# inhomogeneous forms

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## 10 matching pages

##### 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 $\int_{z}^{\infty}\mathrm{Ai}\left(t\right)\mathrm{d}t=\pi\left(\mathrm{Ai}% \left(z\right)\mathrm{Gi}'\left(z\right)-\mathrm{Ai}'\left(z\right)\mathrm{Gi}% \left(z\right)\right),$
9.10.3 $\int_{-\infty}^{z}\mathrm{Bi}\left(t\right)\mathrm{d}t=\int_{0}^{z}\mathrm{Bi}% \left(t\right)\mathrm{d}t=\pi\left(\mathrm{Bi}'\left(z\right)\mathrm{Gi}\left(% z\right)-\mathrm{Bi}\left(z\right)\mathrm{Gi}'\left(z\right)\right)\\ =\pi\left(\mathrm{Bi}\left(z\right)\mathrm{Hi}'\left(z\right)-\mathrm{Bi}'% \left(z\right)\mathrm{Hi}\left(z\right)\right).$
##### 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
1.13.4 $\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=\det\begin{bmatrix}w_{1}(z)&w_{2}(% z)\\ w_{1}^{\prime}(z)&w_{2}^{\prime}(z)\end{bmatrix}=w_{1}(z)w_{2}^{\prime}(z)-w_{% 2}(z)w_{1}^{\prime}(z).$
###### §1.13(iii) Inhomogeneous Equations
The inhomogeneous (or nonhomogeneous) equation …
##### 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 $\mathbf{D}$, 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 $\xi$ 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). …