Bessel equation

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21: 18.39 Applications in the Physical Sciences

… ►The finite system of functions ${\psi }_n$ is orthonormal in $L^2(ℝ,dx)$, see (18.34.7_3). …

22: 29.18 Mathematical Applications

… ►(29.18.5) is the differential equation of spherical Bessel functions (§10.47(i)), and (29.18.6), (29.18.7) agree with the Lamé equation (29.2.1). …

23: 9.13 Generalized Airy Functions

… ►and ${𝒵}_p$ is any linear combination of the modified Bessel functions $I_p$ and ${\mathrm{e}}^{p\pi \mathrm{i}}{K}_p$ (§10.25(ii)). ►Swanson and Headley (1967) define independent solutions $A_n\left(z\right)$ and $B_n\left(z\right)$ of (9.13.1) by … ►

$$-A_n^{\prime }\left(0\right)=p^{1/2}{B}_n^{\prime }\left(0\right)=\frac{p^p}{\mathrm{\Gamma }\left(p\right)}.$$

…

24: 32.10 Special Function Solutions

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§32.10(iii) Third Painlevé Equation

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25: Frank W. J. Olver

… ►Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history. … ►He is particularly known for his extensive work in the study of the asymptotic solution of differential equations, i. …, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e. …, Bessel functions, hypergeometric functions, Legendre functions). … ►

10 Bessel Functions

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26: 10.68 Modulus and Phase Functions

… ►Equations (10.68.8)–(10.68.14) also hold with the symbols $\mathrm{ber}$, $\mathrm{bei}$, $M$, and $\theta$ replaced throughout by $\ker$, $\mathrm{kei}$, $N$, and $\varphi$, respectively. … ►However, numerical tabulations show that if the second of these equations applies and ${\varphi }_1\left(x\right)$ is continuous, then ${\varphi }_1\left(0\right)=-\frac{3}{4}\pi$; compare Abramowitz and Stegun (1964, p. 433).

27: 30.10 Series and Integrals

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28: 30.2 Differential Equations

… ►If $\gamma =0$, Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).

29: 1.8 Fourier Series

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1.8.5 $$\frac{1}{\pi }{\int }_{-\pi }^{\pi }{\left|f(x)\right|}^{2}dx=\frac{1}{2}{\left|a_0\right|}^2+\sum _{n=1}^{\mathrm{\infty }}({\left|a_n\right|}^2+{\left|b_n\right|}^2),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

Erratum (V1.2.0) for Equations (1.8.5), (1.8.6)

Permalink:

http://dlmf.nist.gov/1.8.E5

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

Previously this equation was given as an inequality. For square integrable functions this inequality $\ge$ can be sharpened to $=$.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

►

1.8.6 $$\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{\left|f(x)\right|}^{2}dx=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\left|c_n\right|}^2,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.8(i), Erratum (V1.2.0) for Equations (1.8.5), (1.8.6)

Permalink:

http://dlmf.nist.gov/1.8.E6

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

Previously this equation was given as an inequality. For square integrable functions this inequality $\ge$ can be sharpened to $=$.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

… ►

1.8.6_1 $$\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\overline{g(x)}dx=\frac{1}{2}{a}_0\overline{a_0^{\prime }}+\sum _{n=1}^{\mathrm{\infty }}(a_n\overline{a_n^{\prime }}+b_n\overline{b_n^{\prime }}),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Referenced by:

(1.8.13), (1.8.13), §1.8(i), §1.8(iv), Erratum (V1.2.0) Section 1.8

Permalink:

http://dlmf.nist.gov/1.8.E6_1

Encodings:

pMML, png, TeX

Rearrangement (effective with 1.2.0):

This equation, which was originally (1.8.13), was moved here.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

►

1.8.6_2 $$\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x)\overline{g(x)}dx=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n\overline{c_n^{\prime }}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.8(i), Erratum (V1.2.0) Section 1.8

Permalink:

http://dlmf.nist.gov/1.8.E6_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

… ►

1.8.13Moved to (1.8.6_1).

ⓘ

Referenced by:

(1.8.6_1), §1.8(i), §1.8(iv)

Permalink:

http://dlmf.nist.gov/1.8.E13

Rearrangement (effective with 1.2.0):

This equation has been moved to (1.8.6_1).

See also:

Annotations for §1.8(iv), §1.8 and Ch.1

…

30: 10.16 Relations to Other Functions

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Elementary Functions

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Parabolic Cylinder Functions

… ►Principal values on each side of these equations correspond. ►

Confluent Hypergeometric Functions

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Generalized Hypergeometric Functions

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