limiting form as a Bessel function
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11: 10.7 Limiting Forms
§10.7 Limiting Forms
… ►
10.7.2
…
►For and when combine (10.4.6) and (10.7.7).
For and when and combine (10.4.3), (10.7.3), and (10.7.6).
…
►For the corresponding results for and 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 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 is akin to the notation used for Bessel functions (§10.2(ii)), inasmuch as is an entire function of each of its parameters , , and : 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 functions (§14.1); (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
… ►
10.50.4
►where is given by (10.49.1).
►Results corresponding to (10.50.3) and (10.50.4) for and 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 , they are cumbersome to use when 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 and 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
is called a Hilbert space if every
Cauchy sequence
in
(i.e.,
)
converges in norm to some
, i.e.,
. For an orthonormal set
in a Hilbert space
Bessel’s inequality holds:
…
►Eigenfunctions corresponding to the continuous spectrum are non-
functions.
Let
be the self adjoint extension of a formally self-adjoint
differential operator
of the form (1.18.28) on an unbounded interval
, which we will take as
, and assume that
monotonically as
, and that the eigenfunctions are non-vanishing but bounded in this same limit.
Assume
has no point spectrum, i.e.,
has no eigenfunctions in
, then the spectrum
of
consists only of a continuous spectrum, referred to as
.
In this subsection it is assumed that
. This will be generalized, along with the choice of
, in
§1.18(vii).
…
►By Bessel’s differential equation in the form
(10.13.1) we
have the functions
(
, for
see §10.2(ii))
as eigenfunctions with eigenvalue
of the self-adjoint extension of the differential operator
…
►
A boundary value for the end point
is a linear form
on
of the form
…where
and
are given functions on
, and where the limit has
to exist for all
. Then, if the linear form
is nonzero,
the condition
is called
a boundary condition at
.
Boundary values and boundary conditions for the end point
are defined in a
similar way.
If
then there are no nonzero boundary values at
; if
then the above boundary values at
form a two-dimensional class.
Similarly at
. Any self-adjoint extension of
can be obtained by restricting
to those
for which, if
,
for a chosen
at
and, if
,
for a chosen
at
.
…
18: 10.30 Limiting Forms
§10.30 Limiting Forms
►§10.30(i)
… ►For , when is purely imaginary and , see (10.45.2) and (10.45.7). … ►
10.30.4
,
…
►For see (10.25.3).
19: 33.5 Limiting Forms for Small , Small , or Large
§33.5 Limiting Forms for Small , Small , or Large
►§33.5(i) Small
… ►For the functions , , , see §§10.47(ii), 10.2(ii). … ►§33.5(iii) Small
… ►§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 has a double zero , or more generally is a zero of order , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order .
…The order of the approximating Bessel functions, or modified Bessel functions, is , except in the case when has a double pole at .
…
►In regions in which the function
has a simple pole at and is analytic at (the case in §10.72(i)), asymptotic expansions of the solutions of (10.72.1) for large can be constructed in terms of Bessel functions and modified Bessel functions of order , where is the limiting value of as .
…
►