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11: Bibliography D
  • T. M. Dunster (1990a) Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.
  • 12: 31.7 Relations to Other Functions
    With z = sn 2 ( ζ , k ) and …
    13: 1.13 Differential Equations
    The substitution ξ = 1 / z in (1.13.1) gives …
    1.13.21 { z , ζ } = ( d ξ / d ζ ) 2 { z , ξ } + { ξ , ζ } .
    14: 22.9 Cyclic Identities
    22.9.7 κ = dn ( 2 K ( k ) / 3 , k ) ,
    22.9.8 s 1 , 3 ( 4 ) s 2 , 3 ( 4 ) + s 2 , 3 ( 4 ) s 3 , 3 ( 4 ) + s 3 , 3 ( 4 ) s 1 , 3 ( 4 ) = κ 2 - 1 k 2 ,
    22.9.15 d 1 , 3 ( 2 ) d 2 , 3 ( 2 ) d 3 , 3 ( 2 ) = κ 2 + k 2 - 1 1 - κ 2 ( d 1 , 3 ( 2 ) + d 2 , 3 ( 2 ) + d 3 , 3 ( 2 ) ) ,
    22.9.22 s 1 , 3 ( 2 ) c 1 , 3 ( 2 ) d 2 , 3 ( 2 ) d 3 , 3 ( 2 ) + s 2 , 3 ( 2 ) c 2 , 3 ( 2 ) d 3 , 3 ( 2 ) d 1 , 3 ( 2 ) + s 3 , 3 ( 2 ) c 3 , 3 ( 2 ) d 1 , 3 ( 2 ) d 2 , 3 ( 2 ) = κ 2 + k 2 - 1 1 - κ 2 ( s 1 , 3 ( 2 ) c 1 , 3 ( 2 ) + s 2 , 3 ( 2 ) c 2 , 3 ( 2 ) + s 3 , 3 ( 2 ) c 3 , 3 ( 2 ) ) ,
    15: 15.8 Transformations of Variable
    With ζ = e 2 π i / 3 ( 1 - z ) / ( z - e 4 π i / 3 )
    16: 9.6 Relations to Other Functions
    9.6.1 ζ = 2 3 z 3 / 2 ,
    9.6.10 z = ( 3 2 ζ ) 2 / 3 ,
    9.6.11 J ± 1 / 3 ( ζ ) = 1 2 3 / z ( 3 Ai ( - z ) Bi ( - z ) ) ,
    9.6.13 I ± 1 / 3 ( ζ ) = 1 2 3 / z ( 3 Ai ( z ) + Bi ( z ) ) ,
    9.6.14 I ± 2 / 3 ( ζ ) = 1 2 ( 3 / z ) ( ± 3 Ai ( z ) + Bi ( z ) ) ,
    17: 9.7 Asymptotic Expansions
    9.7.1 ζ = 2 3 z 3 / 2 .
    9.7.13 Bi ( z e ± π i / 3 ) 2 π e ± π i / 6 z 1 / 4 ( cos ( ζ - 1 4 π 1 2 i ln 2 ) k = 0 ( - 1 ) k u 2 k ζ 2 k + sin ( ζ - 1 4 π 1 2 i ln 2 ) k = 0 ( - 1 ) k u 2 k + 1 ζ 2 k + 1 ) , | ph z | 2 3 π - δ ,
    18: 20.11 Generalizations and Analogs
    20.11.2 1 n G ( m , n ) = 1 n k = 0 n - 1 e - π i k 2 m / n = e - π i / 4 m j = 0 m - 1 e π i j 2 n / m = e - π i / 4 m G ( - n , m ) .
    20.11.3 f ( a , b ) = n = - a n ( n + 1 ) / 2 b n ( n - 1 ) / 2 ,
    §20.11(iii) Ramanujan’s Change of Base
    As in §20.11(ii), the modulus k of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in q -series via (20.9.1). … These results are called Ramanujan’s changes of base. …
    19: 27.12 Asymptotic Formulas: Primes
    27.12.5 | π ( x ) - li ( x ) | = O ( x exp ( - c ( ln x ) 1 / 2 ) ) , x .
    27.12.6 | π ( x ) - li ( x ) | = O ( x exp ( - d ( ln x ) 3 / 5 ( ln ln x ) - 1 / 5 ) ) .
    27.12.8 li ( x ) ϕ ( m ) + O ( x exp ( - λ ( α ) ( ln x ) 1 / 2 ) ) , m ( ln x ) α , α > 0 ,
    20: 1.17 Integral and Series Representations of the Dirac Delta
    1.17.10 1 2 π - e - t 2 / ( 4 n ) e i ( x - a ) t d t = n π e - n ( x - a ) 2 .
    1.17.11 δ n ( x - a ) = 1 2 π - e - t 2 / ( 4 n ) e i ( x - a ) t d t ,
    1.17.23 δ ( x - a ) = e - ( x + a ) / 2 k = 0 L k ( x ) L k ( a ) .
    1.17.24 δ ( x - a ) = e - ( x 2 + a 2 ) / 2 π k = 0 H k ( x ) H k ( a ) 2 k k ! .