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11: 23.20 Mathematical Applications
or equivalently, on replacing x by x / z and y by y / z (projective coordinates), into the form
12: 18.39 Applications in the Physical Sciences
The equivalent quadrature weight, w i / w CP ( x i ) , also forms the foundation of a novel inversion of the Stieltjes–Perron moment inversion discussed in §18.40(ii). …
13: 10.41 Asymptotic Expansions for Large Order
§10.41(i) Asymptotic Forms
as z in | ph z | π δ , or equivalently as ζ in | ph ( ζ ) | 2 3 π δ , for fixed ( 0 ) and fixed ν ( > 0 ) . It needs to be noted that the results (10.41.14) and (10.41.15) do not apply when z 0 + or equivalently ζ + . This is because A k ( ζ ) and ζ 1 2 B k ( ζ ) , k = 0 , 1 , , do not form an asymptotic scale (§2.1(v)) as ζ + ; see Olver (1997b, pp. 422–425). …
14: 27.12 Asymptotic Formulas: Primes
π ( x ) li ( x ) changes sign infinitely often as x ; see Littlewood (1914), Bays and Hudson (2000). The Riemann hypothesis25.10(i)) is equivalent to the statement that for every x 2657 , … A Mersenne prime is a prime of the form 2 p 1 . … For example, if 2 n 2 ( mod n ) , then n is composite. … A Carmichael number is a composite number n for which b n b ( mod n ) for all b . …
15: 21.1 Special Notation
g , h positive integers.
S 1 S 2 set of all elements of the form element of  S 1 × element of  S 2 ”.
S 1 / S 2 set of all elements of S 1 , modulo elements of S 2 . Thus two elements of S 1 / S 2 are equivalent if they are both in S 1 and their difference is in S 2 . (For an example see §20.12(ii).)
16: 1.12 Continued Fractions
Equivalence
Two continued fractions are equivalent if they have the same convergents. … A contraction of a continued fraction C is a continued fraction C whose convergents { C n } form a subsequence of the convergents { C n } of C . …The even part of C exists iff b 2 k 0 , k = 1 , 2 , , and up to equivalence is given by …The odd part of C exists iff b 2 k + 1 0 , k = 0 , 1 , 2 , , and up to equivalence is given by …
17: 25.16 Mathematical Applications
The prime number theorem (27.2.3) is equivalent to the statement
25.16.3 ψ ( x ) = x + o ( x ) , x .
The Riemann hypothesis is equivalent to the statement … Euler sums have the form
18: 3.6 Linear Difference Equations
Many special functions satisfy second-order recurrence relations, or difference equations, of the form …or equivalently, … Then w n is said to be a recessive (equivalently, minimal or distinguished) solution as n , and it is unique except for a constant factor. … (This part of the process is equivalent to forward elimination.) … 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). …
19: 26.9 Integer Partitions: Restricted Number and Part Size
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. … Equations (26.9.2)–(26.9.3) are examples of closed forms that can be computed explicitly for any positive integer k . … equivalently, partitions into at most k parts either have exactly k parts, in which case we can subtract one from each part, or they have strictly fewer than k parts. …
§26.9(iv) Limiting Form
20: Mathematical Introduction
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). …
complex plane (excluding infinity).
is equivalent to.
( a , b ] or [ a , b ) half-closed intervals.
mod or modulo m n ( mod p ) means p divides m n , where m , n , and p are positive integers with m > n .
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. …