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1: 9.1 Special Notation
k

nonnegative integer, except in §9.9(iii).

The main functions treated in this chapter are the Airy functions Ai ( z ) and Bi ( z ) , and the Scorer functions Gi ( z ) and Hi ( z ) (also known as inhomogeneous Airy functions). …
2: 14.29 Generalizations
For inhomogeneous versions of the associated Legendre equation, and properties of their solutions, see Babister (1967, pp. 252–264).
3: 9.12 Scorer Functions
9.12.4 Gi ( z ) = Bi ( z ) z Ai ( t ) d t + Ai ( z ) 0 z Bi ( t ) d t ,
9.12.5 Hi ( z ) = Bi ( z ) z Ai ( t ) d t Ai ( z ) z Bi ( t ) d t .
9.12.6 Gi ( 0 ) = 1 2 Hi ( 0 ) = 1 3 Bi ( 0 ) = 1 / ( 3 7 / 6 Γ ( 2 3 ) ) = 0.20497 55424 ,
9.12.7 Gi ( 0 ) = 1 2 Hi ( 0 ) = 1 3 Bi ( 0 ) = 1 / ( 3 5 / 6 Γ ( 1 3 ) ) = 0.14942 94524 .
9.12.11 Gi ( z ) + Hi ( z ) = Bi ( z ) ,
4: 11.13 Methods of Computation
A comprehensive approach is to integrate the defining inhomogeneous differential equations (11.2.7) and (11.2.9) numerically, using methods described in §3.7. … Sequences of values of 𝐇 ν ( z ) and 𝐋 ν ( z ) , with z fixed, can be computed by application of the inhomogeneous difference equations (11.4.23) and (11.4.25). …
5: 1.13 Differential Equations
§1.13(iii) Inhomogeneous Equations
The inhomogeneous (or nonhomogeneous) equation …
Variation of Parameters
6: 3.6 Linear Difference Equations
If d n = 0 , n , then the difference equation is homogeneous; otherwise it is inhomogeneous. …
§3.6(iv) Inhomogeneous Equations
It is applicable equally to the computation of the recessive solution of the homogeneous equation (3.6.3) or the computation of any solution w n of the inhomogeneous equation (3.6.1) for which the conditions of §3.6(iv) are satisfied. … Thus in the inhomogeneous case it may sometimes be necessary to recur backwards to achieve stability. … or for systems of k first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. …
7: 11.9 Lommel Functions
The inhomogeneous Bessel differential equation … For uniform asymptotic expansions, for large ν and fixed μ = 1 , 0 , 1 , 2 , , of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). … …
8: 3.7 Ordinary Differential Equations
If h = 0 the differential equation is homogeneous, otherwise it is inhomogeneous. … … (This can happen only for inhomogeneous equations.) … The latter is especially useful if the endpoint b of 𝒫 is at , or if the differential equation is inhomogeneous. …
9: 9.10 Integrals
9.10.1 z Ai ( t ) d t = π ( Ai ( z ) Gi ( z ) Ai ( z ) Gi ( z ) ) ,
9.10.2 z Ai ( t ) d t = π ( Ai ( z ) Hi ( z ) Ai ( z ) Hi ( z ) ) ,
9.10.3 z Bi ( t ) d t = 0 z Bi ( t ) d t = π ( Bi ( z ) Gi ( z ) Bi ( z ) Gi ( z ) ) = π ( Bi ( z ) Hi ( z ) Bi ( z ) Hi ( z ) ) .
10: 11.2 Definitions
§11.2(ii) Differential Equations
Modified Struve’s Equation