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A team of mathematicians from RUDN University added new symbolic integration functionality to the Sage computerized algebra system by RUDN University.
The
team implemented ideas and methods suggested by the German mathematician
Karl Weierstrass in the 1870s. The results were published in the Journal of Symbolic Computation.
Algebra Mathematics
Photo: RUDN University
The first computer program capable of calculating integrals of elementary functions was developed in the late 1950s. By creating it, the developers confirmed that a computer could not only perform simple calculations but was also able to deal with tasks that required a certain degree of ‘thinking.’ Symbolic integration, i.e. integration that involves letters and abstract symbols instead of numbers, is an example of such a task.
At the same time, scientists realized that neither humans nor computers were able to determine whether a given integral can be taken in elementary functions (provided such a human or computer used the methods studied in a university course of analysis and took a finite number of steps)...
One of the theories developed by the German mathematician Karl Weierstrass in the 1870s reduces the calculation of an integral of an algebraic function to finding a given set of known integrals of all three types. The initial integral is represented as a sum of standard integrals (this construction is knowns as the normal representation of an Abelian integral). The team from RUDN University confirmed that this representation is indicative of whether a given integral can be calculated in elementary functions. To confirm their theory, the mathematicians tested them on simple elliptical integrals using a software package that had been created by the team in 2017. The package helps calculate coefficients of the normal form of an integral. In the future, the team plans to conduct similar studies for a wider range of integrals.
Reference:
“On symbolic integration of algebraic functions” by M.D. Malykh, L.A. Sevastianova and Y. Yu, 11 September 2020, Journal of Symbolic Computation.
DOI: 10.1016/j.jsc.2020.09.002
Source: SciTechDaily
