### Synopses & Reviews

Due to the rapid expansion of the frontiers of physics and engineering, the demand for higher-level mathematics is increasing yearly. This book is designed to provide accessible knowledge of higher-level mathematics demanded in contemporary physics and engineering. Rigorous mathematical structures of important subjects in these fields are fully covered, which will be helpful for readers to become acquainted with certain abstract mathematical concepts. The selected topics are: - Real analysis, Complex analysis, Functional analysis, Lebesgue integration theory, Fourier analysis, Laplace analysis, Wavelet analysis, Differential equations, and Tensor analysis. This book is essentially self-contained, and assumes only standard undergraduate preparation such as elementary calculus and linear algebra. It is thus well suited for graduate students in physics and engineering who are interested in theoretical backgrounds of their own fields. Further, it will also be useful for mathematics students who want to understand how certain abstract concepts in mathematics are applied in a practical situation. The readers will not only acquire basic knowledge toward higher-level mathematics, but also imbibe mathematical skills necessary for contemporary studies of their own fields.

#### Review

From the reviews: "This is a largely self-contained exposition of fundamental topics in the mathematics of physics and engineering, which ... will lead to an understanding of the symbiotic relationship between mathematics and the physical sciences. ... The exercises ... are solved in full immediately after the problem statements. ... It may be most useful for graduate students and as a reference for professionals. Summing Up: Recommended. Upper-division undergraduate through professional collections." (D. Robbins, Choice, Vol. 48 (5), January, 2011)

#### Synopsis

Mathematics for physics and engineering is traditionally covered by textbooks on Mathematical Physics or Applied Mathematics. This book differs from those on pure mathematics and differs from lexicographic collections of methods for solving specific problems. Instead it emphasizes the mathematical concepts underlying manifold physical phenomena.

The readers will not only acquire knowledges required for actual applications but also acquire the minimum mathematical skills necessary to study physics.

The text is coherent and self-contained, states and proves a large number of theorems, lemmas, and corollaries that are relevant to physics and other related sciences. Extensive details on mathematical manipulations are provided.

Each chapter contains a number of examples and practical exercises. Such a large number of examples provides the balance between mathematical formalisms and their applications. Reflecting the current interests, several new topics in developing fields, such as the mathematical background of quantum information theory and topology for the knot theory, are included.

#### Synopsis

Differing from many mathematics texts, this one emphasizes the mathematical concepts underlying manifold physical phenomena. Readers get both the knowledge required in applications, and also the minimum "mathematical skills" necessary in the study of physics.

### About the Author

Tsuneyoshi Nakayama graduated from Hokkaido University in Japan in 1973. He is a professor of Theoretical Condensed Matter Physics in Department of Applied Physics in Hokkaido University from 1986. During this period he stayed Max-Planck Institute, University of Monpellier, University of Cambridge, and The University of Tokyo. He is the co-author of the book "Fractal concepts of condensed matter." Hiroyuki Shima obtained Ph.D from Hokkaido University. He is currently pursuing his studies, with a special interest in critical phenomena in disordered systems and many-body problems in complex systems. He has had a considerable amount of experience in teaching mathematics and physics to undergraduate and graduate students.

### Table of Contents

1. Preliminaries.- 2. Real Sequences and Series.- 3. Real Functions.- 4. Hilbert Spaces.- 5. Orthonormal Polynomials.- 6. Lebesgue Integrals.- 7. Complex Functions.- 8. Singularity and Continuation.- 9. Contour Integrals.- 10. Conformal Mapping.- 11. Fourier Series.- 12. Fourier Transformation.- 13. Laplace Transformation.- 14. Wavelet Transformation.- 15. Ordinary Differential Equations.- 16. System of Ordinary Differential Equations.- 17. Partial Differential Equations.- 18. Cartesian Tensors.- 19. Non-Cartesian Tensors.- 20. Tensor as Mapping.- A. Proof of the Bolzano-Weierstrass Theorem.- B. Dirac's delta-Function.- C. Proof of Weierstrass' Approximation Theorem.- D. Tabulated List of Orthonormal Polynomial Functions.