equivalent expressions
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1: 31.3 Basic Solutions
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§31.3(iii) Equivalent Expressions
… ►There are 192 automorphisms in all, so there are equivalent expressions for each of the 8. …2: Guide to Searching the DLMF
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►To recognize the math symbols and structures, and to accommodate equivalence between various notations and various forms of expression, the search system maps the math part of your queries into a different form.
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3: 18.39 Applications in the Physical Sciences
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►This equivalent quadrature relationship, see Heller et al. (1973), Yamani and Reinhardt (1975), allows extraction of scattering information from the finite dimensional functions of (18.39.53), provided that such information involves potentials, or projections onto functions, exactly expressed, or well approximated, in the finite basis of (18.39.44).
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4: 15.5 Derivatives and Contiguous Functions
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►By repeated applications of (15.5.11)–(15.5.18) any function , in which are integers, can be expressed as a linear combination of and any one of its contiguous functions, with coefficients that are rational functions of , and .
►An equivalent equation to the hypergeometric differential equation (15.10.1) is
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5: 34.4 Definition: Symbol
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►Except in degenerate cases the combination of the triangle inequalities for the four symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths ; see Figure 34.4.1.
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►The symbol can be expressed as the finite sum
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►Equivalently,
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►For alternative expressions for the symbol, written either as a finite sum or as other terminating generalized hypergeometric series of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).
6: 27.2 Functions
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►It can be expressed as a sum over all primes :
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►An equivalent form states that the th prime (when the primes are listed in increasing order) is asymptotic to as :
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27.2.8
►and if is the smallest positive integer such that , then is a primitive root mod .
…It is the special case of the function that counts the number of ways of expressing
as the product of factors, with the order of factors taken into account.
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7: 33.11 Asymptotic Expansions for Large
8: 2.1 Definitions and Elementary Properties
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►Condition (2.1.13) is equivalent to
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►Some asymptotic approximations are expressed in terms of two or more Poincaré asymptotic expansions.
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9: 23.20 Mathematical Applications
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►or equivalently, on replacing by and by (projective coordinates), into the form
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►In terms of the addition law can be expressed
, ; otherwise , where
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10: 33.22 Particle Scattering and Atomic and Molecular Spectra
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►The relativistic motion of spinless particles in a Coulomb field, as encountered in pionic atoms and pion-nucleon scattering (Backenstoss (1970)) is described by a Klein–Gordon equation equivalent to (33.2.1); see Barnett (1981a).
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►The penetrability of repulsive Coulomb potential barriers is normally expressed in terms of the quantity (Mott and Massey (1956, pp. 63–65)).
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