Bessel equation

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1: 10.73 Physical Applications

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§10.73(ii) Spherical Bessel Functions

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2: 10.1 Special Notation

… ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

3: 10.2 Definitions

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§10.2(i) Bessel’s Equation

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§10.2(ii) Standard Solutions

… ►The notation ${𝒞}_{\nu }\left(z\right)$ denotes $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$. ►

§10.2(iii) Numerically Satisfactory Pairs of Solutions

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Table 10.2.1: Numerically satisfactory pairs of solutions of Bessel’s equation.

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Pair	Interval or Region
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4: 10.25 Definitions

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§10.25(i) Modified Bessel’s Equation

… ►This equation is obtained from Bessel’s equation (10.2.1) on replacing $z$ by $\pm \mathrm{i}z$, and it has the same kinds of singularities. … ►

§10.25(ii) Standard Solutions

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§10.25(iii) Numerically Satisfactory Pairs of Solutions

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5: 10.72 Mathematical Applications

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§10.72(i) Differential Equations with Turning Points

►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … ►

6: 10.36 Other Differential Equations

§10.36 Other Differential Equations

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7: 11.9 Lommel Functions

… ►The inhomogeneous Bessel differential equation … ►

11.9.2 $$w=s_{\mu ,\nu }\left(z\right)+A{J}_{\nu }\left(z\right)+B{Y}_{\nu }\left(z\right),$$

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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, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/11.9.E2

Encodings:

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See also:

Annotations for §11.9(i), §11.9 and Ch.11

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11.9.7 $$s_{\mu ,\nu }\left(z\right)=2^{\mu +1}\sum _{k=0}^{\mathrm{\infty }}\frac{(2k+\mu +1)\mathrm{\Gamma }\left(k+\mu +1\right)}{k!(2k+\mu -\nu +1)(2k+\mu +\nu +1)}{J}_{2k+\mu +1}\left(z\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $!$: factorial (as in $n!$), $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/11.9.E7

Encodings:

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Annotations for §11.9(ii), §11.9 and Ch.11

… ►For uniform asymptotic expansions, for large $\nu$ and fixed $\mu =-1,0,1,2,\mathrm{\dots }$, of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). For an error bound for (11.9.9) and an exponentially-improved extension see Nemes (2015b). …

8: 10.13 Other Differential Equations

§10.13 Other Differential Equations

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9: 11.2 Definitions

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§11.2(ii) Differential Equations

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Modified Struve’s Equation

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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:

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

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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:

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See also:

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

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10: 10.47 Definitions and Basic Properties

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§10.47(i) Differential Equations

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§10.47(ii) Standard Solutions

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Equation (10.47.1)

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Equation (10.47.2)

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