Isomorphism Theorems of a Series Sum and the Improper Integral
Қатардың қосындысы мен меншiксiз интегралдың изоморфизмi туралы теоремалар
Теоремы об изоморфизме суммы ряда и несобственного интеграла
Malyshev V.P. Kazhikenova S.S. Makasheva A.M. Kazhikenova A.S.
2024E.A. Buketov Karaganda University Publish house
Bulletin of the Karaganda University. Mathematics Series
2024#113Issue 1128 - 139 pp.
The discrete and continuous dependencies’ relationship question has been investigated. An algorithm for determining the final and total series sums through the equivalence ratio of the series common term an and the an-model function improper integral mean value within the change unit interval based on the extended integral Cauchy convergence criterion has been developed. Examples of determining for the statistical sum in the Boltzmann distribution, for the first time directly expressed through an-model function. This eliminates the need for calculations to accumulate the sum of the series up to a value that is specified by a certain accuracy of this sum. In addition, it allows in this case to vary the energy variation interval with any given accuracy. The conducted studies allow solving both theoretical and practical problems of physics and materials science, directly using the Boltzmann distribution (energy spectrum) to calculate the entropy, which determines the loss of thermal energy in technological processes.
Boltzmann distribution , equivalence relation , improper integral , isomorphism , series sum
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The Laboratory of Entropy and Information Analysis of Complex Physicochemical Systems, Abishev Chemical and Metallurgical Institute, 63 Ermekov street, Karaganda, 100009, Kazakhstan
The Department of Higher Mathematics, Abylkas Saginov Karaganda Technical University, 56 Nursultan Nazarbayev Avenue, Karaganda, 100000, Kazakhstan
The Department of Higher Mathematics, Karaganda Buketov University, 28 Universitetskaya street, Karaganda, 100028, Kazakhstan
The Laboratory of Entropy and Information Analysis of Complex Physicochemical Systems
The Department of Higher Mathematics
The Department of Higher Mathematics
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