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1: 33.5 Limiting Forms for Small ρ , Small | η | , or Large
§33.5 Limiting Forms for Small ρ , Small | η | , or Large
§33.5(i) Small ρ
§33.5(iii) Small | η |
2: 18.15 Asymptotic Approximations
18.15.7 ε M ( ρ , θ ) = { θ O ( ρ 2 M ( 3 / 2 ) ) , c ρ 1 θ π δ , θ α + ( 5 / 2 ) O ( ρ 2 M + α ) , 0 θ c ρ 1 ,
3: 2.11 Remainder Terms; Stokes Phenomenon
Owing to the factor e ρ , that is, e | z | in (2.11.13), F n + p ( z ) is uniformly exponentially small compared with E p ( z ) . … Hence from §7.12(i) erfc ( 1 2 ρ c ( θ ) ) is of the same exponentially-small order of magnitude as the contribution from the other terms in (2.11.15) when ρ is large. On the other hand, when π + δ θ 3 π δ , c ( θ ) is in the left half-plane and erfc ( 1 2 ρ c ( θ ) ) differs from 2 by an exponentially-small quantity. …
4: 33.12 Asymptotic Expansions for Large η
§33.12 Asymptotic Expansions for Large η
§33.12(i) Transition Region
§33.12(ii) Uniform Expansions
With the substitution ρ = 2 η z , Equation (33.2.1) becomes …
5: 33.23 Methods of Computation
§33.23 Methods of Computation
Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. …
§33.23(vii) WKBJ Approximations
WKBJ approximations (§2.7(iii)) for ρ > ρ tp ( η , ) are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq.  (12) ( ρ c ) / c should be ( ρ c ) / ρ ). …
6: 3.2 Linear Algebra
𝐀 2 = ρ ( 𝐀 𝐀 T ) ,
where ρ ( 𝐀 𝐀 T ) is the largest of the absolute values of the eigenvalues of the matrix 𝐀 𝐀 T ; see §3.2(iv). … The sensitivity of the solution vector 𝐱 in (3.2.1) to small perturbations in the matrix 𝐀 and the vector 𝐛 is measured by the condition numberIf 𝐀 is nondefective and λ is a simple zero of p n ( λ ) , then the sensitivity of λ to small perturbations in the matrix 𝐀 is measured by the condition number
7: 27.19 Methods of Computation: Factorization
Type I probabilistic algorithms include the Brent–Pollard rho algorithm (also called Monte Carlo method), the Pollard p 1 algorithm, and the Elliptic Curve Method (ecm). …As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard p 1 , and a 67-digit prime for ecm. … The snfs can be applied only to numbers that are very close to a power of a very small base. …
8: 2.5 Mellin Transform Methods
§2.5(iii) Laplace Transforms with Small Parameters
for any ρ satisfying 1 < ρ < 2 . … where l ( 2 ) is an arbitrary integer and δ is an arbitrary small positive constant. … … To verify (2.5.48) we may use …
9: 28.35 Tables
  • Blanch and Clemm (1969) includes eigenvalues a n ( q ) , b n ( q ) for q = ρ e i ϕ , ρ = 0 ( .5 ) 25 , ϕ = 5 ( 5 ) 90 , n = 0 ( 1 ) 15 ; 4D. Also a n ( q ) and b n ( q ) for q = i ρ , ρ = 0 ( .5 ) 100 , n = 0 ( 2 ) 14 and n = 2 ( 2 ) 16 , respectively; 8D. Double points for n = 0 ( 1 ) 15 ; 8D. Graphs are included.

  • Ince (1932) includes the first zero for ce n , se n for n = 2 ( 1 ) 5 or 6 , q = 0 ( 1 ) 10 ( 2 ) 40 ; 4D. This reference also gives zeros of the first derivatives, together with expansions for small q .

  • 10: 13.7 Asymptotic Expansions for Large Argument
    Here δ denotes an arbitrary small positive constant. …
    13.7.5 | ε n ( z ) | , β 1 | ε n ( z ) | 2 α C n | ( a ) n ( a b + 1 ) n n ! z a + n | exp ( 2 α ρ C 1 | z | ) ,
    ρ = 1 2 | 2 a 2 2 a b + b | + σ ( 1 + 1 4 σ ) ( 1 σ ) 2 ,
    13.7.13 R m , n ( a , b , z ) = { O ( e | z | z m ) , | ph z | π , O ( e z z m ) , π | ph z | 5 2 π δ .