# inhomogeneous

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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 $\operatorname{Ai}\left(z\right)$ and $\operatorname{Bi}\left(z\right)$, and the Scorer functions $\operatorname{Gi}(z)$ and $\operatorname{Hi}(z)$ (also known as inhomogeneous Airy functions). …
##### 2: 9.12 Scorer Functions
9.12.4 $\operatorname{Gi}\left(z\right)=\operatorname{Bi}\left(z\right)\int_{z}^{% \infty}\operatorname{Ai}\left(t\right)\,\mathrm{d}t+\operatorname{Ai}\left(z% \right)\int_{0}^{z}\operatorname{Bi}\left(t\right)\,\mathrm{d}t,$
9.12.5 $\operatorname{Hi}\left(z\right)=\operatorname{Bi}\left(z\right)\int_{-\infty}^% {z}\operatorname{Ai}\left(t\right)\,\mathrm{d}t-\operatorname{Ai}\left(z\right% )\int_{-\infty}^{z}\operatorname{Bi}\left(t\right)\,\mathrm{d}t.$
9.12.6 $\operatorname{Gi}\left(0\right)=\tfrac{1}{2}\operatorname{Hi}\left(0\right)=% \tfrac{1}{3}\operatorname{Bi}\left(0\right)={1\Big{/}\!\left(3^{7/6}\Gamma% \left(\tfrac{2}{3}\right)\right)=0.20497\;55424\ldots,}$
9.12.7 $\operatorname{Gi}'\left(0\right)=\tfrac{1}{2}\operatorname{Hi}'\left(0\right)=% \tfrac{1}{3}\operatorname{Bi}'\left(0\right)=1\Big{/}\left(3^{5/6}\Gamma\left(% \tfrac{1}{3}\right)\right)=0.14942\;94524\ldots.$
##### 3: 14.29 Generalizations
For inhomogeneous versions of the associated Legendre equation, and properties of their solutions, see Babister (1967, pp. 252–264).
##### 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 $\mathbf{H}_{\nu}\left(z\right)$ 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). …
##### 5: 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. …
##### 6: 11.9 Lommel Functions
The inhomogeneous Bessel differential equation … For uniform asymptotic expansions, for large $\nu$ and fixed $\mu=-1,0,1,2,\dots$, of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). … …
##### 7: 1.13 Differential Equations
###### §1.13(iii) Inhomogeneous Equations
The inhomogeneous (or nonhomogeneous) equation …
##### 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 $\mathscr{P}$ is at $\infty$, or if the differential equation is inhomogeneous. …
##### 10: 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 $\int_{-\infty}^{z}\operatorname{Bi}\left(t\right)\,\mathrm{d}t=\int_{0}^{z}% \operatorname{Bi}\left(t\right)\,\mathrm{d}t=\pi\left(\operatorname{Bi}'\left(% z\right)\operatorname{Gi}\left(z\right)-\operatorname{Bi}\left(z\right)% \operatorname{Gi}'\left(z\right)\right)\\ =\pi\left(\operatorname{Bi}\left(z\right)\operatorname{Hi}'\left(z\right)-% \operatorname{Bi}'\left(z\right)\operatorname{Hi}\left(z\right)\right).$