inhomogeneous forms

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1: 11.9 Lommel Functions
The inhomogeneous Bessel differential 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 $\int_{z}^{\infty}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=\pi\left(% \operatorname{Ai}\left(z\right)\operatorname{Gi}'\left(z\right)-\operatorname{% Ai}'\left(z\right)\operatorname{Gi}\left(z\right)\right),$
9.10.3 $\int_{-\infty}^{z}\operatorname{Bi}\left(t\right)\,\mathrm{d}t=\int_{0}^{z}% \operatorname{Bi}\left(t\right)\,\mathrm{d}t=\pi\left(\operatorname{Bi}'\left(% z\right)\operatorname{Gi}\left(z\right)-\operatorname{Bi}\left(z\right)% \operatorname{Gi}'\left(z\right)\right)\\ =\pi\left(\operatorname{Bi}\left(z\right)\operatorname{Hi}'\left(z\right)-% \operatorname{Bi}'\left(z\right)\operatorname{Hi}\left(z\right)\right).$
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 $\mathbf{H}_{\nu}\left(x\right)$ and $\mathbf{L}_{\nu}\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. … Sequences of values of $\mathbf{H}_{\nu}\left(z\right)$ and $\mathbf{L}_{\nu}\left(z\right)$, 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 $d_{n}=0$, $\forall n$, 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 $\mathscr{P}$ is at $\infty$, 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(iii) Inhomogeneous Equations
The inhomogeneous (or nonhomogeneous) equation …