About the Project

%E7%9C%9F%E4%BA%BA%E7%9C%9F%E9%92%B1%E6%A3%8B%E7%89%8C%E6%B8%B8%E6%88%8F%E5%A4%A7%E5%8E%85,%E7%BD%91%E4%B8%8A%E7%8E%B0%E9%87%91%E6%A3%8B%E7%89%8C%E6%B8%B8%E6%88%8F%E5%B9%B3%E5%8F%B0,%E3%80%90%E7%9C%9F%E4%BA%BA%E6%A3%8B%E7%89%8C%E5%AE%98%E6%96%B9%E7%BD%91%E7%AB%99%E2%88%B6789yule.com%E3%80%91%E7%BD%91%E4%B8%8A%E7%9C%9F%E9%92%B1%E6%A3%8B%E7%89%8C%E6%B8%B8%E6%88%8F%E5%A4%A7%E5%8E%85,%E5%9C%A8%E7%BA%BF%E6%A3%8B%E7%89%8C%E6%B8%B8%E6%88%8F%E7%BD%91%E7%AB%99,%E6%89%8B%E6%9C%BA%E6%A3%8B%E7%89%8Capp%E4%B8%8B%E8%BD%BD,%E6%A3%8B%E7%89%8C%E6%B8%B8%E6%88%8Fapp,%E6%A3%8B%E7%89%8C%E5%8D%9A%E5%BD%A9%E5%B9%B3%E5%8F%B0,%E3%80%90%E6%A3%8B%E7%89%8C%E6%B8%B8%E6%88%8F%E5%85%AC%E5%8F%B8%E2%88%B6789yule.com%E3%80%91%E7%BD%91%E5%9D%80ZEAE0CA0kAkn0BEB

AdvancedHelp

Did you mean %E7%9C%9F%E4%BA%BA%E7%9C%9F%E9%92%B1%E6%A3%8B%E7%89%8C%E6%B8%B8%E6%88%8F%E5%A4%A7%E5%8E%85,%E7%BD%91%E4%B8%8A%E7%8E%B0%E9%87%91%E6%A3%8B%E7%89%8C%E6%B8%B8%E6%88%8F%E5%B9%B3%E5%8F%B0,%E3%80%90%E7%9C%9F%E4%BA%BA%E6%A3%8B%E7%89%8C%E5%AE%98%E6%96%B9%E7%BD%91%E7%AB%99%E2%88%caniuse.com%E3%80%91%E7%BD%91%E4%B8%8A%E7%9C%9F%E9%92%B1%E6%A3%8B%E7%89%8C%E6%B8%B8%E6%88%8F%E5%A4%A7%E5%8E%85,%E5%9C%A8%E7%BA%BF%E6%A3%8B%E7%89%8C%E6%B8%B8%E6%88%8F%E7%BD%91%E7%AB%99,%E6%89%8B%E6%9C%BA%E6%A3%8B%E7%89%8Capp%E4%B8%8B%E8%BD%BD,%E6%A3%8B%E7%89%8C%E6%B8%B8%E6%88%8Fapp,%E6%A3%8B%E7%89%8C%E5%8D%9A%E5%BD%A9%E5%B9%B3%E5%8F%B0,%E3%80%90%E6%A3%8B%E7%89%8C%E6%B8%B8%E6%88%8F%E5%85%AC%E5%8F%B8%E2%88%caniuse.com%E3%80%91%E7%BD%91%E5%9D%80ZEAE0CA0kAkn0BEB ?

(0.081 seconds)

1—10 of 612 matching pages

1: 34.6 Definition: 9 ⁒ j Symbol
§34.6 Definition: 9 ⁒ j Symbol
β–ΊThe 9 ⁒ j symbol may be defined either in terms of 3 ⁒ j symbols or equivalently in terms of 6 ⁒ j symbols: β–Ί
34.6.1 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = all  ⁒ m r ⁒ s ( j 11 j 12 j 13 m 11 m 12 m 13 ) ⁒ ( j 21 j 22 j 23 m 21 m 22 m 23 ) ⁒ ( j 31 j 32 j 33 m 31 m 32 m 33 ) ⁒ ( j 11 j 21 j 31 m 11 m 21 m 31 ) ⁒ ( j 12 j 22 j 32 m 12 m 22 m 32 ) ⁒ ( j 13 j 23 j 33 m 13 m 23 m 33 ) ,
β–Ί
34.6.2 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = j ( 1 ) 2 ⁒ j ⁒ ( 2 ⁒ j + 1 ) ⁒ { j 11 j 21 j 31 j 32 j 33 j } ⁒ { j 12 j 22 j 32 j 21 j j 23 } ⁒ { j 13 j 23 j 33 j j 11 j 12 } .
β–ΊThe 9 ⁒ j symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …
2: 19.2 Definitions
β–ΊThe principal values of K ⁑ ( k ) and E ⁑ ( k ) are even functions. … β–Ί
§19.2(iv) A Related Function: R C ⁑ ( x , y )
β–ΊFormulas involving Ξ  ⁑ ( Ο• , Ξ± 2 , k ) that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using R C ⁑ ( x , y ) . … β–ΊWhen x and y are positive, R C ⁑ ( x , y ) is an inverse circular function if x < y and an inverse hyperbolic function (or logarithm) if x > y : …For the special cases of R C ⁑ ( x , x ) and R C ⁑ ( 0 , y ) see (19.6.15). …
3: 3.4 Differentiation
β–ΊThe B k n are the differentiated Lagrangian interpolation coefficients: …where A k n is as in (3.3.10). … β–Ί
B 2 7 = 1 240 ⁒ ( 72 + 36 ⁒ t 267 ⁒ t 2 80 ⁒ t 3 + 90 ⁒ t 4 + 12 ⁒ t 5 7 ⁒ t 6 ) ,
β–Ίwhere C is a simple closed contour described in the positive rotational sense such that C and its interior lie in the domain of analyticity of f , and x 0 is interior to C . Taking C to be a circle of radius r centered at x 0 , we obtain …
4: 1.12 Continued Fractions
β–Ί C n is called the n th approximant or convergent to C . … β–ΊDefine … β–ΊA 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 . Conversely, C is called an extension of C . … β–ΊThen the convergents C n satisfy …
5: 26.6 Other Lattice Path Numbers
β–Ί
Delannoy Number D ⁑ ( m , n )
β–Ί D ⁑ ( m , n ) is the number of paths from ( 0 , 0 ) to ( m , n ) that are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . … β–Ί
Table 26.6.1: Delannoy numbers D ⁑ ( m , n ) .
β–Ί β–Ίβ–Ίβ–Ί
m n
0 1 2 3 4 5 6 7 8 9 10
β–Ί
β–Ί
Table 26.6.2: Motzkin numbers M ⁑ ( n ) .
β–Ί β–Ίβ–Ίβ–Ί
n M ⁑ ( n ) n M ⁑ ( n ) n M ⁑ ( n ) n M ⁑ ( n ) n M ⁑ ( n )
0 1 4 9 8 323 12 15511 16 8 53467
β–Ί
β–Ί
Table 26.6.3: Narayana numbers N ⁑ ( n , k ) .
β–Ί β–Ίβ–Ίβ–Ί
n k
0 1 2 3 4 5 6 7 8 9 10
β–Ί
6: 26.5 Lattice Paths: Catalan Numbers
β–Ί C ⁑ ( n ) is the Catalan number. …(Sixty-six equivalent definitions of C ⁑ ( n ) are given in Stanley (1999, pp. 219–229).) … β–Ί
26.5.3 C ⁑ ( n + 1 ) = k = 0 n C ⁑ ( k ) ⁒ C ⁑ ( n k ) ,
β–Ί
26.5.4 C ⁑ ( n + 1 ) = 2 ⁒ ( 2 ⁒ n + 1 ) n + 2 ⁒ C ⁑ ( n ) ,
β–Ί
26.5.7 lim n C ⁑ ( n + 1 ) C ⁑ ( n ) = 4 .
7: 34.14 Tables
§34.14 Tables
β–ΊTables of exact values of the squares of the 3 ⁒ j and 6 ⁒ j symbols in which all parameters are 8 are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of 3 ⁒ j , 6 ⁒ j , and 9 ⁒ j symbols on pp. … β–ΊSome selected 9 ⁒ j symbols are also given. … 16-17; for 9 ⁒ j symbols on p. … β–Ί 310–332, and for the 9 ⁒ j symbols on pp. …
8: Bibliography H
β–Ί
  • P. I. HadΕΎi (1975b) Integrals containing the Fresnel functions S ⁒ ( x ) and C ⁒ ( x ) . Bul. Akad. Ε tiince RSS Moldoven. 1975 (3), pp. 48–60, 93 (Russian).
  • β–Ί
  • B. A. Hargrave and B. D. Sleeman (1977) Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.
  • β–Ί
  • J. R. Herndon (1961b) Algorithm 56: Complete elliptic integral of the second kind. Comm. ACM 4 (4), pp. 180–181.
  • β–Ί
  • L. E. Hoisington and G. Breit (1938) Calculation of Coulomb wave functions for high energies. Phys. Rev. 54 (8), pp. 627–628.
  • β–Ί
  • K. Horata (1991) On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.
  • 9: Bibliography K
    β–Ί
  • K. W. J. Kadell (1994) A proof of the q -Macdonald-Morris conjecture for B ⁒ C n . Mem. Amer. Math. Soc. 108 (516), pp. vi+80.
  • β–Ί
  • M. Kaneko (1997) Poly-Bernoulli numbers. J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.
  • β–Ί
  • E. Kanzieper (2002) Replica field theories, Painlevé transcendents, and exact correlation functions. Phys. Rev. Lett. 89 (25), pp. (250201–1)–(250201–4).
  • β–Ί
  • T. H. Koornwinder (1992) Askey-Wilson Polynomials for Root Systems of Type B ⁒ C . In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, pp. 189–204.
  • β–Ί
  • E. Kreyszig (1957) On the zeros of the Fresnel integrals. Canad. J. Math. 9, pp. 118–131.
  • 10: 27.2 Functions
    β–ΊIt is the special case k = 2 of the function d k ⁑ ( n ) that counts the number of ways of expressing n as the product of k factors, with the order of factors taken into account. …Note that Οƒ 0 ⁑ ( n ) = d ⁑ ( n ) . … β–ΊTable 27.2.2 tabulates the Euler totient function Ο• ⁑ ( n ) , the divisor function d ⁑ ( n ) ( = Οƒ 0 ⁑ ( n ) ), and the sum of the divisors Οƒ ⁑ ( n ) ( = Οƒ 1 ⁑ ( n ) ), for n = 1 ⁒ ( 1 ) ⁒ 52 . … β–Ί
    Table 27.2.2: Functions related to division.
    β–Ί β–Ίβ–Ίβ–Ίβ–Ί
    n Ο• ⁑ ( n ) d ⁑ ( n ) Οƒ ⁑ ( n ) n Ο• ⁑ ( n ) d ⁑ ( n ) Οƒ ⁑ ( n ) n Ο• ⁑ ( n ) d ⁑ ( n ) Οƒ ⁑ ( n ) n Ο• ⁑ ( n ) d ⁑ ( n ) Οƒ ⁑ ( n )
    1 1 1 1 14 6 4 24 27 18 4 40 40 16 8 90
    10 4 4 18 23 22 2 24 36 12 9 91 49 42 3 57
    β–Ί