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inhomogeneous forms

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1: 11.9 Lommel Functions
The inhomogeneous Bessel differential equation …
2: 11.2 Definitions
§11.2(ii) Differential Equations
Modified Struve’s Equation
3: 11.10 Anger–Weber Functions
The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation …
4: 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 ) ) .
5: 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. … 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. … 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). …
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: 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 remaining two equations are supplied by boundary conditions of the formThe latter is especially useful if the endpoint b of 𝒫 is at , or if the differential equation is inhomogeneous. …
8: Bibliography M
  • H. Maass (1971) Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.
  • A. J. MacLeod (1994) Computation of inhomogeneous Airy functions. J. Comput. Appl. Math. 53 (1), pp. 109–116.
  • P. Maroni (1995) An integral representation for the Bessel form. J. Comput. Appl. Math. 57 (1-2), pp. 251–260.
  • 9: 1.13 Differential Equations
    1.13.4 𝒲 { w 1 ( z ) , w 2 ( z ) } = det [ w 1 ( z ) w 2 ( z ) w 1 ( z ) w 2 ( z ) ] = w 1 ( z ) w 2 ( z ) - w 2 ( z ) w 1 ( z ) .
    §1.13(iii) Inhomogeneous Equations
    The inhomogeneous (or nonhomogeneous) equation …
    Variation of Parameters
    §1.13(vii) Closed-Form Solutions
    10: 2.8 Differential Equations with a Parameter
    Many special functions satisfy an equation of the form …The form of the asymptotic expansion depends on the nature of the transition points in D , that is, points at which f ( z ) has a zero or singularity. … The transformed equation has the formFor error bounds, extensions to pure imaginary or complex u , an extension to inhomogeneous differential equations, and examples, see Olver (1997b, Chapter 10). … For error bounds, more delicate error estimates, extensions to complex ξ and u , zeros, connection formulas, extensions to inhomogeneous equations, and examples, see Olver (1997b, Chapters 11, 13), Olver (1964b), Reid (1974a, b), Boyd (1987), and Baldwin (1991). …