Preliminaries -- I Real Analysis -- Real Sequences and Series -- Real Functions -- II Functional Analysis -- Hilbert Spaces -- Orthonormal Polynomials -- Lebesgue Integrals -- III Complex Analysis -- Complex Functions -- Singularity and Continuation -- Contour Integrals -- Conformal Mapping -- IV Fourier Analysis -- Fourier Series -- Fourier Transformation -- Laplace Transformation -- Wavelet Transformation -- V Differential Equations -- Ordinary Differential Equations -- System of Ordinary Differential Equations -- Partial Differential Equations -- VI Tensor Analyses -- Cartesian Tensors -- Non-Cartesian Tensors -- Tensor as Mapping.

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.