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: 10.13 Other Differential Equations

§10.13 Other Differential Equations

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8: 11.9 Lommel Functions

… ►The inhomogeneous Bessel differential equation … ►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). … …

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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10: 28.10 Integral Equations

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§28.10(ii) Equations with Bessel-Function Kernels

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28.10.9 $${\int }_0^{\pi /2}{J}_0\left(2\sqrt{q({\cos }^2\tau -{\sin }^2\zeta )}\right){\mathrm{ce}}_{2n}(\tau ,q)d\tau =w_{\text{II}}(\frac{1}{2}\pi ;a_{2n}\left(q\right),q){\mathrm{ce}}_{2n}(\zeta ,q),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $a_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $q=h^2$: parameter, $n$: integer and $w_{\text{II}}(z;a,q)$: fundamental solution

Permalink:

http://dlmf.nist.gov/28.10.E9

Encodings:

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See also:

Annotations for §28.10(ii), §28.10 and Ch.28

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28.10.10 $${\int }_0^{\pi }{J}_0\left(2\sqrt{q}(\cos \tau +\cos \zeta )\right){\mathrm{ce}}_n(\tau ,q)d\tau =w_{\text{II}}(\pi ;a_n\left(q\right),q){\mathrm{ce}}_n(\zeta ,q).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $a_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $n$: integer and $w_{\text{II}}(z;a,q)$: fundamental solution

Permalink:

http://dlmf.nist.gov/28.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(ii), §28.10 and Ch.28

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