Weber function

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11: 10.1 Special Notation

… ►The main functions treated in this chapter are the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$; Hankel functions $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$; modified Bessel functions $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$; spherical Bessel functions ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$; modified spherical Bessel functions ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, ${𝗄}_n\left(z\right)$; Kelvin functions ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

12: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

… ►

13.21.2 $$W_{\kappa ,\mu }\left(x\right)=\sqrt{x}\mathrm{\Gamma }\left(\kappa +\frac{1}{2}\right)\left(\sin \left(\kappa \pi -\mu \pi \right)J_{2\mu }\left(2\sqrt{x\kappa }\right)-\cos \left(\kappa \pi -\mu \pi \right)Y_{2\mu }\left(2\sqrt{x\kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(2\sqrt{x\kappa }\right)O\left({\kappa }^{-\frac{1}{2}}\right)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\sin z$: sine function and $x$: real variable

Referenced by:

§13.21(i)

Permalink:

http://dlmf.nist.gov/13.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(i), §13.21 and Ch.13

►

13.21.3 $$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{-\pi \mathrm{i}}\right)=\frac{\pi \sqrt{x}}{\mathrm{\Gamma }\left(\kappa +\frac{1}{2}\right)}{\mathrm{e}}^{\mu \pi \mathrm{i}}\left(H_{2\mu }^{(1)}\left(2\sqrt{x\kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(2\sqrt{x\kappa }\right)O\left({\kappa }^{-\frac{1}{2}}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.21.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(i), §13.21 and Ch.13

… ►

13.21.15 $$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{-\pi \mathrm{i}}\right)=c(\kappa ,\mu )\mathrm{\Psi }(\kappa ,\mu ,x)\left(H_{2\mu }^{(1)}\left(\sqrt{\zeta \kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(\sqrt{\zeta \kappa }\right)O\left({\kappa }^{-1}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $c(\kappa ,\mu )$: function and $\mathrm{\Psi }(\kappa ,\mu ,x)$: function

Permalink:

http://dlmf.nist.gov/13.21.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(ii), §13.21 and Ch.13

►

13.21.16 $$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{\pi \mathrm{i}}\right)={\mathrm{e}}^{-2\mu \pi \mathrm{i}}c(\kappa ,\mu )\mathrm{\Psi }(\kappa ,\mu ,x)\left(H_{2\mu }^{(2)}\left(\sqrt{\zeta \kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(\sqrt{\zeta \kappa }\right)O\left({\kappa }^{-1}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $c(\kappa ,\mu )$: function and $\mathrm{\Psi }(\kappa ,\mu ,x)$: function

Permalink:

http://dlmf.nist.gov/13.21.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(ii), §13.21 and Ch.13

… ►For the functions $J_{2\mu }$, $Y_{2\mu }$, $H_{2\mu }^{(1)}$, and $H_{2\mu }^{(2)}$ see §10.2(ii), and for the $\mathrm{env}$ functions associated with $J_{2\mu }$ and $Y_{2\mu }$ see §2.8(iv). …

13: 10.21 Zeros

… ►When all of their zeros are simple, the $m$th positive zeros of these functions are denoted by $j_{\nu ,m}$, $j_{\nu ,m}^{\prime }$, $y_{\nu ,m}$, and $y_{\nu ,m}^{\prime }$ respectively, except that $z=0$ is counted as the first zero of $J_0^{\prime }\left(z\right)$. … ►

10.21.33 $$y_{\nu ,m}\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k}{{\nu }^{2k/3}},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $y_{\nu ,m}$: zeros of the Bessel function $Y_{\nu }\left(x\right)$ , $m$: integer, $k$: nonnegative integer, $\nu$: complex parameter and $\alpha$: coefficients

Permalink:

http://dlmf.nist.gov/10.21.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

… ►

10.21.35 $$Y_{\nu }^{\prime }\left(y_{\nu ,m}\right)\sim {(-1)}^{m-1}\frac{{(2/\nu )}^{\frac{2}{3}}}{\pi M\left(b_m\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\beta }_k}{{\nu }^{2k/3}},$$

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $M\left(z\right)$: Airy modulus function, $b_k$: $k$th zero of Airy $\mathrm{Bi}$ , $y_{\nu ,m}$: zeros of the Bessel function $Y_{\nu }\left(x\right)$ , $m$: integer, $k$: nonnegative integer, $\nu$: complex parameter and $\beta$: coefficients

Permalink:

http://dlmf.nist.gov/10.21.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

… ►

10.21.37 $$y_{\nu ,m}^{\prime }\sim \nu \sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k^{\prime }}{{\nu }^{2k/3}},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $y_{\nu ,m}$: zeros of the Bessel function $Y_{\nu }\left(x\right)$ , $m$: integer, $k$: nonnegative integer, $\nu$: complex parameter and ${\alpha }^{\prime }$: coefficients

Referenced by:

§10.21(vii)

Permalink:

http://dlmf.nist.gov/10.21.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

… ►

10.21.39 $$Y_{\nu }\left(y_{\nu ,m}^{\prime }\right)\sim {(-1)}^{m-1}\frac{{(2/\nu )}^{\frac{1}{3}}}{\pi N\left(b_m^{\prime }\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\beta }_k^{\prime }}{{\nu }^{2k/3}}.$$

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $N\left(z\right)$: Airy modulus function, $y_{\nu ,m}$: zeros of the Bessel function $Y_{\nu }\left(x\right)$ , $b_k^{\prime }$: $k$th zero of Airy ${\mathrm{Bi}}^{\prime }$ , $m$: integer, $k$: nonnegative integer, $\nu$: complex parameter and $\beta$: coefficients

Referenced by:

§10.21(vii)

Permalink:

http://dlmf.nist.gov/10.21.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

…

14: 33.20 Expansions for Small $|ϵ|$

… ►For the functions $J$, $Y$, $I$, and $K$ see §§10.2(ii), 10.25(ii). … ►

33.20.8 $${𝖧}_k(\mathrm{\ell };r)=\sum _{p=2k}^{3k}{(2r)}^{(p+1)/2}{C}_{k,p}{Y}_{2\mathrm{\ell }+1+p}\left(\sqrt{8r}\right),$$ $r>0$,

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $C_{k,p}$: coefficients and ${𝖧}_k(\mathrm{\ell };r)$: coefficient

Permalink:

http://dlmf.nist.gov/33.20.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §33.20(iii), §33.20 and Ch.33

… ►The functions $Y$ and $K$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by (33.20.6). …

15: 10.2 Definitions

… ►

Bessel Function of the Second Kind (Weber’s Function)

►

10.2.3 $$Y_{\nu }\left(z\right)=\frac{J_{\nu }\left(z\right)\cos \left(\nu \pi \right)-J_{-\nu }\left(z\right)}{\sin \left(\nu \pi \right)}.$$

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Defines:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.2

Referenced by:

§10.11, §10.15, §10.19(i), §10.2(ii), §10.22(ii), §10.24, §10.4, §10.47(ii), §10.7(i), §10.8, §11.10(v), §11.4(i)

Permalink:

http://dlmf.nist.gov/10.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

… ►

10.2.4 $$Y_n\left(z\right)={\frac{1}{\pi }\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}+{\frac{{(-1)}^n}{\pi }\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =-n},$$ $n=0,\pm 1,\pm 2,\mathrm{\dots }$.

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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, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.15, §10.2(ii), §10.22(iv), §10.4

Permalink:

http://dlmf.nist.gov/10.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

… ►Except where indicated otherwise, it is assumed throughout the DLMF that the symbols $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, and $H_{\nu }^{(2)}\left(z\right)$ denote the principal values of these functions. … ►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 $\nu$. …

16: 2.8 Differential Equations with a Parameter

… ►

2.8.30 $$W_{n,4}(u,\xi )={|\xi |}^{1/2}{Y}_{\nu }\left(u{|\xi |}^{1/2}\right)\left(\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)-|\xi |Y_{\nu +1}\left(u{|\xi |}^{1/2}\right)\left(\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+1}}+O\left(\frac{1}{u^{2n-2}}\right)\right).$$

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter

Referenced by:

§2.8(iv), §2.8(iv)

Permalink:

http://dlmf.nist.gov/2.8.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iv), §2.8 and Ch.2

… ►

2.8.32 $$J_{\nu }(x)+Y_{\nu }(x)=0.$$

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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 and $\nu$: real nonnegative constant

Permalink:

http://dlmf.nist.gov/2.8.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iv), §2.8 and Ch.2

►Define … ►

2.8.34 $$\mathrm{env}{J}_{\nu }(x)=\mathrm{env}{Y}_{\nu }(x)={\left(J_{\nu }^2(x)+Y_{\nu }^2(x)\right)}^{1/2},$$ .

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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, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$ , $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\nu$: real nonnegative constant and $X_{\nu }$: smallest positive root

Permalink:

http://dlmf.nist.gov/2.8.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iv), §2.8 and Ch.2

… ►

2.8.36 $$W_{n,4}(u,\xi )={|\xi |}^{1/2}{Y}_{\nu }\left(u{|\xi |}^{1/2}\right)\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}-|\xi |Y_{\nu +1}\left(u{|\xi |}^{1/2}\right)\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+1}}+{|\xi |}^{1/2}\mathrm{env}{Y}_{\nu }\left(u{|\xi |}^{1/2}\right)O\left(\frac{1}{u^{2n-1}}\right),$$

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter

Permalink:

http://dlmf.nist.gov/2.8.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iv), §2.8 and Ch.2

…

17: 10.20 Uniform Asymptotic Expansions for Large Order

… ►

10.20.5 $$Y_{\nu }\left(\nu z\right)\sim -{\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Bi}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Bi}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{B_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

A&S Ref:

9.3.36

Permalink:

http://dlmf.nist.gov/10.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

… ►

10.20.8 $$Y_{\nu }^{\prime }\left(\nu z\right)\sim \frac{2}{z}{\left(\frac{1-z^2}{4\zeta }\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Bi}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{4}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{C_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Bi}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{2}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{D_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $C_k(\zeta )$: coefficients and $D_k(\zeta )$: coefficients

A&S Ref:

9.3.44

Permalink:

http://dlmf.nist.gov/10.20.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

…

18: Bibliography H

… ►

Diffraction and Weber functions. SIAM J. Appl. Math. 57 (6), pp. 1702–1715.

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External Links:

ISSN 0036-1399, Document, MathReview, ZentralBlatt

Referenced by:

§12.13(ii), §12.17.

Encodings:

BibTeX

…

19: 11.2 Definitions

… ►

11.2.5 $${𝐊}_{\nu }\left(z\right)={𝐇}_{\nu }\left(z\right)-Y_{\nu }\left(z\right),$$

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Defines:

${𝐊}_{\nu }\left(z\right)$: Struve function

Symbols:

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

Referenced by:

§11.13(iii), §11.13(iv), §11.2(i), §11.4(i), §11.5(i), §11.6(i), §11.6(i), §11.6(iii)

Permalink:

http://dlmf.nist.gov/11.2.E5

Encodings:

pMML, png, TeX

See also:

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

… ►

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

►

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

ⓘ

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

…

20: Bibliography K

… ►

Tables of Weber functions. Vol. I. Mathematical Tables Series, Vol. 15, Pergamon Press, London-New York.

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Notes:

Translated by Prasenjit Basu

External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

…