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1: 1.13 Differential Equations
Variation of Parameters
2: 10.15 Derivatives with Respect to Order
10.15.2 Y ν ( z ) ν = cot ( ν π ) ( J ν ( z ) ν π Y ν ( z ) ) csc ( ν π ) J ν ( z ) ν π J ν ( z ) .
10.15.3 J ν ( z ) ν | ν = n = π 2 Y n ( z ) + n ! 2 ( 1 2 z ) n k = 0 n 1 ( 1 2 z ) k J k ( z ) k ! ( n k ) .
10.15.4 Y ν ( z ) ν | ν = n = π 2 J n ( z ) + n ! 2 ( 1 2 z ) n k = 0 n 1 ( 1 2 z ) k Y k ( z ) k ! ( n k ) ,
10.15.5 J ν ( z ) ν | ν = 0 = π 2 Y 0 ( z ) , Y ν ( z ) ν | ν = 0 = π 2 J 0 ( z ) .
10.15.6 J ν ( x ) ν | ν = 1 2 = 2 π x ( Ci ( 2 x ) sin x Si ( 2 x ) cos x ) ,
3: 9.12 Scorer Functions
Solutions of this equation are the Scorer functions and can be found by the method of variation of parameters1.13(iii)). …
4: 10.40 Asymptotic Expansions for Large Argument
10.40.11 | R ( ν , z ) | 2 | a ( ν ) | 𝒱 z , ( t ) exp ( | ν 2 1 4 | 𝒱 z , ( t 1 ) ) ,
5: 10.17 Asymptotic Expansions for Large Argument
10.17.14 | R ± ( ν , z ) | 2 | a ( ν ) | 𝒱 z , ± i ( t ) exp ( | ν 2 1 4 | 𝒱 z , ± i ( t 1 ) ) ,
6: 2.3 Integrals of a Real Variable
Then … In 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 … Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … (In other words, differentiation of (2.3.8) with respect to the parameter λ (or μ ) is legitimate.) … When the parameter x is large the contributions from the real and imaginary parts of the integrand in …
7: Bibliography M
  • E. L. Mansfield and H. N. Webster (1998) On one-parameter families of Painlevé III. Stud. Appl. Math. 101 (3), pp. 321–341.
  • R. J. Muirhead (1978) Latent roots and matrix variates: A review of some asymptotic results. Ann. Statist. 6 (1), pp. 5–33.
  • 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.
  • 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.
  • J. Murzewski and A. Sowa (1972) Tables of the functions of the parabolic cylinder for negative integer parameters. Zastos. Mat. 13, pp. 261–273.
  • 8: 10.23 Sums
    10.23.1 𝒞 ν ( λ z ) = λ ± ν k = 0 ( 1 ) k ( λ 2 1 ) k ( 1 2 z ) k k ! 𝒞 ν ± k ( z ) , | λ 2 1 | < 1 .
    10.23.2 𝒞 ν ( u ± v ) = k = 𝒞 ν k ( u ) J k ( v ) , | v | < | u | .
    10.23.14 z ν f ( z ) = a 0 J ν ( z ) + 2 k = 1 a k J ν + k ( z )
    10.23.20 1 2 f ( x ) + 1 2 f ( x + ) = m = 1 a m J ν ( j ν , m x ) ,
    provided that f ( t ) is of bounded variation1.4(v)) on an interval [ a , b ] with 0 < a < x < b < 1 . …
    9: 1.14 Integral Transforms
    Suppose that f ( t ) is absolutely integrable on ( , ) and of bounded variation in a neighborhood of t = u 1.4(v)). … If f ( t ) is absolutely integrable on [ 0 , ) and of bounded variation1.4(v)) in a neighborhood of t = u , then … If 0 | f ( t ) | d t < , g ( t ) is of bounded variation on ( 0 , ) and g ( t ) 0 as t , then … Suppose f ( t ) is a real- or complex-valued function and s is a real or complex parameter. … Suppose the integral (1.14.32) is absolutely convergent on the line s = σ and f ( x ) is of bounded variation in a neighborhood of x = u . …
    10: 2.8 Differential Equations with a Parameter
    §2.8 Differential Equations with a Parameter
    In addition, 𝒱 𝒬 j ( A 1 ) and 𝒱 𝒬 j ( A n ) must be bounded on 𝚫 j ( α j ) . … These results are valid when 𝒱 α 1 , α 2 ( | ξ | 1 / 2 B 0 ) and 𝒱 α 1 , α 2 ( | ξ | 1 / 2 B n 1 ) are finite. … These results are valid when 𝒱 0 , α 2 ( ξ 1 / 2 B 0 ) and 𝒱 0 , α 2 ( ξ 1 / 2 B n 1 ) are finite. … These results are valid when 𝒱 α 1 , 0 ( | ξ | 1 / 2 B 0 ) and 𝒱 α 1 , 0 ( | ξ | 1 / 2 B n 1 ) are finite. …