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1: 32.12 Asymptotic Approximations for Complex Variables

… ►See Boutroux (1913), Kapaev and Kitaev (1993), Takei (1995), Costin (1999), Joshi and Kitaev (2001), Kapaev (2004), and Olde Daalhuis (2005b). … ►

§32.12(iii) Third Painlevé Equation

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2: 10.4 Connection Formulas

… ►Other solutions of (10.2.1) include $J_{-\nu }\left(z\right)$, $Y_{-\nu }\left(z\right)$, $H_{-\nu }^{(1)}\left(z\right)$, and $H_{-\nu }^{(2)}\left(z\right)$. … ►

$$H_{-n}^{(1)}\left(z\right)={(-1)}^n{H}_n^{(1)}\left(z\right),$$

►

$$H_{-n}^{(2)}\left(z\right)={(-1)}^n{H}_n^{(2)}\left(z\right).$$

… ►

$$J_{\nu }\left(z\right)=\frac{1}{2}\left(H_{\nu }^{(1)}\left(z\right)+H_{\nu }^{(2)}\left(z\right)\right),$$

… ►

$$H_{-\nu }^{(1)}\left(z\right)={\mathrm{e}}^{\nu \pi \mathrm{i}}{H}_{\nu }^{(1)}\left(z\right),$$

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3: 10.5 Wronskians and Cross-Products

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10.5.3 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(1)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(1)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(1)}\left(z\right)=2\mathrm{i}/(\pi z),$$

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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), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

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10.5.4 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(2)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(2)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(2)}\left(z\right)=-2\mathrm{i}/(\pi z),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E4

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pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

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10.5.5 $$𝒲\left\{H_{\nu }^{(1)}\left(z\right),H_{\nu }^{(2)}\left(z\right)\right\}=H_{\nu +1}^{(1)}\left(z\right)H_{\nu }^{(2)}\left(z\right)-H_{\nu }^{(1)}\left(z\right)H_{\nu +1}^{(2)}\left(z\right)=-4\mathrm{i}/(\pi z).$$

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Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.17

Permalink:

http://dlmf.nist.gov/10.5.E5

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See also:

Annotations for §10.5 and Ch.10

4: 10.11 Analytic Continuation

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10.11.3 $$\sin \left(\nu \pi \right)H_{\nu }^{(1)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=-\sin \left((m-1)\nu \pi \right)H_{\nu }^{(1)}\left(z\right)-{\mathrm{e}}^{-\nu \pi \mathrm{i}}\sin \left(m\nu \pi \right)H_{\nu }^{(2)}\left(z\right),$$

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Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.37

Referenced by:

§10.17(i)

Permalink:

http://dlmf.nist.gov/10.11.E3

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See also:

Annotations for §10.11 and Ch.10

►

10.11.4 $$\sin \left(\nu \pi \right)H_{\nu }^{(2)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{\nu \pi \mathrm{i}}\sin \left(m\nu \pi \right)H_{\nu }^{(1)}\left(z\right)+\sin \left((m+1)\nu \pi \right)H_{\nu }^{(2)}\left(z\right).$$

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Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.38

Referenced by:

§10.11, §10.17(i)

Permalink:

http://dlmf.nist.gov/10.11.E4

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See also:

Annotations for §10.11 and Ch.10

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10.11.7 $$H_n^{(1)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn-1}((m-1)H_n^{(1)}\left(z\right)+m{H}_n^{(2)}\left(z\right)),$$

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Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $n$: integer and $z$: complex variable

Referenced by:

§10.17(i)

Permalink:

http://dlmf.nist.gov/10.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.11 and Ch.10

►

10.11.8 $$H_n^{(2)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn}(m{H}_n^{(1)}\left(z\right)+(m+1)H_n^{(2)}\left(z\right)).$$

ⓘ

Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $n$: integer and $z$: complex variable

Referenced by:

§10.11, §10.17(i)

Permalink:

http://dlmf.nist.gov/10.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.11 and Ch.10

… ►

$H_{\nu }^{(1)}\left(\overline{z}\right)=\overline{H_{\nu }^{(2)}\left(z\right)},$ $H_{\nu }^{(2)}\left(\overline{z}\right)=\overline{H_{\nu }^{(1)}\left(z\right)}.$

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5: 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 ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. … ►Abramowitz and Stegun (1964): $j_n(z)$, $y_n(z)$, $h_n^{(1)}(z)$, $h_n^{(2)}(z)$, for ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$, respectively, when $n\ge 0$. ►Jeffreys and Jeffreys (1956): ${\mathrm{Hs}}_{\nu }(z)$ for $H_{\nu }^{(1)}\left(z\right)$, ${\mathrm{Hi}}_{\nu }(z)$ for $H_{\nu }^{(2)}\left(z\right)$, ${\mathrm{Kh}}_{\nu }(z)$ for $(2/\pi )K_{\nu }\left(z\right)$. …

6: 10.2 Definitions

… ►

Bessel Functions of the Third Kind (Hankel Functions)

►These solutions of (10.2.1) are denoted by $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$, and their defining properties are given by … ►The principal branches of $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ are two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. … ►For fixed $z$ $(\ne 0)$ each branch of $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ is entire in $\nu$. … ►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. …

7: Bibliography C

… ►

On Temme’s algorithm for the modified Bessel function of the third kind. ACM Trans. Math. Software 6 (4), pp. 581–586.

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External Links:

ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(iv).

Encodings:

BibTeX

… ►

A table of elliptic integrals of the third kind. Math. Comp. 51 (183), pp. 267–280, S1–S5.

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External Links:

ISSN 0025-5718, Document, Supplement, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§19.29(i).

Encodings:

BibTeX

… ►

Symmetry in c, d, n of Jacobian elliptic functions. J. Math. Anal. Appl. 299 (1), pp. 242–253.

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External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§19.25(v), §19.25(v), §22.13(i), §22.6(ii), §22.6(iii), §22.8(i), §22.8(ii), Profile Bille C. Carlson.

Encodings:

BibTeX

… ►

The third Painlevé equation and associated special polynomials. J. Phys. A 36 (36), pp. 9507–9532.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§18.38(iii), §32.8(iii), §32.9(i).

Encodings:

BibTeX

… ►

Tables of Jacobian Elliptic Functions Whose Arguments are Rational Fractions of the Quarter Period. National Physical Laboratory Mathematical Tables, Vol. 7, Her Majesty’s Stationery Office, London.

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External Links:

ISBN 00-791-1158-0, MathReview (John Todd), ZentralBlatt

Referenced by:

§22.20(vii), §22.21.

Encodings:

BibTeX

…

8: 10.20 Uniform Asymptotic Expansions for Large Order

… ►

10.20.6 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(\nu z\right)\\ H_{\nu }^{(2)}\left(\nu z\right)\end{array}\}\sim 2{\mathrm{e}}^{\mp \pi \mathrm{i}/3}{\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{B_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

A&S Ref:

9.3.37

Referenced by:

§10.20(iii), §10.41(v)

Permalink:

http://dlmf.nist.gov/10.20.E6

Encodings:

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See also:

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

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10.20.9 $$\begin{array}{c}H_{\nu }^{(1)}{}_{}{}^{\prime }\left(\nu z\right)\\ H_{\nu }^{(2)}{}_{}{}^{\prime }\left(\nu z\right)\end{array}\}\sim \frac{4{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}}{z}{\left(\frac{1-z^2}{4\zeta }\right)}^{\frac{1}{4}}\left(\frac{{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{4}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{C_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{2}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{D_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $C_k(\zeta )$: coefficients and $D_k(\zeta )$: coefficients

A&S Ref:

9.3.45

Referenced by:

§10.20(ii)

Permalink:

http://dlmf.nist.gov/10.20.E9

Encodings:

pMML, png, TeX

See also:

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

… ►For further results see Dunster (2001a), Wang and Wong (2002), and Paris (2004). …

9: 10.52 Limiting Forms

… ►

10.52.2 $$-{𝗒}_n\left(z\right),\mathrm{i}{𝗁}_n^{(1)}\left(z\right),-\mathrm{i}{𝗁}_n^{(2)}\left(z\right),{(-1)}^n{𝗂}_n^{(2)}\left(z\right),(2/\pi ){𝗄}_n\left(z\right)\sim (2n-1)!!/z^{n+1}.$$

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!!$: double factorial, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, ${𝗁}_n^{(1)}\left(z\right)$: spherical Bessel function of the third kind, ${𝗁}_n^{(2)}\left(z\right)$: spherical Bessel function of the third kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.5

Permalink:

http://dlmf.nist.gov/10.52.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.52(i), §10.52 and Ch.10

… ►

${𝗁}_n^{(1)}\left(z\right)\sim {\mathrm{i}}^{-n-1}{z}^{-1}{\mathrm{e}}^{\mathrm{i}z},$

►

${𝗁}_n^{(2)}\left(z\right)\sim {\mathrm{i}}^{n+1}{z}^{-1}{\mathrm{e}}^{-\mathrm{i}z},$

…

10: 10.47 Definitions and Basic Properties

… ► ${𝗃}_n\left(z\right)$ and ${𝗒}_n\left(z\right)$ are the spherical Bessel functions of the first and second kinds, respectively; ${𝗁}_n^{(1)}\left(z\right)$ and ${𝗁}_n^{(2)}\left(z\right)$ are the spherical Bessel functions of the third kind. … ►Many properties of ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$, ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, and ${𝗄}_n\left(z\right)$ follow straightforwardly from the above definitions and results given in preceding sections of this chapter. For example, $z^{-n}{𝗃}_n\left(z\right)$, $z^{n+1}{𝗒}_n\left(z\right)$, $z^{n+1}{𝗁}_n^{(1)}\left(z\right)$, $z^{n+1}{𝗁}_n^{(2)}\left(z\right)$, $z^{-n}{𝗂}_n^{(1)}\left(z\right)$, $z^{n+1}{𝗂}_n^{(2)}\left(z\right)$, and $z^{n+1}{𝗄}_n\left(z\right)$ are all entire functions of $z$. … ►

10.47.15 ${𝗁}_n^{(1)}\left(-z\right)={(-1)}^n{𝗁}_n^{(2)}\left(z\right),$ ${𝗁}_n^{(2)}\left(-z\right)={(-1)}^n{𝗁}_n^{(1)}\left(z\right).$

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Symbols:

${𝗁}_n^{(1)}\left(z\right)$: spherical Bessel function of the third kind, ${𝗁}_n^{(2)}\left(z\right)$: spherical Bessel function of the third kind, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.47.E15

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §10.47(v), §10.47 and Ch.10

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