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1: 16.8 Differential Equations

§16.8 Differential Equations

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§16.8(i) Classification of Singularities

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2: 11.2 Definitions

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11.2.11 $$w={𝐇}_{\nu }\left(x\right)+A{J}_{\nu }\left(x\right)+B{Y}_{\nu }\left(x\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, ${𝐇}_{\nu }\left(z\right)$: Struve function, $x$: real variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Referenced by:

§11.2(iii)

Permalink:

http://dlmf.nist.gov/11.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(iii), §11.2 and Ch.11

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11.2.12 $$w={𝐊}_{\nu }\left(x\right)+A{J}_{\nu }\left(x\right)+B{Y}_{\nu }\left(x\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, ${𝐊}_{\nu }\left(z\right)$: Struve function, $x$: real variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Referenced by:

§11.2(iii)

Permalink:

http://dlmf.nist.gov/11.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(iii), §11.2 and Ch.11

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11.2.13 $$w={𝐇}_{\nu }\left(z\right)+A{J}_{\nu }\left(z\right)+B{H}_{\nu }^{(1)}\left(z\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), ${𝐇}_{\nu }\left(z\right)$: Struve function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Referenced by:

§11.2(iii)

Permalink:

http://dlmf.nist.gov/11.2.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(iii), §11.2 and Ch.11

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11.2.16 $$w={𝐋}_{\nu }\left(z\right)+A{K}_{\nu }\left(z\right)+B{I}_{\nu }\left(z\right),$$

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Referenced by:

§11.2(iii)

Permalink:

http://dlmf.nist.gov/11.2.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(iii), §11.2 and Ch.11

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11.2.17 $$w={𝐌}_{\nu }\left(z\right)+A{K}_{\nu }\left(z\right)+B{I}_{\nu }\left(z\right).$$

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Symbols:

${𝐌}_{\nu }\left(z\right)$: modified Struve function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Referenced by:

§11.2(iii)

Permalink:

http://dlmf.nist.gov/11.2.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(iii), §11.2 and Ch.11

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3: Bibliography L

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Evaluating infinite integrals involving Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 217–231.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 269–282.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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4: 3.2 Linear Algebra

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§3.2(iii) Condition of Linear Systems

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5: 2.9 Difference Equations

… ►For an introduction to, and references for, the general asymptotic theory of linear difference equations of arbitrary order, see Wimp (1984, Appendix B). …

6: 1.13 Differential Equations

… ►For extensions of these results to linear homogeneous differential equations of arbitrary order see Spigler (1984). …

7: 11.11 Asymptotic Expansions of Anger–Weber Functions

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11.11.8 $${𝐀}_{\nu }\left(\lambda \nu \right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(\lambda )}{{\nu }^{2k+1}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$,

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Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_k(\lambda )$: expansion function

Sources:

Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)

Referenced by:

(11.11.18), (11.11.19), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E8

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which originally was $\nu \to \mathrm{\infty }$, , has been replaced to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.14 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim \frac{1}{\pi \nu (\lambda -1)},$$ $\lambda >1$, $|\mathrm{ph}\nu |\le \pi -\delta$,

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Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter

Source:

Olver (1997b, pp. 103 and 352)

Referenced by:

§11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E14

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint has been extended to include $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-06 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.15 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim {\left(\frac{2}{\pi \nu }\right)}^{1/2}{\left(\frac{1+\sqrt{1-{\lambda }^2}}{\lambda }\right)}^{\nu }\frac{{\mathrm{e}}^{-\nu \sqrt{1-{\lambda }^2}}}{{(1-{\lambda }^2)}^{1/4}},$$ , $|\mathrm{ph}\nu |\le \frac{\pi }{2}-\delta$.

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Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter

Source:

Olver (1997b, pp. 103 and 352)

Permalink:

http://dlmf.nist.gov/11.11.E15

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint has been extended to include $|\mathrm{ph}\nu |\le \frac{\pi }{2}-\delta$.

Suggested 2021-04-06 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.18 $${𝐉}_{\nu }\left(\nu \right)\sim \frac{2^{1/3}}{3^{2/3}\mathrm{\Gamma }\left(\frac{2}{3}\right){\nu }^{1/3}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$,

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\nu$: real or complex order and $\delta$: arbitrary small positive constant

Source:

Olver (1997b, pp. 103 and 352)

Proof sketch:

Dervivable using (11.10.15), (11.10.16), (11.11.8), (11.11.10), (11.11.16) and the expansions in Watson (1944, §8.42).

Referenced by:

§11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E18

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which originally was $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.19 $${𝐄}_{\nu }\left(\nu \right)\sim \frac{2^{1/3}}{3^{7/6}\mathrm{\Gamma }\left(\frac{2}{3}\right){\nu }^{1/3}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\nu$: real or complex order and $\delta$: arbitrary small positive constant

Source:

Olver (1997b, pp. 103 and 352)

Proof sketch:

Dervivable using (11.10.15), (11.10.16), (11.11.8), (11.11.10), (11.11.16) and the expansions in Watson (1944, §8.42).

Referenced by:

§11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E19

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which originally was $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

8: 11.6 Asymptotic Expansions

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11.6.1 $${𝐊}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.29

Referenced by:

§11.6(i), §11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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11.6.5 $${𝐇}_{\nu }\left(z\right),{𝐋}_{\nu }\left(z\right)\sim \frac{z}{\pi \nu \sqrt{2}}{\left(\frac{\mathrm{e}z}{2\nu }\right)}^{\nu },$$ $|\mathrm{ph}\nu |\le \pi -\delta$.

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order and $\delta$: arbitrary small positive constant

Referenced by:

(11.11.7)

Permalink:

http://dlmf.nist.gov/11.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(ii), §11.6 and Ch.11

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11.6.6 $${𝐊}_{\nu }\left(\lambda \nu \right)\sim \frac{{(\frac{1}{2}\lambda \nu )}^{\nu -1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{k!c_k(\lambda )}{{\nu }^k},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $c_k(\lambda )$: expansion function

A&S Ref:

12.1.34

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(iii), §11.6 and Ch.11

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11.6.7 $${𝐌}_{\nu }\left(\lambda \nu \right)\sim -\frac{{(\frac{1}{2}\lambda \nu )}^{\nu -1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{k!c_k(\mathrm{i}\lambda )}{{\nu }^k},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐌}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $c_k(\lambda )$: expansion function

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(iii), §11.6 and Ch.11

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11.6.9 $${𝐋}_{\nu }\left(\lambda \nu \right)\sim I_{\nu }\left(\lambda \nu \right),$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(iii), §11.6 and Ch.11

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9: Bibliography S

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Computation of infinite integrals involving Bessel functions of arbitrary order by the $\overline{D}$-transformation. J. Comput. Appl. Math. 78 (1), pp. 125–130.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

…

10: 2.7 Differential Equations

… ►For corresponding definitions, together with examples, for linear differential equations of arbitrary order see §§16.8(i)–16.8(ii). …