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11: 23.20 Mathematical Applications
12: 18.39 Applications in the Physical Sciences
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►The equivalent quadrature weight, , also forms the foundation of a novel inversion of the Stieltjes–Perron moment inversion discussed in §18.40(ii).
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13: 10.41 Asymptotic Expansions for Large Order
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§10.41(i) Asymptotic Forms
… ►as in , or equivalently as in , for fixed and fixed . ►It needs to be noted that the results (10.41.14) and (10.41.15) do not apply when or equivalently . This is because and , do not form an asymptotic scale (§2.1(v)) as ; see Olver (1997b, pp. 422–425). …14: 27.12 Asymptotic Formulas: Primes
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changes sign infinitely often as ; see Littlewood (1914), Bays and Hudson (2000).
►The Riemann hypothesis (§25.10(i)) is equivalent to the statement that for every ,
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►A Mersenne prime is a prime of the form
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►For example, if , then is composite.
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►A Carmichael number is a composite number for which for all .
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15: 21.1 Special Notation
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positive integers. | |
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set of all elements of the form “”. | |
set of all elements of , modulo elements of . Thus two elements of are equivalent if they are both in and their difference is in . (For an example see §20.12(ii).) | |
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16: 1.12 Continued Fractions
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Equivalence
►Two continued fractions are equivalent if they have the same convergents. … ►A contraction of a continued fraction is a continued fraction whose convergents form a subsequence of the convergents of . …The even part of exists iff , , and up to equivalence is given by …The odd part of exists iff , , and up to equivalence is given by …17: 25.16 Mathematical Applications
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►The prime number theorem (27.2.3) is equivalent to the statement
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25.16.3
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►The Riemann hypothesis is equivalent to the statement
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►Euler sums have the form
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18: 3.6 Linear Difference Equations
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►Many special functions satisfy second-order recurrence relations, or difference equations, of the form
…or equivalently,
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►Then is said to be a recessive (equivalently, minimal or distinguished) solution as , and it is unique except for a constant factor.
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►(This part of the process is equivalent to forward elimination.)
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►For further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6).
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19: 26.9 Integer Partitions: Restricted Number and Part Size
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►The conjugate partition is obtained by reflecting the Ferrers graph across the main diagonal or, equivalently, by representing each integer by a column of dots.
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►Equations (26.9.2)–(26.9.3) are examples of closed forms that can be computed explicitly for any positive integer .
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►equivalently, partitions into at most parts either have exactly parts, in which case we can subtract one from each part, or they have strictly fewer than parts.
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§26.9(iv) Limiting Form
…20: Mathematical Introduction
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►Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19).
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►For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed.
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complex plane (excluding infinity). | |
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is equivalent to. | |
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or | half-closed intervals. |
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or modulo | means divides , where , , and are positive integers with . |
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