant expresses an ambiguity inherent in the construction of antiderivatives.
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take "quadrature" to include higher-dimensional integration.
In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards."
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.
In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra who developed it in 1968.
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
In mathematics, an expression is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic operations and function composition. Commonly, the allowed functions are nth root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context.
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative that is, itself, not an elementary function. A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. This theorem also provides a basis for the Risch algorithm for determining which elementary functions have elementary antiderivatives.
In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms of D-modules theory. More precisely, a holonomic function is an element of a holonomic module of smooth functions. Holonomic functions can also be described as differentiably finite functions, also known as D-finite functions. When a power series in the variables is the Taylor expansion of a holonomic function, the sequence of its coefficients, in one or several indices, is also called holonomic. Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivariate recurrences satisfied by the whole sequence and by suitable specializations of it. The situation simplifies in the univariate case: any univariate sequence that satisfies a linear homogeneous recurrence relation with polynomial coefficients, or equivalently a linear homogeneous difference equation with polynomial coefficients, is holonomic.
In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in 1833 to 1841, places an important restriction on antiderivatives that can be expressed as elementary functions.
Xcas is a user interface to Giac, which is an open source computer algebra system (CAS) for Windows, macOS and Linux among many other platforms. Xcas is written in C++. Giac can be used directly inside software written in C++.
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function. The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area.
In calculus, Cavalieri's quadrature formula, named for 17th-century Italian mathematician Bonaventura Cavalieri, is the integral
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