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1: 1.13 Differential Equations
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Variation of Parameters
2: 10.15 Derivatives with Respect to Order
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10.15.3 J Ξ½ ⁑ ( z ) Ξ½ | Ξ½ = n = Ο€ 2 ⁒ Y n ⁑ ( z ) + n ! 2 ⁒ ( 1 2 ⁒ z ) n ⁒ k = 0 n 1 ( 1 2 ⁒ z ) k ⁒ J k ⁑ ( z ) k ! ⁒ ( n k ) .
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10.15.4 Y Ξ½ ⁑ ( z ) Ξ½ | Ξ½ = n = Ο€ 2 ⁒ J n ⁑ ( z ) + n ! 2 ⁒ ( 1 2 ⁒ z ) n ⁒ k = 0 n 1 ( 1 2 ⁒ z ) k ⁒ Y k ⁑ ( z ) k ! ⁒ ( n k ) ,
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10.15.5 J Ξ½ ⁑ ( z ) Ξ½ | Ξ½ = 0 = Ο€ 2 ⁒ Y 0 ⁑ ( z ) , Y Ξ½ ⁑ ( z ) Ξ½ | Ξ½ = 0 = Ο€ 2 ⁒ J 0 ⁑ ( z ) .
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10.15.6 J Ξ½ ⁑ ( x ) Ξ½ | Ξ½ = 1 2 = 2 Ο€ ⁒ x ⁒ ( Ci ⁑ ( 2 ⁒ x ) ⁒ sin ⁑ x Si ⁑ ( 2 ⁒ x ) ⁒ cos ⁑ x ) ,
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10.15.7 J Ξ½ ⁑ ( x ) Ξ½ | Ξ½ = 1 2 = 2 Ο€ ⁒ x ⁒ ( Ci ⁑ ( 2 ⁒ x ) ⁒ cos ⁑ x + Si ⁑ ( 2 ⁒ x ) ⁒ sin ⁑ x ) ,
3: 18.39 Applications in the Physical Sciences
β–ΊWhile s in the basis of (18.39.44) is simply a variational parameter, care must be taken, or the relationship between the results of the matrix variational approximation and the Pollaczek polynomials is lost, although this has no effect on the use of the variational approximations Reinhardt (2021a, b). …
4: 9.12 Scorer Functions
β–ΊSolutions of this equation are the Scorer functions and can be found by the method of variation of parameters1.13(iii)). …
5: 10.40 Asymptotic Expansions for Large Argument
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10.40.11 | R β„“ ⁑ ( Ξ½ , z ) | 2 ⁒ | a β„“ ⁑ ( Ξ½ ) | ⁒ 𝒱 z , ⁑ ( t β„“ ) ⁒ exp ⁑ ( | Ξ½ 2 1 4 | ⁒ 𝒱 z , ⁑ ( t 1 ) ) ,
6: 10.17 Asymptotic Expansions for Large Argument
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10.17.14 | R β„“ ± ⁑ ( Ξ½ , z ) | 2 ⁒ | a β„“ ⁑ ( Ξ½ ) | ⁒ 𝒱 z , ± i ⁒ ⁑ ( t β„“ ) ⁒ exp ⁑ ( | Ξ½ 2 1 4 | ⁒ 𝒱 z , ± i ⁒ ⁑ ( t 1 ) ) ,
7: 2.3 Integrals of a Real Variable
β–ΊThen … β–ΊIn both cases the n th error term is bounded in absolute value by x n ⁒ 𝒱 a , b ⁑ ( q ( n 1 ) ⁑ ( t ) ) , where the variational operator 𝒱 a , b is defined by … β–ΊThen the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … β–Ί(In other words, differentiation of (2.3.8) with respect to the parameter Ξ» (or ΞΌ ) is legitimate.) … β–ΊWhen the parameter x is large the contributions from the real and imaginary parts of the integrand in …
8: Bibliography M
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  • E. L. Mansfield and H. N. Webster (1998) On one-parameter families of Painlevé III. Stud. Appl. Math. 101 (3), pp. 321–341.
  • β–Ί
  • R. J. Muirhead (1978) Latent roots and matrix variates: A review of some asymptotic results. Ann. Statist. 6 (1), pp. 5–33.
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  • M. E. Muldoon (1981) The variation with respect to order of zeros of Bessel functions. Rend. Sem. Mat. Univ. Politec. Torino 39 (2), pp. 15–25.
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  • H. P. Mulholland and S. Goldstein (1929) The characteristic numbers of the Mathieu equation with purely imaginary parameter. Phil. Mag. Series 7 8 (53), pp. 834–840.
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  • J. Murzewski and A. Sowa (1972) Tables of the functions of the parabolic cylinder for negative integer parameters. Zastos. Mat. 13, pp. 261–273.
  • 9: 10.23 Sums
    β–Ί
    10.23.1 π’ž Ξ½ ⁑ ( Ξ» ⁒ z ) = Ξ» ± Ξ½ ⁒ k = 0 ( βˆ“ 1 ) k ⁒ ( Ξ» 2 1 ) k ⁒ ( 1 2 ⁒ z ) k k ! ⁒ π’ž Ξ½ ± k ⁑ ( z ) , | Ξ» 2 1 | < 1 .
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    10.23.2 π’ž Ξ½ ⁑ ( u ± v ) = k = π’ž Ξ½ βˆ“ k ⁑ ( u ) ⁒ J k ⁑ ( v ) , | v | < | u | .
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    10.23.14 z ν ⁒ f ⁑ ( z ) = a 0 ⁒ J ν ⁑ ( z ) + 2 ⁒ k = 1 a k ⁒ J ν + k ⁑ ( z )
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    10.23.20 1 2 ⁒ f ⁑ ( x ) + 1 2 ⁒ f ⁑ ( x + ) = m = 1 a m ⁒ J ν ⁑ ( j ν , m ⁒ x ) ,
    β–Ίprovided that f ⁑ ( t ) is of bounded variation1.4(v)) on an interval [ a , b ] with 0 < a < x < b < 1 . …
    10: 1.14 Integral Transforms
    β–ΊSuppose that f ⁑ ( t ) is absolutely integrable on ( , ) and of bounded variation in a neighborhood of t = u 1.4(v)). … β–ΊIf f ⁑ ( t ) is absolutely integrable on [ 0 , ) and of bounded variation1.4(v)) in a neighborhood of t = u , then … β–ΊSuppose f ⁑ ( t ) is a real- or complex-valued function and s is a real or complex parameter. … β–ΊIf x Οƒ 1 ⁒ f ⁑ ( x ) is integrable on ( 0 , ) for all Οƒ in a < Οƒ < b , then the integral (1.14.32) converges and β„³ ⁑ f ⁑ ( s ) is an analytic function of s in the vertical strip a < ⁑ s < b . … β–ΊSuppose the integral (1.14.32) is absolutely convergent on the line ⁑ s = Οƒ and f ⁑ ( x ) is of bounded variation in a neighborhood of x = u . …