Algebraic integration

In mathematics, algebraic or symbolic integration or quadrature is the term used to describe the calculation of integrals using exact term transformations , in contrast to approximate numerical quadrature .

Algebraic integration is one of the most important use cases of computer algebra systems . Functions for determining an antiderivative are implemented in these programs . The most important rules here are the substitution rule and partial integration . However, these techniques quickly reach the limits of their usability. The function does not have a closed representation of its antiderivative. For these cases there are also the techniques of Fourier transformation and the residual theorem , which are also mastered by modern computer algebra systems. In addition, computer algebra systems use so-called error functions, for example the Gaussian error function , to form antiderivatives that do not have a closed representation. ${\ displaystyle \ textstyle \ exp \ left (- {\ frac {x ^ {2}} {2}} \ right)}$

There is a method, called the Risch algorithm , which can decide for many classes of integrands whether an integral exists and then determines it. Such algorithms are still being developed, because the Risch algorithm is limited to indefinite integrals. The vast majority of the integrals of interest to physicists, theoretical chemists and engineers are, however, certain integrals, often with reference to the Laplace transform , Fourier transform or Mellin transform . An alternative to the Risch algorithm uses a combination of computer algebra system and pattern recognition as well as knowledge of special functions, in particular the incomplete gamma function. Although this approach is heuristic rather than algorithmic, it is an effective method of computing certain integrals, especially those that appear in engineering practice. This method was first implemented by the developers of the Maple computer algebra system, and later adopted by systems such as Mathematica , MuPAD, and others.

example

A simple example is given using the polynomial function . So is ${\ displaystyle x ^ {2}}$

{\ displaystyle \ int x ^ {2} \, \ mathrm {d} x = {\ frac {x ^ {3}} {3}} + C}

the symbolic result for the indefinite integral, where is a constant of integration. For the definite integral ${\ displaystyle C}$

{\ displaystyle \ int _ {- 1} ^ {1} x ^ {2} \, dx = {\ frac {2} {3}} = 0 {,} 6666 \ ldots}

is the symbolic value and the numeric one is 0.6666…. The number of decimal places is infinite. ${\ displaystyle {\ tfrac {2} {3}}}$

Individual evidence

↑ Wolfram: Erf ; Retrieved April 27, 2010
↑ KO Geddes, ML Glasser, RA Moore and TC Scott, Evaluation of Classes of Definite Integrals Involving Elementary Functions via Differentiation of Special Functions , AAECC (Applicable Algebra in Engineering, Communication and Computing), vol. 1, (1990), pp. 149-165, doi : 10.1007 / BF01810298
↑ KO Geddes and TC Scott, Recipes for Classes of Definite Integrals Involving Exponentials and Logarithms , Proceedings of the 1989 Computers and Mathematics conference, (held at MIT June 12, 1989), edited by E. Kaltofen and SM Watt, Springer-Verlag, New York, (1989), pp. 192-201. see http://portal.acm.org/citation.cfm?id=93094

[1] Wolfram: Erf ; Retrieved April 27, 2010

[2] KO Geddes, ML Glasser, RA Moore and TC Scott, Evaluation of Classes of Definite Integrals Involving Elementary Functions via Differentiation of Special Functions , AAECC (Applicable Algebra in Engineering, Communication and Computing), vol. 1, (1990), pp. 149-165, doi : 10.1007 / BF01810298

[3] KO Geddes and TC Scott, Recipes for Classes of Definite Integrals Involving Exponentials and Logarithms , Proceedings of the 1989 Computers and Mathematics conference, (held at MIT June 12, 1989), edited by E. Kaltofen and SM Watt, Springer-Verlag, New York, (1989), pp. 192-201. see http://portal.acm.org/citation.cfm?id=93094