Titlebook: Higher Mathematics for Physics and Engineering; Tsuneyoshi Nakayama,Hiroyuki Shima Textbook 2010 Springer-Verlag Berlin Heidelberg 2010 An

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Real Functionsnot generally preserve the nature of their constituents; e.g., a sequence of “continuous” functions can converge into a “discontinuous” function. In this chapter, we show that this is not true in cases of uniform convergence (Sect. 3.2.2), which is a special class of convergence that preserves the c

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Hilbert Spaces of the vector space, and the completeness property (Sect. 4.1.6), which guarantees the self-consistency of the space. Most of the mathematical topics covered in this volume are AQ1 based on Hilbert spaces. In particular, L. spaces and l. spaces (Sect. 4.3), which are specific classes of Hilbert spa

RAG

Orthonormal Polynomialshe sense of the L. space. In this chapter we present three important approaches for the construction of orthonormal polynomials, based, respectively, on the Weierstrass theorem (Sect. 5.1.1), the Rodrigues formula (Sect. 5.2.1), and generating functions (Sect. 5.2.7). We shall find that various orth

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Lebesgue Integralsconcept of length that allows us to quantify the length of a set that is composed of, for instance, an infinite number of infinitesimal points with a highly discontinuous distribution. Thus, the Lebesgue integral is an effective tool for integrating highly discontinuous functions that cannot be inte

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Contour Integralsng closed contours. This utility of singularities is based on the residue theorem (Sect. 9.1.1), argument principle (Sect. 9.4), and principal value integrals (Sect. 9.5.1), all of which correlate the nature of singularities within and/or on the contour with the relevant complex integrals.

GENRE

废墟

Fourier Seriesak up an arbitrary periodic function into a set of simple terms that can be solved individually and then recombined in order to obtain the solution to the original problem with the desired level of accuracy. In this chapter, we place particular emphasis on the mean convergence property of a Fourier

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Fourier Transformationhe transformation enables us to measure certain correlations of a function with itself or with other functions; thus a Fourier transform can be applied to probability theory, signal analysis, etc. In this chapter we also provide the essence of a discrete Fourier transform (Sect. 12.3), which refers