histogram construction
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1: 18.40 Methods of Computation
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►These quadrature weights and abscissas will then allow construction of a convergent sequence of approximations to , as will be considered in the following paragraphs.
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Histogram Approach
… ►Equation (18.40.7) provides step-histogram approximations to , as shown in Figure 18.40.1 for and , shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein. ► … ► …2: 9.15 Mathematical Applications
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►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point.
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3: 27.17 Other Applications
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►Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers.
…Apostol and Zuckerman (1951) uses congruences to construct magic squares.
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4: 18.39 Applications in the Physical Sciences
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5: Joyce E. Conlon
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►In 1999 she joined the NIST Mathematical and Computational Sciences Division, where she contributed to the DLMF project, especially in the construction of the bibliography.
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6: Tom H. Koornwinder
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►Currently he is on the editorial board for Constructive Approximation, and is editor for the volume on Multivariable Special Functions in the ongoing Askey–Bateman book project.
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7: Bruce R. Miller
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►In particular, he developed the LaTeXML system used for converting the LaTeX source documents into XML and MathML from which the DLMF website is constructed.
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8: Peter L. Walker
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►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004.
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9: 10.72 Mathematical Applications
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►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter.
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►In regions in which (10.72.1) has a simple turning point , that is, and are analytic (or with weaker conditions if is a real variable) and is a simple zero of , asymptotic expansions of the solutions for large can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order (§9.6(i)).
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►If has a double zero , or more generally is a zero of order , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order .
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►In regions in which the function has a simple pole at and is analytic at (the case in §10.72(i)), asymptotic expansions of the solutions of (10.72.1) for large can be constructed in terms of Bessel functions and modified Bessel functions of order , where is the limiting value of as .
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►Then for large asymptotic approximations of the solutions can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on and ).
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