Titlebook: Differential Equations and Data Analysis; Aleksei Beltukov Textbook 2025 The Editor(s) (if applicable) and The Author(s), under exclusive

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Exponential Function,n it is applied to complex numbers and matrices. To avoid that confusion in later chapters, we show how the exponential function is properly defined by its Taylor series expansion and how that definition naturally extends its domain to complex and matrix inputs.

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ODE and Linear Algebra,structure is codified in the structure theorem which, together with the principle of superposition, provides a blueprint for constructing general solutions of linear ODE; no such blueprint exists for nonlinear ODE. As a concrete illustration of linearity in action, we study an RC-circuit driven firs

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Discrete Fourier Transform,, where it is practically synonymous with digital signal processing (DSP). After reviewing orthogonal expansions, we show that DFT is one, and that it is also a way of approximating functions with sums of real harmonics or, equivalently, complex exponentials. The role of DFT in data analysis is illu

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Fourier Series,reased. In the limit, as the number of grid points tends to infinity, the finite sum becomes a Fourier series which represents the function exactly. Fourier series representation is thus DFT approximation brought to its logical conclusion. Previously, we used DFT to approximate the solution of a fir

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Fourier and Laplace Transforms,n limit of a Fourier expansion of a function. We then show how to solve linear ODE with constant coefficients using the Fourier transform. More importantly, we show that all information about solutions of such ODE is encoded in transfer functions. Finding transfer functions is often the main goal of

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https://doi.org/10.1007/978-3-031-62257-1Ordinary Differential Equations; Modeling with Differential Equations; Modeling with ODEs; Applied Diff

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