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11: 10.47 Definitions and Basic Properties
§10.47(i) Differential Equations
§10.47(ii) Standard Solutions
Equation (10.47.1)
Equation (10.47.2)
12: 11.10 Anger–Weber Functions
The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation
13: 18.34 Bessel Polynomials
§18.34(iii) Other Properties
14: 10.15 Derivatives with Respect to Order
10.15.1 J ± ν ( z ) ν = ± J ± ν ( z ) ln ( 1 2 z ) ( 1 2 z ) ± ν k = 0 ( - 1 ) k ψ ( k + 1 ± ν ) Γ ( k + 1 ± ν ) ( 1 4 z 2 ) k k ! ,
15: 10.45 Functions of Imaginary Order
With z = x , and ν replaced by i ν , the modified Bessel’s equation (10.25.1) becomes …
16: 10.24 Functions of Imaginary Order
With z = x and ν replaced by i ν , Bessel’s equation (10.2.1) becomes …
17: 12.18 Methods of Computation
These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …
18: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2004b) Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments. ACM Trans. Math. Software 30 (2), pp. 145–158.
  • 19: 29.18 Mathematical Applications
    (29.18.5) is the differential equation of spherical Bessel functions (§10.47(i)), and (29.18.6), (29.18.7) agree with the Lamé equation (29.2.1). …
    20: 13.6 Relations to Other Functions
    13.6.11_1 M ( ν + 1 2 , 2 ν + 1 + n , 2 z ) = Γ ( ν ) e z ( z / 2 ) - ν k = 0 n ( - n ) k ( 2 ν ) k ( ν + k ) ( 2 ν + 1 + n ) k k ! I ν + k ( z ) ,
    13.6.11_2 M ( ν + 1 2 , 2 ν + 1 - n , 2 z ) = Γ ( ν - n ) e z ( z / 2 ) n - ν k = 0 n ( - 1 ) k ( - n ) k ( 2 ν - 2 n ) k ( ν - n + k ) ( 2 ν + 1 - n ) k k ! I ν + k - n ( z ) .