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Kelvin functions

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1: 11.8 Analogs to Kelvin Functions
§11.8 Analogs to Kelvin Functions
2: 10.64 Integral Representations
§10.64 Integral Representations
Schläfli-Type Integrals
10.64.1 ber n ( x 2 ) = ( - 1 ) n π 0 π cos ( x sin t - n t ) cosh ( x sin t ) d t ,
10.64.2 bei n ( x 2 ) = ( - 1 ) n π 0 π sin ( x sin t - n t ) sinh ( x sin t ) d t .
3: 10.71 Integrals
§10.71(i) Indefinite Integrals
§10.71(ii) Definite Integrals
§10.71(iii) Compendia
For infinite double integrals involving Kelvin functions see Prudnikov et al. (1986b, pp. 630–631). …
4: 10.70 Zeros
§10.70 Zeros
5: 10.62 Graphs
§10.62 Graphs
6: 10.61 Definitions and Basic Properties
§10.61(i) Definitions
§10.61(ii) Differential Equations
§10.61(iii) Reflection Formulas for Arguments
§10.61(iv) Reflection Formulas for Orders
§10.61(v) Orders ± 1 2
7: 10.69 Uniform Asymptotic Expansions for Large Order
§10.69 Uniform Asymptotic Expansions for Large Order
10.69.2 ber ν ( ν x ) + i bei ν ( ν x ) e ν ξ ( 2 π ν ξ ) 1 / 2 ( x e 3 π i / 4 1 + ξ ) ν k = 0 U k ( ξ - 1 ) ν k ,
All fractional powers take their principal values. …
8: 10.65 Power Series
§10.65 Power Series
§10.65(iii) Cross-Products and Sums of Squares
10.65.6 ber ν 2 x + bei ν 2 x = ( 1 2 x ) 2 ν k = 0 1 Γ ( ν + k + 1 ) Γ ( ν + 2 k + 1 ) ( 1 4 x 2 ) 2 k k ! ,
§10.65(iv) Compendia
For further power series summable in terms of Kelvin functions and their derivatives see Hansen (1975).
9: 10.63 Recurrence Relations and Derivatives
§10.63(i) ber ν x , bei ν x , ker ν x , kei ν x
Let f ν ( x ) , g ν ( x ) denote any one of the ordered pairs: …
2 kei x = - ker 1 x + kei 1 x .
§10.63(ii) Cross-Products
10: 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(ii) Cross-Products and Sums of Squares in the Case ν = 0
10.67.9 ber 2 x + bei 2 x e x 2 2 π x ( 1 + 1 4 2 1 x + 1 64 1 x 2 - 33 256 2 1 x 3 - 1797 8192 1 x 4 + ) ,
10.67.10 ber x bei x - ber x bei x e x 2 2 π x ( 1 2 + 1 8 1 x + 9 64 2 1 x 2 + 39 512 1 x 3 + 75 8192 2 1 x 4 + ) ,