Geodesic
On analytical complexity of antiderivatives
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 6, pp. 694-698 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is shown that the class of all functions of two variables of finite analytical complexity is not closed under integration. It also follows that the class of all functions of finite analytical complexity in the case of three or more variables is not closed under integration. For the case of three or more variables explicit examples of finite complexity functions with infinite complexity antiderivatives are constructed.
Keywords: analytical complexity, integration, finite complexity functions. 
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Maria A. Stepanova. On analytical complexity of antiderivatives. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 6, pp. 694-698. http://geodesic.mathdoc.fr/item/JSFU_2019_12_6_a4/

[1] V.K. Beloshapka, “Analytical complexity: Development of the topic”, Russian Journal of Mathematical Physics, 19:4 (2012), 428–439 | DOI | MR | Zbl

[2] V.K. Beloshapka, “On Integration of Functions of Complexity One”, Journal of Siberian Federal University. Mathematics $\$ Physics, 12:4 (2019), 1–7 | MR

[3] J. Liouville, “Sur la détermination des intégrales dont la valeur est algébrique”, Journal de l'ecole polytechnique, XIV (1833), Sec. 23 | MR

[4] J.F. Ritt, Integration in finite term, Liouville's theory of elementary methods, Colombia University Press, NY, 1948 | MR

[5] V.K. Beloshapka, “Decomposition of functions of finite analytical complexity”, Journal of Siberian Federal University. Mathematics $\$ Physics, 11:6 (2018), 680–685 | DOI | MR

[6] V.K. Beloshapka, “Analytic complexity of functions of two variables”, Russian Journal of Mathematical Physics, 14:3 (2007), 243–249 | DOI | MR | Zbl

[7] V.K. Beloshapka, “Analytic Complexity of Functions of Several Variables”, Mat. Zametki, 100:6 (2016), 781–789 | DOI | MR | Zbl