inhomogeneous forms

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1: 11.9 Lommel Functions

… ►The inhomogeneous Bessel differential equation … ► …

2: 11.2 Definitions

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§11.2(ii) Differential Equations

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Modified Struve’s Equation

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3: 11.10 Anger–Weber Functions

… ►The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation …

4: 9.10 Integrals

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9.10.1 $${\int }_z^{\mathrm{\infty }}\mathrm{Ai}\left(t\right)dt=\pi \left(\mathrm{Ai}\left(z\right){\mathrm{Gi}}^{\prime }\left(z\right)-{\mathrm{Ai}}^{\prime }\left(z\right)\mathrm{Gi}\left(z\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.12.4) and its differentiated form with the first of (9.10.11)

Referenced by:

(9.10.2)

Permalink:

http://dlmf.nist.gov/9.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(i), §9.10 and Ch.9

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9.10.2 $${\int }_{-\mathrm{\infty }}^z\mathrm{Ai}\left(t\right)dt=\pi \left(\mathrm{Ai}\left(z\right){\mathrm{Hi}}^{\prime }\left(z\right)-{\mathrm{Ai}}^{\prime }\left(z\right)\mathrm{Hi}\left(z\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.12.11) and its differentiated form with (9.10.1), then apply (9.2.7) and (9.10.11)

Permalink:

http://dlmf.nist.gov/9.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(i), §9.10 and Ch.9

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9.10.3 $${\int }_{-\mathrm{\infty }}^z\mathrm{Bi}\left(t\right)dt={\int }_0^z\mathrm{Bi}\left(t\right)dt=\pi \left({\mathrm{Bi}}^{\prime }\left(z\right)\mathrm{Gi}\left(z\right)-\mathrm{Bi}\left(z\right){\mathrm{Gi}}^{\prime }\left(z\right)\right)=\pi \left(\mathrm{Bi}\left(z\right){\mathrm{Hi}}^{\prime }\left(z\right)-{\mathrm{Bi}}^{\prime }\left(z\right)\mathrm{Hi}\left(z\right)\right).$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.12.5) and its differentiated form with (9.2.7).

Permalink:

http://dlmf.nist.gov/9.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(i), §9.10 and Ch.9

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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 ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. … ►Sequences of values of ${𝐇}_{\nu }\left(z\right)$ and ${𝐋}_{\nu }\left(z\right)$, 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 form … ►The latter is especially useful if the endpoint $b$ of $𝒫$ is at $\mathrm{\infty }$, or if the differential equation is inhomogeneous. …

8: Bibliography M

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Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.

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External Links:

Document, MathReview (K.-B. Gundlach), ZentralBlatt

Referenced by:

§35.4(ii).

Encodings:

BibTeX

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Computation of inhomogeneous Airy functions. J. Comput. Appl. Math. 53 (1), pp. 109–116.

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External Links:

ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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An integral representation for the Bessel form. J. Comput. Appl. Math. 57 (1-2), pp. 251–260.

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External Links:

ISSN 0377-0427, Document, MathReview (Hans-Jürgen Glaeske), ZentralBlatt

Referenced by:

§18.34(ii).

Encodings:

BibTeX

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9: 1.13 Differential Equations

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§1.13(iii) Inhomogeneous Equations

►The inhomogeneous (or nonhomogeneous) equation … ►

Variation of Parameters

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§1.13(vii) Closed-Form Solutions

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§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

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