1. bookVolume 29 (2021): Issue 4 (December 2021)
Journal Details
License
Format
Journal
eISSN
1898-9934
First Published
09 Jun 2008
Publication timeframe
4 times per year
Languages
English
access type Open Access

Improper Integral. Part I

Published Online: 09 Jul 2022
Volume & Issue: Volume 29 (2021) - Issue 4 (December 2021)
Page range: 201 - 220
Accepted: 30 Sep 2021
Journal Details
License
Format
Journal
eISSN
1898-9934
First Published
09 Jun 2008
Publication timeframe
4 times per year
Languages
English
Summary

In this article, we deal with Riemann’s improper integral [1], using the Mizar system [2], [3]. Improper integrals with finite values are discussed in [5] by Yamazaki et al., but in general, improper integrals do not assume that they are finite. Therefore, we have formalized general improper integrals that does not limit the integral value to a finite value. In addition, each theorem in [5] assumes that the domain of integrand includes a closed interval, but since the improper integral should be discusses based on the half-open interval, we also corrected it.

Keywords

MSC

[1] Tom M. Apostol. Mathematical Analysis. Addison-Wesley, 1969. Search in Google Scholar

[2] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17. Open DOISearch in Google Scholar

[3] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.604425130069070 Open DOISearch in Google Scholar

[4] Noboru Endou, Yasunari Shidama, and Masahiko Yamazaki. Integrability and the integral of partial functions from ℝ into ℝ. Formalized Mathematics, 14(4):207–212, 2006. doi:10.2478/v10037-006-0023-y. Open DOISearch in Google Scholar

[5] Masahiko Yamazaki, Hiroshi Yamazaki, and Yasunari Shidama. Extended Riemann integral of functions of real variable and one-sided Laplace transform. Formalized Mathematics, 16(4):311–317, 2008. doi:10.2478/v10037-008-0038-7. Open DOISearch in Google Scholar

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