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11: 1.3 Determinants, Linear Operators, and Spectral Expansions
The cofactor A j k of a j k is … For real-valued a j k , … where ω 1 , ω 2 , , ω n are the n th roots of unity (1.11.21). … If 𝐷 n [ a j , k ] tends to a limit L as n , then we say that the infinite determinant 𝐷 [ a j , k ] converges and 𝐷 [ a j , k ] = L . … The corresponding eigenvectors 𝐚 1 , , 𝐚 n can be chosen such that they form a complete orthonormal basis in 𝐄 n . …
12: 28.6 Expansions for Small q
Leading terms of the power series for a m ( q ) and b m ( q ) for m 6 are: … The coefficients of the power series of a 2 n ( q ) , b 2 n ( q ) and also a 2 n + 1 ( q ) , b 2 n + 1 ( q ) are the same until the terms in q 2 n 2 and q 2 n , respectively. … Numerical values of the radii of convergence ρ n ( j ) of the power series (28.6.1)–(28.6.14) for n = 0 , 1 , , 9 are given in Table 28.6.1. Here j = 1 for a 2 n ( q ) , j = 2 for b 2 n + 2 ( q ) , and j = 3 for a 2 n + 1 ( q ) and b 2 n + 1 ( q ) . …
§28.6(ii) Functions ce n and se n
13: 10.75 Tables
  • Achenbach (1986) tabulates J 0 ( x ) , J 1 ( x ) , Y 0 ( x ) , Y 1 ( x ) , x = 0 ( .1 ) 8 , 20D or 18–20S.

  • Makinouchi (1966) tabulates all values of j ν , m and y ν , m in the interval ( 0 , 100 ) , with at least 29S. These are for ν = 0 ( 1 ) 5 , 10, 20; ν = 3 2 , 5 2 ; ν = m / n with m = 1 ( 1 ) n 1 and n = 3 ( 1 ) 8 , except for ν = 1 2 .

  • Abramowitz and Stegun (1964, Chapter 11) tabulates 0 x J 0 ( t ) d t , 0 x Y 0 ( t ) d t , x = 0 ( .1 ) 10 , 10D; 0 x t 1 ( 1 J 0 ( t ) ) d t , x t 1 Y 0 ( t ) d t , x = 0 ( .1 ) 5 , 8D.

  • Leung and Ghaderpanah (1979), tabulates all zeros of the principal value of K n ( z ) , for n = 2 ( 1 ) 10 , 29S.

  • Abramowitz and Stegun (1964, Chapter 11) tabulates e x 0 x I 0 ( t ) d t , e x x K 0 ( t ) d t , x = 0 ( .1 ) 10 , 7D; e x 0 x t 1 ( I 0 ( t ) 1 ) d t , x e x x t 1 K 0 ( t ) d t , x = 0 ( .1 ) 5 , 6D.

  • 14: 26.12 Plane Partitions
    26.12.9 ( h = 1 r j = 1 s h + j + t 1 h + j 1 ) 2 ;
    26.12.10 ( h = 1 r j = 1 s h + j + t 1 h + j 1 ) ( h = 1 r + 1 j = 1 s h + j + t 1 h + j 1 ) ;
    26.12.11 ( h = 1 r + 1 j = 1 s h + j + t 1 h + j 1 ) ( h = 1 r j = 1 s + 1 h + j + t 1 h + j 1 ) .
    The notation π B ( r , s , t ) denotes the sum over all plane partitions contained in B ( r , s , t ) , and | π | denotes the number of elements in π . … where σ 2 ( j ) is the sum of the squares of the divisors of j . …
    15: 3.6 Linear Difference Equations
    Given numerical values of w 0 and w 1 , the solution w n of the equation …These errors have the effect of perturbing the solution by unwanted small multiples of w n and of an independent solution g n , say. … The unwanted multiples of g n now decay in comparison with w n , hence are of little consequence. … The latter method is usually superior when the true value of w 0 is zero or pathologically small. … beginning with e 0 = w 0 . …
    16: 5.10 Continued Fractions
    where
    a 0 = 1 12 ,
    a 1 = 1 30 ,
    a 2 = 53 210 ,
    For exact values of a 7 to a 11 and 40S values of a 0 to a 40 , see Char (1980). …
    17: Bibliography
  • S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.
  • D. E. Amos (1983c) Uniform asymptotic expansions for exponential integrals E n ( x ) and Bickley functions Ki n ( x ) . ACM Trans. Math. Software 9 (4), pp. 467–479.
  • K. Aomoto (1987) Special value of the hypergeometric function F 2 3 and connection formulae among asymptotic expansions. J. Indian Math. Soc. (N.S.) 51, pp. 161–221.
  • V. I. Arnol’d (1972) Normal forms of functions near degenerate critical points, the Weyl groups A k , D k , E k and Lagrangian singularities. Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).
  • V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
  • 18: 3.7 Ordinary Differential Equations
    The path is partitioned at P + 1 points labeled successively z 0 , z 1 , , z P , with z 0 = a , z P = b . … Write τ j = z j + 1 z j , j = 0 , 1 , , P , expand w ( z ) and w ( z ) in Taylor series (§1.10(i)) centered at z = z j , and apply (3.7.2). … If, for example, β 0 = β 1 = 0 , then on moving the contributions of w ( z 0 ) and w ( z P ) to the right-hand side of (3.7.13) the resulting system of equations is not tridiagonal, but can readily be made tridiagonal by annihilating the elements of 𝐀 P that lie below the main diagonal and its two adjacent diagonals. … The values λ k are the eigenvalues and the corresponding solutions w k of the differential equation are the eigenfunctions. … where h = z n + 1 z n and …
    19: 24.20 Tables
    Abramowitz and Stegun (1964, Chapter 23) includes exact values of k = 1 m k n , m = 1 ( 1 ) 100 , n = 1 ( 1 ) 10 ; k = 1 k n , k = 1 ( 1 ) k 1 k n , k = 0 ( 2 k + 1 ) n , n = 1 , 2 , , 20D; k = 0 ( 1 ) k ( 2 k + 1 ) n , n = 1 , 2 , , 18D. Wagstaff (1978) gives complete prime factorizations of N n and E n for n = 20 ( 2 ) 60 and n = 8 ( 2 ) 42 , respectively. … For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).
    20: 3.2 Linear Algebra
    where u j = c j , j = 1 , 2 , , n 1 , d 1 = b 1 , and …Forward elimination for solving 𝐀 𝐱 = 𝐟 then becomes y 1 = f 1 , …and back substitution is x n = y n / d n , followed by … Define the Lanczos vectors 𝐯 j and coefficients α j and β j by 𝐯 0 = 𝟎 , a normalized vector 𝐯 1 (perhaps chosen randomly), α 1 = 𝐯 1 T 𝐀 𝐯 1 , β 1 = 0 , and for j = 1 , 2 , , n 1 by the recursive scheme … Start with 𝐯 0 = 𝟎 , vector 𝐯 1 such that 𝐯 1 T 𝐒 𝐯 1 = 1 , α 1 = 𝐯 1 T 𝐀 𝐯 1 , β 1 = 0 . …