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11: 10.3 Graphics

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§10.3(ii) Real Order, Complex Variable

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Figure 10.3.10: $H_0^{(1)}\left(x+\mathrm{i}y\right)$, $-10\le x\le 5$, $-2.8\le y\le 4$. … Magnify 3D Help

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Figure 10.3.12: $H_1^{(1)}\left(x+\mathrm{i}y\right)$, $-10\le x\le 5$, $-2.8\le y\le 4$. … Magnify 3D Help

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Figure 10.3.14: $H_5^{(1)}\left(x+\mathrm{i}y\right)$, $-20\le x\le 10$, $-4\le y\le 4$. … Magnify 3D Help

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Figure 10.3.16: $H_{5.5}^{(1)}\left(x+\mathrm{i}y\right)$, $-20\le x\le 10$, $-4\le y\le 4$. … Magnify 3D Help

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12: 10.8 Power Series

§10.8 Power Series

… ►When $\nu$ is not an integer the corresponding expansions for $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, and $H_{\nu }^{(2)}\left(z\right)$ are obtained by combining (10.2.2) with (10.2.3), (10.4.7), and (10.4.8). … ►The corresponding results for $H_n^{(1)}\left(z\right)$ and $H_n^{(2)}\left(z\right)$ are obtained via (10.4.3) with $\nu =n$. …

13: 11.1 Special Notation

… ►For the functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$, $I_{\nu }\left(z\right)$, and $K_{\nu }\left(z\right)$ see §§10.2(ii), 10.25(ii). …

14: 10.17 Asymptotic Expansions for Large Argument

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§10.17(i) Hankel’s Expansions

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§10.17(iii) Error Bounds for Real Argument and Order

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§10.17(v) Exponentially-Improved Expansions

… ►For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).

15: 10.76 Approximations

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§10.76(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions

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16: 10.74 Methods of Computation

… ►For evaluation of the Hankel functions $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ for complex values of $\nu$ and $z$ based on the integral representations (10.9.18) see Remenets (1973). … ► ►

§10.74(vi) Zeros and Associated Values

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Hankel Transform

… ►The spherical Bessel transform is the Hankel transform (10.22.76) in the case when $\nu$ is half an odd positive integer. …

17: 10.19 Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

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§10.19(i) Asymptotic Forms

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10.19.2 $$Y_{\nu }\left(z\right)\sim -\mathrm{i}{H}_{\nu }^{(1)}\left(z\right)\sim \mathrm{i}{H}_{\nu }^{(2)}\left(z\right)\sim -\sqrt{\frac{2}{\pi \nu }}{\left(\frac{\mathrm{e}z}{2\nu }\right)}^{-\nu }.$$

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Symbols:

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

A&S Ref:

9.3.1

Referenced by:

§10.41(i)

Permalink:

http://dlmf.nist.gov/10.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.19(i), §10.19 and Ch.10

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§10.19(iii) Transition Region

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18: 10.27 Connection Formulas

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10.27.7 $$I_{\nu }\left(z\right)=\frac{1}{2}{\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}\left(H_{\nu }^{(1)}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right)+H_{\nu }^{(2)}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right)\right),$$ $-\pi \le \pm \mathrm{ph}z\le \frac{1}{2}\pi$.

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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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.27.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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10.27.8

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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, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.4

Referenced by:

§10.16, §10.26(ii), §10.27, §10.27, §10.32(ii), §10.39, §10.40(ii), §10.40(iii), §10.41(i), §10.42, §10.43(ii), §10.44(i), §10.44(ii), §10.44(iii), §10.47(iv), (9.7.16), §9.7(iii)

Permalink:

http://dlmf.nist.gov/10.27.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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19: 10.50 Wronskians and Cross-Products

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$$𝒲\left\{{𝗁}_n^{(1)}\left(z\right),{𝗁}_n^{(2)}\left(z\right)\right\}=-2\mathrm{i}{z}^{-2}.$$

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20: 10.54 Integral Representations

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${𝗁}_n^{(1)}\left(z\right)={\displaystyle \frac{{(-\mathrm{i})}^{n+1}}{\pi }}{\displaystyle {\int }_{\mathrm{i}\mathrm{\infty }}^{(1+)}}{\mathrm{e}}^{\mathrm{i}zt}{Q}_n\left(t\right)dt,$

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${𝗁}_n^{(2)}\left(z\right)={\displaystyle \frac{{(-\mathrm{i})}^{n+1}}{\pi }}{\displaystyle {\int }_{\mathrm{i}\mathrm{\infty }}^{(-1+)}}{\mathrm{e}}^{\mathrm{i}zt}{Q}_n\left(t\right)dt,$

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