fundamental unit cell
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11: 4.23 Inverse Trigonometric Functions
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►In (4.23.3) the integration path may not intersect .
… and have branch points at ; the other four functions have branch points at .
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►Care needs to be taken on the cuts, for example, if then .
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§4.23(v) Fundamental Property
… ►where and in (4.23.34) and (4.23.35), and in (4.23.36). …12: 13.14 Definitions and Basic Properties
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13.14.10
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13.14.11
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§13.14(v) Numerically Satisfactory Solutions
►Fundamental pairs of solutions of (13.14.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are … ►A fundamental pair of solutions that is numerically satisfactory in the sector near the origin is …13: 36.7 Zeros
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►Near , and for small and , the modulus has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose and repeat distances are given by
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36.7.6
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14: 1.4 Calculus of One Variable
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►For historical reasons, is also sometimes referred to as a density, as, for example, the mass per unit length at point , see Shohat and Tamarkin (1970, p vii).
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►If, for example, , the Heaviside unit step-function (1.16.14), then the corresponding measure is , where is the Dirac -function of §1.17, such that, for a continuous function on , for and otherwise.
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Fundamental Theorem of Calculus
…15: 3.12 Mathematical Constants
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►The fundamental constant
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16: 8.24 Physical Applications
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►With more general values of , supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).
17: 10.73 Physical Applications
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►and on separation of variables we obtain solutions of the form , from which a solution satisfying prescribed boundary conditions may be constructed.
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►on assuming a time dependence of the form .
…It is fundamental in the study of electromagnetic wave transmission.
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18: 27.14 Unrestricted Partitions
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►A fundamental problem studies the number of ways can be written as a sum of positive integers , that is, the number of solutions of
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27.14.10
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27.14.14
►where and is given by (27.14.11).
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►The 24th power of in (27.14.12) with is an infinite product that generates a power series in with integer coefficients called Ramanujan’s tau function
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19: 18.38 Mathematical Applications
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►Classical OP’s play a fundamental role in Gaussian quadrature.
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►also the case of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane.
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►The symbol (34.2.6), with an alternative expression as a terminating of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials.
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►The symbol (34.4.3), with an alternative expression as a terminating balanced of unit argument, can be expressend in terms of Racah polynomials (18.26.3).
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18.38.10
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