imaginary-modulus%20transformations

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11: 9.9 Zeros

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9.9.14 $${\beta }_k={\mathrm{e}}^{\pi \mathrm{i}/3}T\left(\frac{3}{8}\pi (4k-1)+\frac{3}{4}\mathrm{i}\ln 2\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\beta }_k$: $k$th complex zero of Airy $\mathrm{Bi}$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $T$: expansion

Source:

Olver (1954, (A 22))

Permalink:

http://dlmf.nist.gov/9.9.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.15 $${\mathrm{Bi}}^{\prime }\left({\beta }_k\right)={(-1)}^k\sqrt{2}{\mathrm{e}}^{-\pi \mathrm{i}/6}V\left(\frac{3}{8}\pi (4k-1)+\frac{3}{4}\mathrm{i}\ln 2\right),$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\beta }_k$: $k$th complex zero of Airy $\mathrm{Bi}$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $V$: expansion

Source:

Olver (1954, (A 22))

Permalink:

http://dlmf.nist.gov/9.9.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.16 $${\beta }_k^{\prime }={\mathrm{e}}^{\pi \mathrm{i}/3}U\left(\frac{3}{8}\pi (4k-3)+\frac{3}{4}\mathrm{i}\ln 2\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\beta }_k^{\prime }$: $k$th complex zero of Airy ${\mathrm{Bi}}^{\prime }$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $U$: expansion

Source:

Olver (1954, (A 22))

Permalink:

http://dlmf.nist.gov/9.9.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.17 $$\mathrm{Bi}\left({\beta }_k^{\prime }\right)={(-1)}^{k-1}\sqrt{2}{\mathrm{e}}^{\pi \mathrm{i}/6}W\left(\frac{3}{8}\pi (4k-3)+\frac{3}{4}\mathrm{i}\ln 2\right).$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\beta }_k^{\prime }$: $k$th complex zero of Airy ${\mathrm{Bi}}^{\prime }$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $W$: expansion

Source:

Olver (1954, (A 22))

Permalink:

http://dlmf.nist.gov/9.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

… ►

Table 9.9.3: Complex zeros of $\mathrm{Bi}$.

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	${\mathrm{e}}^{-\pi \mathrm{i}/3}{\beta }_k$		${\mathrm{Bi}}^{\prime }\left({\beta }_k\right)$
…

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…

12: 9.6 Relations to Other Functions

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9.6.2 $$\mathrm{Ai}\left(z\right)={\pi }^{-1}\sqrt{z/3}{K}_{\pm 1/3}\left(\zeta \right)=\frac{1}{3}\sqrt{z}\left(I_{-1/3}\left(\zeta \right)-I_{1/3}\left(\zeta \right)\right)=\frac{1}{2}\sqrt{z/3}{\mathrm{e}}^{2\pi \mathrm{i}/3}{H}_{1/3}^{(1)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)=\frac{1}{2}\sqrt{z/3}{\mathrm{e}}^{\pi \mathrm{i}/3}{H}_{-1/3}^{(1)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)=\frac{1}{2}\sqrt{z/3}{\mathrm{e}}^{-2\pi \mathrm{i}/3}{H}_{1/3}^{(2)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right)=\frac{1}{2}\sqrt{z/3}{\mathrm{e}}^{-\pi \mathrm{i}/3}{H}_{-1/3}^{(2)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right),$$

ⓘ

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), $\pi$: 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, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Derivable from those given in Miller (1946, p. B17) and Olver (1997b, pp. 392–393).

A&S Ref:

10.4.14 (partial)

Referenced by:

(9.6.21)

Permalink:

http://dlmf.nist.gov/9.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(i), §9.6 and Ch.9

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9.6.3 $${\mathrm{Ai}}^{\prime }\left(z\right)=-{\pi }^{-1}(z/\sqrt{3})K_{\pm 2/3}\left(\zeta \right)=(z/3)\left(I_{2/3}\left(\zeta \right)-I_{-2/3}\left(\zeta \right)\right)=\frac{1}{2}(z/\sqrt{3}){\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{2/3}^{(1)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)=\frac{1}{2}(z/\sqrt{3}){\mathrm{e}}^{-5\pi \mathrm{i}/6}{H}_{-2/3}^{(1)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)=\frac{1}{2}(z/\sqrt{3}){\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{2/3}^{(2)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right)=\frac{1}{2}(z/\sqrt{3}){\mathrm{e}}^{5\pi \mathrm{i}/6}{H}_{-2/3}^{(2)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right),$$

ⓘ

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), $\pi$: 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, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Derivable from those given in Miller (1946, p. B17) and Olver (1997b, pp. 392–393).

A&S Ref:

10.4.16 (with opposite sign!)

Referenced by:

(9.6.22)

Permalink:

http://dlmf.nist.gov/9.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(i), §9.6 and Ch.9

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9.6.4 $$\mathrm{Bi}\left(z\right)=\sqrt{z/3}\left(I_{1/3}\left(\zeta \right)+I_{-1/3}\left(\zeta \right)\right)=\frac{1}{2}\sqrt{z/3}\left({\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{1/3}^{(1)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right)+{\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{1/3}^{(2)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)\right)=\frac{1}{2}\sqrt{z/3}\left({\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{-1/3}^{(1)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right)+{\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{-1/3}^{(2)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)\right),$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: 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), $\pi$: 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, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Derivable from those given in Miller (1946, p. B17) and Olver (1997b, pp. 392–393).

A&S Ref:

10.4.18 (partial)

Referenced by:

(9.6.23), (9.7.16), §9.7(iii)

Permalink:

http://dlmf.nist.gov/9.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(i), §9.6 and Ch.9

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9.6.5 $${\mathrm{Bi}}^{\prime }\left(z\right)=(z/\sqrt{3})\left(I_{2/3}\left(\zeta \right)+I_{-2/3}\left(\zeta \right)\right)=\frac{1}{2}(z/\sqrt{3})\left({\mathrm{e}}^{\pi \mathrm{i}/3}{H}_{2/3}^{(1)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right)+{\mathrm{e}}^{-\pi \mathrm{i}/3}{H}_{2/3}^{(2)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)\right)=\frac{1}{2}(z/\sqrt{3})\left({\mathrm{e}}^{-\pi \mathrm{i}/3}{H}_{-2/3}^{(1)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right)+{\mathrm{e}}^{\pi \mathrm{i}/3}{H}_{-2/3}^{(2)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)\right),$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: 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), $\pi$: 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, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Derivable from those given in Miller (1946, p. B17) and Olver (1997b, pp. 392–393).

A&S Ref:

10.4.20 (partial)

Referenced by:

(9.6.24), (9.7.16), (9.7.17), §9.7(iii), §9.7(iv)

Permalink:

http://dlmf.nist.gov/9.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(i), §9.6 and Ch.9

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9.6.6 $$\mathrm{Ai}\left(-z\right)=(\sqrt{z}/3)\left(J_{1/3}\left(\zeta \right)+J_{-1/3}\left(\zeta \right)\right)=\frac{1}{2}\sqrt{z/3}\left({\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{1/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{1/3}^{(2)}\left(\zeta \right)\right)=\frac{1}{2}\sqrt{z/3}\left({\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{-1/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{-1/3}^{(2)}\left(\zeta \right)\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Derivable from those given in Miller (1946, p. B17) and Olver (1997b, pp. 392–393).

A&S Ref:

10.4.15 (partial)

Permalink:

http://dlmf.nist.gov/9.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(i), §9.6 and Ch.9

…

13: 15.12 Asymptotic Approximations

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15.12.9 $${(z+1)}^{3\lambda /2}{(2\lambda )}^{c-1}𝐅(\genfrac{}{}{0pt}{}{a+\lambda ,b+2\lambda }{c};-z)={\lambda }^{-1/3}\left({\mathrm{e}}^{\pi \mathrm{i}(a-c+\lambda +(1/3))}\mathrm{Ai}\left({\mathrm{e}}^{-2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)+{\mathrm{e}}^{\pi \mathrm{i}(c-a-\lambda -(1/3))}\mathrm{Ai}\left({\mathrm{e}}^{2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)\right)\left(a_0(\zeta )+O({\lambda }^{-1})\right)+{\lambda }^{-2/3}\left({\mathrm{e}}^{\pi \mathrm{i}(a-c+\lambda +(2/3))}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{-2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)+{\mathrm{e}}^{\pi \mathrm{i}(c-a-\lambda -(2/3))}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)\right)\left(a_1(\zeta )+O({\lambda }^{-1})\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\zeta$: change of variable, $\beta$ and $a_j(\zeta )$: function

Referenced by:

§15.12(iii)

Permalink:

http://dlmf.nist.gov/15.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

… ►By combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for $F(a+e_1\lambda ,b+e_2\lambda ;c+e_3\lambda ;z)$ can be obtained with $e_j=\pm 1$ or $0$, $j=1,2,3$. …

14: 10.61 Definitions and Basic Properties

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10.61.1 $${\mathrm{ber}}_{\nu }x+\mathrm{i}{\mathrm{bei}}_{\nu }x=J_{\nu }\left(x{\mathrm{e}}^{3\pi \mathrm{i}/4}\right)={\mathrm{e}}^{\nu \pi \mathrm{i}}{J}_{\nu }\left(x{\mathrm{e}}^{-\pi \mathrm{i}/4}\right)={\mathrm{e}}^{\nu \pi \mathrm{i}/2}{I}_{\nu }\left(x{\mathrm{e}}^{\pi \mathrm{i}/4}\right)={\mathrm{e}}^{3\nu \pi \mathrm{i}/2}{I}_{\nu }\left(x{\mathrm{e}}^{-3\pi \mathrm{i}/4}\right),$$

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Defines:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function and ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: 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, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.9.1

Referenced by:

§10.63(ii), §10.65(i), §10.65(ii), §10.67(i), §10.69

Permalink:

http://dlmf.nist.gov/10.61.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.61(i), §10.61 and Ch.10

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10.61.2 $${\ker }_{\nu }x+\mathrm{i}{\mathrm{kei}}_{\nu }x={\mathrm{e}}^{-\nu \pi \mathrm{i}/2}{K}_{\nu }\left(x{\mathrm{e}}^{\pi \mathrm{i}/4}\right)=\frac{1}{2}\pi \mathrm{i}{H}_{\nu }^{(1)}\left(x{\mathrm{e}}^{3\pi \mathrm{i}/4}\right)=-\frac{1}{2}\pi \mathrm{i}{\mathrm{e}}^{-\nu \pi \mathrm{i}}{H}_{\nu }^{(2)}\left(x{\mathrm{e}}^{-\pi \mathrm{i}/4}\right).$$

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Defines:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function and ${\ker }_{\nu }\left(x\right)$: Kelvin function

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, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.9.2

Referenced by:

§10.63(ii), §10.65(ii), §10.67(i), §10.69

Permalink:

http://dlmf.nist.gov/10.61.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.61(i), §10.61 and Ch.10

…

15: 19.7 Connection Formulas

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Reciprocal-Modulus Transformation

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Imaginary-Modulus Transformation

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Imaginary-Argument Transformation

… ►For two further transformations of this type see Erdélyi et al. (1953b, p. 316). …

16: 14.23 Values on the Cut

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14.23.1 $$P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)={\mathrm{e}}^{\mp \mu \pi \mathrm{i}/2}{𝖯}_{\nu }^{\mu }\left(x\right),$$

ⓘ

Symbols:

: Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.4 $${𝖯}_{\nu }^{\mu }\left(x\right)={\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}{P}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right),$$

ⓘ

Symbols:

: Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.3.2 (corrected)

Referenced by:

§14.23, §14.23, §14.3(iv)

Permalink:

http://dlmf.nist.gov/14.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

►

14.23.5 $${𝖰}_{\nu }^{\mu }\left(x\right)=\frac{1}{2}\mathrm{\Gamma }\left(\nu +\mu +1\right)\left({\mathrm{e}}^{-\mu \pi \mathrm{i}/2}{𝑸}_{\nu }^{\mu }\left(x+\mathrm{i}0\right)+{\mathrm{e}}^{\mu \pi \mathrm{i}/2}{𝑸}_{\nu }^{\mu }\left(x-\mathrm{i}0\right)\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.3.4 (modified)

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

… ►

14.23.6 $${𝖰}_{\nu }^{\mu }\left(x\right)={\mathrm{e}}^{\mp \mu \pi \mathrm{i}/2}\mathrm{\Gamma }\left(\nu +\mu +1\right){𝑸}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)\pm \frac{1}{2}\pi \mathrm{i}{\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}{P}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23, §14.23

Permalink:

http://dlmf.nist.gov/14.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

… ►

14.23.7 $${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\frac{1}{2}{\mathrm{e}}^{3\mu \pi \mathrm{i}/2}{Q}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x-\mathrm{i}0\right)+\frac{1}{2}{\mathrm{e}}^{-3\mu \pi \mathrm{i}/2}{Q}_{-\frac{1}{2}-\mathrm{i}\tau }^{-\mu }\left(x+\mathrm{i}0\right).$$

ⓘ

Symbols:

${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\tau$: real variable and $\mu$: general order

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

17: 36.2 Catastrophes and Canonical Integrals

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36.2.6 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)=2\sqrt{\pi /3}\exp \left(\mathrm{i}\left(\frac{4}{27}{z}^3+\frac{1}{3}xz-\frac{1}{4}\pi \right)\right){\int }_{\mathrm{\infty }\exp \left(-7\pi \mathrm{i}/12\right)}^{\mathrm{\infty }\exp \left(\pi \mathrm{i}/12\right)}\exp \left(\mathrm{i}\left(u^6+2z{u}^4+(z^2+x)u^2+\frac{y^2}{12u^2}\right)\right)du,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real parameter, $z$: real parameter and $x$: real parameter

Referenced by:

§36.2(i), §36.2(ii), §36.5(iii)

Permalink:

http://dlmf.nist.gov/36.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

… ►

36.2.8 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)=4\sqrt{\pi /6}\exp \left(\mathrm{i}\left(\frac{1}{27}{z}^3+\frac{1}{6}z(y+x)+\frac{1}{4}\pi \right)\right){\int }_{\mathrm{\infty }\exp \left(5\pi \mathrm{i}/12\right)}^{\mathrm{\infty }\exp \left(\pi \mathrm{i}/12\right)}\exp \left(\mathrm{i}\left(2u^6+2z{u}^4+\left(\frac{1}{2}{z}^2+x+y\right)u^2-\frac{{(y-x)}^2}{24u^2}\right)\right)du,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real parameter, $z$: real parameter and $x$: real parameter

Referenced by:

§36.2(i), §36.2(ii), §36.5(iii)

Permalink:

http://dlmf.nist.gov/36.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

… ►

36.2.9 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)=\frac{2\pi }{3^{1/3}}{\int }_{\mathrm{\infty }\exp \left(5\pi \mathrm{i}/6\right)}^{\mathrm{\infty }\exp \left(\pi \mathrm{i}/6\right)}\exp \left(\mathrm{i}(s^3+xs)\right)\mathrm{Ai}\left(\frac{zs+y}{3^{1/3}}\right)ds.$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real parameter, $z$: real parameter, $s$: variable and $x$: real parameter

Referenced by:

§36.2(i), §36.2(ii)

Permalink:

http://dlmf.nist.gov/36.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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36.2.28 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-z\right)}=2\pi \sqrt{\frac{\pi z}{27}}\exp \left(\frac{2}{27}\mathrm{i}{z}^3\right)\left(J_{-1/6}\left(\frac{2}{27}{z}^3\right)+\mathrm{i}{J}_{1/6}\left(\frac{2}{27}{z}^3\right)\right),$$ $z\ge 0$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E28

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36

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36.2.29 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,-z\right)}=\frac{2^{1/3}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right){\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-\frac{z}{2^{2/3}}\right),$$ .

ⓘ

Symbols:

$\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E29

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36

18: 9.5 Integral Representations

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9.5.3 $$\mathrm{Bi}\left(x\right)=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\exp \left(-\frac{1}{3}{t}^3+xt\right)dt+\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\sin \left(\frac{1}{3}{t}^3+xt\right)dt.$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b)$: half-closed interval, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $(a,b]$: half-closed interval, $\sin z$: sine function and $x$: real variable

Proof sketch:

Use (9.5.5) with the substitions: for the paths $(-\mathrm{\infty },0]$ use $t=-\tau$, for the paths $[0,\mathrm{\infty }{\mathrm{e}}^{\pm \pi \mathrm{i}/3})$ use $t={\mathrm{e}}^{\pm \pi \mathrm{i}/3}\tau$. For the resulting integrals use (4.14.1) and (4.14.2).

A&S Ref:

10.4.33

Referenced by:

(9.12.19)

Permalink:

http://dlmf.nist.gov/9.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.5(i), §9.5 and Ch.9

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9.5.4 $$\mathrm{Ai}\left(z\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{\mathrm{\infty }{\mathrm{e}}^{-\pi \mathrm{i}/3}}^{\mathrm{\infty }{\mathrm{e}}^{\pi \mathrm{i}/3}}\exp \left(\frac{1}{3}{t}^3-zt\right)dt,$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

Source:

Olver (1997b, p. 53)

Referenced by:

§36.2(ii), §9.14, §9.17(iii), (9.5.5)

Permalink:

http://dlmf.nist.gov/9.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.5(ii), §9.5 and Ch.9

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9.5.5 $$\mathrm{Bi}\left(z\right)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }{\mathrm{e}}^{\pi \mathrm{i}/3}}\exp \left(\frac{1}{3}{t}^3-zt\right)dt+\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }{\mathrm{e}}^{-\pi \mathrm{i}/3}}\exp \left(\frac{1}{3}{t}^3-zt\right)dt.$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.2.10) and (9.5.4).

Referenced by:

(9.5.3)

Permalink:

http://dlmf.nist.gov/9.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.5(ii), §9.5 and Ch.9

…

19: 9.18 Tables

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Zhang and Jin (1996, p. 337) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

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Harvard University (1945) tabulates the real and imaginary parts of $h_1(z)$, $h_1^{\prime }(z)$, $h_2(z)$, $h_2^{\prime }(z)$ for $-x_0\le \mathrm{\Re }z\le x_0$, $0\le \mathrm{\Im }z\le y_0$, , with interval 0.1 in $\mathrm{\Re }z$ and $\mathrm{\Im }z$. Precision is 8D. Here $h_1(z)=-2^{4/3}{3}^{1/6}\mathrm{i}\mathrm{Ai}\left({\mathrm{e}}^{-\pi \mathrm{i}/3}z\right)$, $h_2(z)=2^{4/3}{3}^{1/6}\mathrm{i}\mathrm{Ai}\left({\mathrm{e}}^{\pi \mathrm{i}/3}z\right)$.

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Miller (1946) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$. Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

►

Sherry (1959) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; 20S.

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Zhang and Jin (1996, p. 339) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$; 8D.

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20: 10.69 Uniform Asymptotic Expansions for Large Order

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10.69.2 $${\mathrm{ber}}_{\nu }\left(\nu x\right)+\mathrm{i}{\mathrm{bei}}_{\nu }\left(\nu x\right)\sim \frac{{\mathrm{e}}^{\nu \xi }}{{(2\pi \nu \xi )}^{1/2}}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{\nu }\sum _{k=0}^{\mathrm{\infty }}\frac{U_k({\xi }^{-1})}{{\nu }^k},$$

ⓘ

Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin 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, $x$: real variable, $\nu$: complex parameter, $U_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.37

Referenced by:

§10.69

Permalink:

http://dlmf.nist.gov/10.69.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

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10.69.3 $${\ker }_{\nu }\left(\nu x\right)+\mathrm{i}{\mathrm{kei}}_{\nu }\left(\nu x\right)\sim {\mathrm{e}}^{-\nu \xi }{\left(\frac{\pi }{2\nu \xi }\right)}^{1/2}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{-\nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k({\xi }^{-1})}{{\nu }^k},$$

ⓘ

Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin 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, $x$: real variable, $\nu$: complex parameter, $U_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.38

Permalink:

http://dlmf.nist.gov/10.69.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

►

10.69.4 $${\mathrm{ber}}_{\nu }^{\prime }\left(\nu x\right)+\mathrm{i}{\mathrm{bei}}_{\nu }^{\prime }\left(\nu x\right)\sim \frac{{\mathrm{e}}^{\nu \xi }}{x}{\left(\frac{\xi }{2\pi \nu }\right)}^{1/2}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{\nu }\sum _{k=0}^{\mathrm{\infty }}\frac{V_k({\xi }^{-1})}{{\nu }^k},$$

ⓘ

Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin 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, $x$: real variable, $\nu$: complex parameter, $V_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.39

Referenced by:

§10.69

Permalink:

http://dlmf.nist.gov/10.69.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

►

10.69.5 $${\ker }_{\nu }^{\prime }\left(\nu x\right)+\mathrm{i}{\mathrm{kei}}_{\nu }^{\prime }\left(\nu x\right)\sim -\frac{{\mathrm{e}}^{-\nu \xi }}{x}{\left(\frac{\pi \xi }{2\nu }\right)}^{1/2}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{-\nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{V_k({\xi }^{-1})}{{\nu }^k},$$

ⓘ

Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin 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, $x$: real variable, $\nu$: complex parameter, $V_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.40

Permalink:

http://dlmf.nist.gov/10.69.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

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