About the Project
NIST

kernel functions

AdvancedHelp

(0.001 seconds)

1—10 of 29 matching pages

1: 10.62 Graphs
§10.62 Graphs
For the modulus functions M ( x ) and N ( x ) see §10.68(i) with ν = 0 . …
See accompanying text
Figure 10.62.2: ker x , kei x , ker x , kei x , 0 x 8 . Magnify
See accompanying text
Figure 10.62.4: e x / 2 ker x , e x / 2 kei x , e x / 2 N ( x ) , 0 x 8 . Magnify
2: 10.70 Zeros
§10.70 Zeros
10.70.1 μ - 1 16 t + μ - 1 32 t 2 + ( μ - 1 ) ( 5 μ + 19 ) 1536 t 3 + 3 ( μ - 1 ) 2 512 t 4 + .
zeros of  ker ν x 2 ( t + f ( - t ) ) , t = ( m - 1 2 ν - 5 8 ) π ,
In the case ν = 0 , numerical tabulations (Abramowitz and Stegun (1964, Table 9.12)) indicate that each of (10.70.2) corresponds to the m th zero of the function on the left-hand side. …
3: 31.10 Integral Equations and Representations
Kernel Functions
Fuchs–Frobenius solutions W m ( z ) = κ ~ m z - α H ( 1 / a , q m ; α , α - γ + 1 , α - β + 1 , δ ; 1 / z ) are represented in terms of Heun functions w m ( z ) = ( 0 , 1 ) Hf m ( a , q m ; α , β , γ , δ ; z ) by (31.10.1) with W ( z ) = W m ( z ) , w ( z ) = w m ( z ) , and with kernel chosen from …
Kernel Functions
31.10.19 𝒦 ( u , v , w ) = u 1 - γ v 1 - δ w 1 - ϵ 𝒞 1 - γ ( u σ 1 ) 𝒞 1 - δ ( v σ 2 ) 𝒞 1 - ϵ ( i w σ 1 + σ 2 ) ,
4: 10.67 Asymptotic Expansions for Large Argument
§10.67(i) ber ν x , bei ν x , ker ν x , kei ν x , and Derivatives
10.67.13 ker 2 x + kei 2 x π 2 x e - x 2 ( 1 - 1 4 2 1 x + 1 64 1 x 2 + 33 256 2 1 x 3 - 1797 8192 1 x 4 + ) ,
10.67.14 ker x kei x - ker x kei x - π 2 x e - x 2 ( 1 2 - 1 8 1 x + 9 64 2 1 x 2 - 39 512 1 x 3 + 75 8192 2 1 x 4 + ) ,
10.67.15 ker x ker x + kei x kei x - π 2 x e - x 2 ( 1 2 + 3 8 1 x - 15 64 2 1 x 2 + 45 512 1 x 3 + 315 8192 2 1 x 4 + ) ,
10.67.16 ( ker x ) 2 + ( kei x ) 2 π 2 x e - x 2 ( 1 + 3 4 2 1 x + 9 64 1 x 2 - 75 256 2 1 x 3 + 2475 8192 1 x 4 + ) .
5: 10.61 Definitions and Basic Properties
10.61.2 ker ν x + i kei ν x = e - ν π i / 2 K ν ( x e π i / 4 ) = 1 2 π i H ν ( 1 ) ( x e 3 π i / 4 ) = - 1 2 π i e - ν π i H ν ( 2 ) ( x e - π i / 4 ) .
10.61.3 x 2 d 2 w d x 2 + x d w d x - ( i x 2 + ν 2 ) w = 0 , w = ber ν x + i bei ν x , ber - ν x + i bei - ν x ker ν x + i kei ν x , ker - ν x + i kei - ν x .
10.61.4 x 4 d 4 w d x 4 + 2 x 3 d 3 w d x 3 - ( 1 + 2 ν 2 ) ( x 2 d 2 w d x 2 - x d w d x ) + ( ν 4 - 4 ν 2 + x 4 ) w = 0 , w = ber ± ν x , bei ± ν x , ker ± ν x , kei ± ν x .
10.61.11 ker 1 2 ( x 2 ) = kei - 1 2 ( x 2 ) = - 2 - 3 4 π x e - x sin ( x - π 8 ) ,
10.61.12 kei 1 2 ( x 2 ) = - ker - 1 2 ( x 2 ) = - 2 - 3 4 π x e - x cos ( x - π 8 ) .
6: 10.63 Recurrence Relations and Derivatives
§10.63(i) ber ν x , bei ν x , ker ν x , kei ν x
2 kei x = - ker 1 x + kei 1 x .
7: 10.69 Uniform Asymptotic Expansions for Large Order
10.69.3 ker ν ( ν x ) + i kei ν ( ν x ) e - ν ξ ( π 2 ν ξ ) 1 / 2 ( x e 3 π i / 4 1 + ξ ) - ν k = 0 ( - 1 ) k U k ( ξ - 1 ) ν k ,
10.69.5 ker ν ( ν x ) + i kei ν ( ν x ) - e - ν ξ x ( π ξ 2 ν ) 1 / 2 ( x e 3 π i / 4 1 + ξ ) - ν k = 0 ( - 1 ) k V k ( ξ - 1 ) ν k ,
8: 28.10 Integral Equations
§28.10(i) Equations with Elementary Kernels
§28.10(ii) Equations with Bessel-Function Kernels
9: 10.68 Modulus and Phase Functions
10: 10.65 Power Series
10.65.3 ker n x = 1 2 ( 1 2 x ) - n k = 0 n - 1 ( n - k - 1 ) ! k ! cos ( 3 4 n π + 1 2 k π ) ( 1 4 x 2 ) k - ln ( 1 2 x ) ber n x + 1 4 π bei n x + 1 2 ( 1 2 x ) n k = 0 ψ ( k + 1 ) + ψ ( n + k + 1 ) k ! ( n + k ) ! cos ( 3 4 n π + 1 2 k π ) ( 1 4 x 2 ) k ,