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The Partition Method for a Power Series Expansion: Theory and Applications explores how the method known as 'the partition method for a power series expansion', which was developed by the author, can be applied to a host of previously intractable problems in mathematics and physics.
In particular, this book describes how the method can be used to determine the Bernoulli, cosecant, and reciprocal logarithm numbers, which appear as the coefficients of the resulting power series expansions, then also extending the method to more complicated situations where the coefficients become polynomials or mathematical functions. From these examples, a general theory for the method is presented, which enables a programming methodology to be established.
Finally, the programming techniques of previous chapters are used to derive power series expansions for complex generating functions arising in the theory of partitions and in lattice models of statistical mechanics.
- Explains the partition method by presenting elementary applications involving the Bernoulli, cosecant, and reciprocal logarithm numbers
- Compares generating partitions via the BRCP algorithm with the standard lexicographic approaches
- Describes how to program the partition method for a power series expansion and the BRCP algorithm
Mathematicians, theoretical physicists, computer scientists and software engineers
- Chapter 1: Introduction
- 1.1. Cosecant Expansion
- 1.2. Reciprocal Logarithm Numbers
- 1.3. Bernoulli and Related Polynomials
- Chapter 2: More Advanced Applications
- 2.1. Bell Polynomials of the First Kind
- 2.2. Generalized Cosecant and Secant Numbers
- 2.3. Generalized Reciprocal Logarithm Numbers
- 2.4. Generalization of Elliptic Integrals
- Chapter 3: Generating Partitions
- Chapter 4: General Theory
- Chapter 5: Programming the Partition Method for a Power Series Expansion
- Chapter 6: Operator Approach
- Chapter 7: Classes of Partitions
- Chapter 8: The Partition-Number Generating Function and Its Inverted Form
- 8.1. Generalization of the Inverted Form of P(z)
- Chapter 9: Generalization of the Partition-Number Generating Function
- Chapter 10: Conclusion
- Appendix A: Regularization
- Appendix B: Computer Programs
- No. of pages:
- © Academic Press 2017
- 19th January 2017
- Academic Press
- Hardcover ISBN:
- eBook ISBN:
Dr Victor Kowalenko is a Senior Research Fellow in the Department of Mathematics and Statistics, University of Melbourne, Australia. Since 2009, he has been associated with the ARC Centre of Excellence in Mathematics and Statistics of Complex Systems. He began his research career by joining the DSTO’s railgun project in Maribyrnong in the early 1980’s before transferring to the DSTO facility at Fishermen’s Bend to work on aeronautical systems. He then returned to the Department of Physics, University of Melbourne as one of the inaugural Australian Research Fellows to work on particle-anti-particle plasmas and general relativistic magnetohydrodynamics. It was here that he introduced the partition method for a power expansion. Between 2001 and 2003, when he was a Senior Research Fellow in the School of Computer Science and Software Engineering, Monash University, he was able to develop the method further and to extend it to intractable problems in mathematics and physics.
Department of Mathematics and Statistics, University of Melbourne, Australia
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