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limiting form as a Bessel function

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11: 10.7 Limiting Forms
§10.7 Limiting Forms
For H ν ( 1 ) ( z ) and H ν ( 2 ) ( z ) when ν > 0 combine (10.4.6) and (10.7.7). For H i ν ( 1 ) ( z ) and H i ν ( 2 ) ( z ) when ν and ν 0 combine (10.4.3), (10.7.3), and (10.7.6). … For the corresponding results for H ν ( 1 ) ( z ) and H ν ( 2 ) ( z ) see (10.2.5) and (10.2.6).
12: 10.25 Definitions
§10.25(i) Modified Bessel’s Equation
Its solutions are called modified Bessel functions or Bessel functions of imaginary argument.
§10.25(ii) Standard Solutions
Branch Conventions
13: 11.9 Lommel Functions
§11.9 Lommel Functions
The inhomogeneous Bessel differential equation … the right-hand side being replaced by its limiting form when μ ± ν is an odd negative integer. …
§11.9(ii) Expansions in Series of Bessel Functions
14: Mathematical Introduction
The mathematical content of the NIST Handbook of Mathematical Functions has been produced over a ten-year period. … This is because 𝐅 is akin to the notation used for Bessel functions10.2(ii)), inasmuch as 𝐅 is an entire function of each of its parameters a , b , and c :​ this results in fewer restrictions and simpler equations. … Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). … Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. … For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed. …
15: 10.50 Wronskians and Cross-Products
§10.50 Wronskians and Cross-Products
𝒲 { 𝗁 n ( 1 ) ( z ) , 𝗁 n ( 2 ) ( z ) } = 2 i z 2 .
10.50.4 𝗃 0 ( z ) 𝗃 n ( z ) + 𝗒 0 ( z ) 𝗒 n ( z ) = cos ( 1 2 n π ) k = 0 n / 2 ( 1 ) k a 2 k ( n + 1 2 ) z 2 k + 2 + sin ( 1 2 n π ) k = 0 ( n 1 ) / 2 ( 1 ) k a 2 k + 1 ( n + 1 2 ) z 2 k + 3 ,
where a k ( n + 1 2 ) is given by (10.49.1). Results corresponding to (10.50.3) and (10.50.4) for 𝗂 n ( 1 ) ( z ) and 𝗂 n ( 2 ) ( z ) are obtainable via (10.47.12).
16: 11.13 Methods of Computation
§11.13(i) Introduction
Subsequent subsections treat the computation of Struve functions. The treatment of Lommel and Anger–Weber functions is similar. … Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of z , they are cumbersome to use when | z | is large owing to slowness of convergence and cancellation. … Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that 𝐇 ν ( x ) and 𝐋 ν ( x ) can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. …
17: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
An inner product space V is called a Hilbert space if every Cauchy sequence { v n } in V (i.e., lim m , n v m v n = 0 ) converges in norm to some v V , i.e., lim n v v n = 0 . For an orthonormal set { v n } in a Hilbert space V Bessel’s inequality holds:Eigenfunctions corresponding to the continuous spectrum are non- L 2 functions. Let T be the self adjoint extension of a formally self-adjoint differential operator of the form (1.18.28) on an unbounded interval X , which we will take as X = [ 0 , + ) , and assume that q ( x ) 0 monotonically as x , and that the eigenfunctions are non-vanishing but bounded in this same limit. Assume T has no point spectrum, i.e., T has no eigenfunctions in L 2 ( X ) , then the spectrum 𝝈 of T consists only of a continuous spectrum, referred to as 𝝈 c . In this subsection it is assumed that 𝝈 c = [ 0 , ) . This will be generalized, along with the choice of X , in §1.18(vii).By Bessel’s differential equation in the form (10.13.1) we have the functions x J ν ( x λ ) ( λ 0 , for J ν see §10.2(ii)) as eigenfunctions with eigenvalue λ of the self-adjoint extension of the differential operator A boundary value for the end point a is a linear form on 𝒟 ( * ) of the formwhere α and β are given functions on X , and where the limit has to exist for all f . Then, if the linear form is nonzero, the condition ( f ) = 0 is called a boundary condition at a . Boundary values and boundary conditions for the end point b are defined in a similar way. If n 1 = 1 then there are no nonzero boundary values at a ; if n 1 = 2 then the above boundary values at a form a two-dimensional class. Similarly at b . Any self-adjoint extension of can be obtained by restricting * to those f 𝒟 ( * ) for which, if n 1 = 2 , 1 ( f ) = 0 for a chosen 1 at a and, if n 2 = 2 , 2 ( f ) = 0 for a chosen 2 at b .
18: 10.30 Limiting Forms
§10.30 Limiting Forms
§10.30(i) z 0
For K ν ( x ) , when ν is purely imaginary and x 0 + , see (10.45.2) and (10.45.7). … For K ν ( z ) see (10.25.3).
19: 33.5 Limiting Forms for Small ρ , Small | η | , or Large
§33.5 Limiting Forms for Small ρ , Small | η | , or Large
§33.5(i) Small ρ
For the functions 𝗃 , 𝗒 , J , Y see §§10.47(ii), 10.2(ii). …
§33.5(iii) Small | η |
§33.5(iv) Large
20: 10.72 Mathematical Applications
Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … If f ( z ) has a double zero z 0 , or more generally z 0 is a zero of order m , m = 2 , 3 , 4 , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order 1 / ( m + 2 ) . …The order of the approximating Bessel functions, or modified Bessel functions, is 1 / ( λ + 2 ) , except in the case when g ( z ) has a double pole at z 0 . … In regions in which the function f ( z ) has a simple pole at z = z 0 and ( z z 0 ) 2 g ( z ) is analytic at z = z 0 (the case λ = 1 in §10.72(i)), asymptotic expansions of the solutions w of (10.72.1) for large u can be constructed in terms of Bessel functions and modified Bessel functions of order ± 1 + 4 ρ , where ρ is the limiting value of ( z z 0 ) 2 g ( z ) as z z 0 . …