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Advanced Mathematical Analysis Course: Complex Analysis, Functional Analysis, Numerical Analysis, Fourier Analysis, Nonlinear Analysis
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En stock
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0.47 kg
0.47 kg
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Sí
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Nuevo
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Amazon
Amazon
Sobre este producto
- General Introduction: The book begins with an introduction outlining the course objectives, which aim to equip students with a robust theoretical understanding of the various areas of mathematical analysis covered. The proposed methodology combines theoretical presentations with practical exercises, allowing for the gradual assimilation of concepts, from foundational topics to the most advanced subjects.
- Complex Analysis: The fundamentals of complex analysis are covered, including complex numbers, holomorphic functions, and key theorems such as Cauchy’s integral formula and the residue theorem. Concrete applications, particularly in the computation of integrals and conformal mappings, are also explored.
- Functional Analysis: This chapter covers normed vector spaces, Banach and Hilbert spaces, as well as operator theory. The Hahn-Banach theorem, Banach-Steinhaus theorem, and the closed graph theorem are presented with specific applications to functional analysis, providing a solid foundation for the study of dynamical systems and PDEs.
- Numerical Analysis: The focus is on numerical methods for solving equations, function approximation, and solving differential equations. Euler’s and Runge-Kutta methods, along with interpolation techniques and error analysis, are explained in detail, emphasizing their importance in modeling and numerical simulation.
- Fourier Analysis: This chapter addresses Fourier series and transforms, including convergence conditions and applications in signal processing and solving differential equations. Discrete Fourier analysis is also covered, with a focus on the Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT) algorithm, which are essential in digital signal processing.
- Nonlinear Analysis: Nonlinear dynamical systems, stability, bifurcations, and chaos are explored, providing an introduction to the complex behaviors observed in many physical systems. Variational methods and fixed-point theorems are applied to concrete problems in continuum mechanics and nonlinear analysis.
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