Symbolic Integration
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calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, symbolic integration is the problem of finding a formula for the
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
, or ''indefinite integral'', of a given function ''f''(''x''), i.e. to find a differentiable function ''F''(''x'') such that :\frac = f(x). This is also denoted :F(x) = \int f(x) \, dx.


Discussion

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 Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
, 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 In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
. The reverse question of finding the integral is much more difficult. Many expressions which are relatively simple do not have integrals that can be expressed in closed form. See
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
and
nonelementary integral In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an ''elementary function'' (i.e. a function constructed from a finite number of quotients of constan ...
for more details. A procedure called the
Risch algorithm 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 w ...
exists which is capable of determining whether the integral of an elementary function (function built from a finite number of exponentials,
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
s, 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, 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 James Harold Davenport (born 26 September 1953) is a British computer scientist who works in computer algebra. Having done his PhD and early research at the Computer Laboratory, University of Cambridge, he is the Hebron and Medlock Professor ...
; the general case was solved and implemented in
Axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy o ...
by Manuel Bronstein. 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. 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 In computer science, pattern matching is the act of checking a given sequence of tokens for the presence of the constituents of some pattern. In contrast to pattern recognition, the match usually has to be exact: "either it will or will not be ...
and other manipulations, was pioneered by developers of the
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system and then later emulated by
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimi ...
,
Axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy o ...
, MuPAD and other systems.


Recent advances

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 In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
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 The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
,
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
, sine, cosine, inverse trigonometric functions, inverse hyperbolic functions. They include also most common special functions such as Airy function, error function,
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
s 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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, when restricted to holonomic functions, represented by their differential equation and initial conditions. This includes the computation of antiderivatives and
definite integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
s (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
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. See also the Dynamic Dictionary of Mathematical functions.http://ddmf.msr-inria.inria.fr ''Dynamic Dictionary of Mathematical functions''


Example

For example: :\int x^2\,dx = \frac + C is a symbolic result for an indefinite integral (here C is a
constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connect ...
), :\int_^1 x^2\,dx = \left frac\right^1= \frac - \frac=\frac is a symbolic result for a definite integral, and :\int_^1 x^2\,dx \approx 0.6667 is a numerical result for the same definite integral.


See also

* * * * * * *


References

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External links

* {{MathWorld, urlname=RischAlgorithm, title=Risch Algorithm, author=Bhatt, Bhuvanesh
Wolfram Integrator
— Free online symbolic integration with
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Computer algebra Differential algebra