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11: 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 } .
12: 15.8 Transformations of Variable
With ζ = e 2 π i / 3 ( 1 - z ) / ( z - e 4 π i / 3 )
13: 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. …
14: 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 ) ) ,
15: 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 ,
16: 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 ! .
17: 9.7 Asymptotic Expansions
9.7.1 ζ = 2 3 z 3 / 2 .
9.7.6 Ai ( z ) - z 1 / 4 e - ζ 2 π k = 0 ( - 1 ) k v k ζ k , | ph z | π - δ ,
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: 23.6 Relations to Other Functions
With z = π u / ( 2 ω 1 ) , …
19: 8.27 Approximations
  • DiDonato (1978) gives a simple approximation for the function F ( p , x ) = x - p e x 2 / 2 x e - t 2 / 2 t p d t (which is related to the incomplete gamma function by a change of variables) for real p and large positive x . This takes the form F ( p , x ) = 4 x / h ( p , x ) , approximately, where h ( p , x ) = 3 ( x 2 - p ) + ( x 2 - p ) 2 + 8 ( x 2 + p ) and is shown to produce an absolute error O ( x - 7 ) as x .

  • 20: 9.5 Integral Representations
    9.5.7 Ai ( z ) = e - ζ π 0 exp ( - z 1 / 2 t 2 ) cos ( 1 3 t 3 ) d t , | ph z | < π .
    9.5.8 Ai ( z ) = e - ζ ζ - 1 / 6 π ( 48 ) 1 / 6 Γ ( 5 6 ) 0 e - t t - 1 / 6 ( 2 + t ζ ) - 1 / 6 d t , | ph z | < 2 3 π .