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11: 5.9 Integral Representations

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5.9.2 $$\frac{1}{\mathrm{\Gamma }\left(z\right)}=\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^t{t}^{-z}dt,$$

ⓘ

Symbols:

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

A&S Ref:

6.1.4 (in a slightly different form.)

Referenced by:

§5.21, §5.9(i)

Permalink:

http://dlmf.nist.gov/5.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

… ► $t^{-z}$ has its principal value where $t$ crosses the positive real axis, and is continuous. … ►

5.9.8 $$\mathrm{\Gamma }\left(1+\frac{1}{n}\right)\cos \left(\frac{\pi }{2n}\right)={\int }_0^{\mathrm{\infty }}\cos \left(t^n\right)dt,$$ $n=2,3,4,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Referenced by:

§5.9(i)

Permalink:

http://dlmf.nist.gov/5.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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5.9.9 $$\mathrm{\Gamma }\left(1+\frac{1}{n}\right)\sin \left(\frac{\pi }{2n}\right)={\int }_0^{\mathrm{\infty }}\sin \left(t^n\right)dt,$$ $n=2,3,4,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $n$: nonnegative integer

Referenced by:

§5.9(i)

Permalink:

http://dlmf.nist.gov/5.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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5.9.15 $$\psi \left(z\right)=\ln z-\frac{1}{2z}-2{\int }_0^{\mathrm{\infty }}\frac{tdt}{(t^2+z^2)({\mathrm{e}}^{2\pi t}-1)}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

6.3.21

Referenced by:

§5.9(ii)

Permalink:

http://dlmf.nist.gov/5.9.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §5.9(ii), §5.9 and Ch.5

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12: 21.2 Definitions

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21.2.1 $$\theta \left(𝐳\right|𝛀)=\sum _{𝐧\in {\mathbb{Z}}^g}{\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}𝐧\cdot 𝛀\cdot 𝐧+𝐧\cdot 𝐳\right)}.$$

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Defines:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $g$: positive integer and $𝛀$: a Riemann matrix

Referenced by:

§21.2(ii)

Permalink:

http://dlmf.nist.gov/21.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(i), §21.2 and Ch.21

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21.2.8 $$\theta \left(z\right|\mathrm{\Omega })={\theta }_3\left(\pi z\right|\mathrm{\Omega }),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left(𝐳\right|𝛀)$: Riemann theta function and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/21.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(iii), §21.2 and Ch.21

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21.2.9 $${\theta }_1\left(\pi z\right|\mathrm{\Omega })=-\theta \left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}}{\frac{1}{2}}\right]\left(z\right|\mathrm{\Omega }),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/21.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(iii), §21.2 and Ch.21

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21.2.10 $${\theta }_2\left(\pi z\right|\mathrm{\Omega })=\theta \left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}}{0}\right]\left(z\right|\mathrm{\Omega }),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/21.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(iii), §21.2 and Ch.21

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21.2.11 $${\theta }_3\left(\pi z\right|\mathrm{\Omega })=\theta \left[\genfrac{}{}{0.0pt}{}{0}{0}\right]\left(z\right|\mathrm{\Omega }),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/21.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(iii), §21.2 and Ch.21

…

13: 6.2 Definitions and Interrelations

… ►where the path does not cross the negative real axis or pass through the origin. … ►

6.2.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{t+z}dt,$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

5.1.1 (in modified form)

Referenced by:

§2.5(iii), §6.18(i), §6.7(iii)

Permalink:

http://dlmf.nist.gov/6.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

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6.2.10 $$\mathrm{si}\left(z\right)=-{\int }_z^{\mathrm{\infty }}\frac{\sin t}{t}dt=\mathrm{Si}\left(z\right)-\frac{1}{2}\pi .$$

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Defines:

$\mathrm{si}\left(z\right)$: sine integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral and $z$: complex variable

A&S Ref:

5.2.5 (in modified form) 5.2.26 (in modified form)

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(ii), §6.2 and Ch.6

… ►where the path does not cross the negative real axis or pass through the origin. … ►

6.2.19 $$\mathrm{Si}\left(z\right)=\frac{1}{2}\pi -\mathrm{f}\left(z\right)\cos z-\mathrm{g}\left(z\right)\sin z,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{Si}\left(z\right)$: sine integral and $z$: complex variable

A&S Ref:

5.2.8

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(iii), §6.2 and Ch.6

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14: Bibliography G

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Algorithm 222: Incomplete beta function ratios. Comm. ACM 7 (3), pp. 143–144.

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Notes:

Variable-precision Algol program. For a Certification see same journal 7 (1964), p. 244.

External Links:

ISSN 0001-0782, Document, Certification

Referenced by:

§8.28(iv).

Encodings:

BibTeX

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Recurrence relations for cross-products of Bessel functions. Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.

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External Links:

ISSN 0033-5614, Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.6(iii).

Encodings:

BibTeX

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On the exceptional zeros of cross-products of derivatives of spherical Bessel functions. Z. Angew. Math. Phys. 36 (3), pp. 491–494.

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External Links:

ISSN 0044-2275, Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§10.58.

Encodings:

BibTeX

… ►

GNU Scientific Library The GNU Project.

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Notes:

The GNU Scientific Library is a large general purpose numerical software library with broad coverage of elementary and special functions. Implementation is in double precision.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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On Mill’s ratio. Proc. Cambridge Philos. Soc. 67, pp. 363–364.

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External Links:

MathReview (C. G. Esseen), ZentralBlatt

Referenced by:

§7.8.

Encodings:

BibTeX

…

15: 10.50 Wronskians and Cross-Products

§10.50 Wronskians and Cross-Products

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10.50.4 $${𝗃}_0\left(z\right){𝗃}_n\left(z\right)+{𝗒}_0\left(z\right){𝗒}_n\left(z\right)=\cos \left(\frac{1}{2}n\pi \right)\sum _{k=0}^{\lfloor n/2\rfloor }{(-1)}^k\frac{a_{2k}(n+\frac{1}{2})}{z^{2k+2}}+\sin \left(\frac{1}{2}n\pi \right)\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{(-1)}^k\frac{a_{2k+1}(n+\frac{1}{2})}{z^{2k+3}},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\lfloor x\rfloor$: floor of $x$ , $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient

A&S Ref:

10.1.33 (modified)

Referenced by:

§10.50, §10.50

Permalink:

http://dlmf.nist.gov/10.50.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.50 and Ch.10

…

16: 6.7 Integral Representations

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6.7.9 $$\mathrm{si}\left(z\right)=-{\int }_0^{\pi /2}{\mathrm{e}}^{-z\cos t}\cos \left(z\sin t\right)dt,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{si}\left(z\right)$: sine integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

5.2.10

Permalink:

http://dlmf.nist.gov/6.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(ii), §6.7 and Ch.6

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6.7.10 $$\mathrm{Ein}\left(z\right)-\mathrm{Cin}\left(z\right)={\int }_0^{\pi /2}{\mathrm{e}}^{-z\cos t}\sin \left(z\sin t\right)dt,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\cos z$: cosine function, $\mathrm{Cin}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

5.2.11 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(ii), §6.7 and Ch.6

… ►

6.7.12 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)={\mathrm{e}}^{-\mathrm{i}z}{\int }_z^{\mathrm{\infty }}\frac{{\mathrm{e}}^{\mathrm{i}t}}{t}dt,$$ $|\mathrm{ph}z|\le \pi .$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.7(iii)

Permalink:

http://dlmf.nist.gov/6.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(iii), §6.7 and Ch.6

►The path of integration does not cross the negative real axis or pass through the origin. …

17: 8.21 Generalized Sine and Cosine Integrals

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8.21.1 $$\mathrm{ci}(a,z)\pm \mathrm{i}\mathrm{si}(a,z)={\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}a}\mathrm{\Gamma }(a,z{\mathrm{e}}^{\mp \frac{1}{2}\pi \mathrm{i}}),$$

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Defines:

$\mathrm{ci}(a,z)$: generalized cosine integral and $\mathrm{si}(a,z)$: generalized sine integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.21(ii)

Permalink:

http://dlmf.nist.gov/8.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(i), §8.21 and Ch.8

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8.21.3 $${\int }_0^{\mathrm{\infty }}{t}^{a-1}{\mathrm{e}}^{\pm \mathrm{i}t}dt={\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}a}\mathrm{\Gamma }\left(a\right),$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part and $a$: parameter

Referenced by:

§8.21(iv)

Permalink:

http://dlmf.nist.gov/8.21.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(ii), §8.21 and Ch.8

… ►In these representations the integration paths do not cross the negative real axis, and in the case of (8.21.4) and (8.21.5) the paths also exclude the origin. … ►

8.21.12 $$\mathrm{Si}(a,\mathrm{\infty })=\mathrm{\Gamma }\left(a\right)\sin \left(\frac{1}{2}\pi a\right),$$ $a\ne -1,-3,-5,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Si}(a,z)$: generalized sine integral, $\sin z$: sine function and $a$: parameter

Referenced by:

§8.21(v)

Permalink:

http://dlmf.nist.gov/8.21.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(v), §8.21 and Ch.8

►

8.21.13 $$\mathrm{Ci}(a,\mathrm{\infty })=\mathrm{\Gamma }\left(a\right)\cos \left(\frac{1}{2}\pi a\right),$$ $a\ne 0,-2,-4,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}(a,z)$: generalized cosine integral and $a$: parameter

Referenced by:

§8.21(v)

Permalink:

http://dlmf.nist.gov/8.21.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(v), §8.21 and Ch.8

…

18: 13.23 Integrals

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13.23.6 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{M}_{\kappa ,\mu }\left(t^{-1}\right)dt=\frac{z^{-\kappa -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{I}_{2\mu }\left(2\sqrt{z}\right),$$ $\mathrm{\Re }z>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

►

13.23.7 $$\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{W}_{\kappa ,\mu }\left(t^{-1}\right)dt=\frac{2z^{-\kappa -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{K}_{2\mu }\left(2\sqrt{z}\right),$$ $\mathrm{\Re }z>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

… ►

13.23.8 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{\int }_0^{\mathrm{\infty }}\cos \left(2xt\right){\mathrm{e}}^{-\frac{1}{2}{t}^2}{t}^{-2\mu -1}{M}_{\kappa ,\mu }\left(t^2\right)dt=\frac{\sqrt{\pi }{\mathrm{e}}^{-\frac{1}{2}{x}^2}{x}^{\mu +\kappa -1}}{2\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{W}_{\frac{1}{2}\kappa -\frac{3}{2}\mu ,\frac{1}{2}\kappa +\frac{1}{2}\mu }\left(x^2\right),$$ $\mathrm{\Re }\left(\kappa +\mu \right)>-\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric 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, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.23.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(ii), §13.23 and Ch.13

… ►

13.23.14 $$f(x)=\frac{1}{\pi \mathrm{i}\sqrt{x}}{\int }_{{\mu }_1-\mathrm{i}\mathrm{\infty }}^{{\mu }_1+\mathrm{i}\mathrm{\infty }}\mu g(\mu )\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)W_{\kappa ,\mu }\left(x\right)d\mu .$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.23.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(iv), §13.23 and Ch.13

…

19: 10.19 Asymptotic Expansions for Large Order

… ►

10.19.1 $$J_{\nu }\left(z\right)\sim \frac{1}{\sqrt{2\pi \nu }}{\left(\frac{\mathrm{e}z}{2\nu }\right)}^{\nu },$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.3.1

Referenced by:

§10.41(i), §33.9(i)

Permalink:

http://dlmf.nist.gov/10.19.E1

Encodings:

pMML, png, TeX

See also:

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

►

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

ⓘ

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

… ►

10.19.5 $$\xi =\nu (\tan \beta -\beta )-\frac{1}{4}\pi ,$$

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Defines:

$\xi$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tan z$: tangent function, $\nu$: complex parameter and $\beta$

Permalink:

http://dlmf.nist.gov/10.19.E5

Encodings:

pMML, png, TeX

See also:

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

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10.19.9 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim \frac{2^{\frac{4}{3}}}{{\nu }^{\frac{1}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{P_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{5}{3}}}{\nu }{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{Q_k(a)}{{\nu }^{2k/3}},$$

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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, $\nu$: complex parameter, $P_k(a)$: polynomial coefficient and $Q_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.0.27):

Originally the polynomials $P_k(a)$, $Q_k(a)$ were incorrectly described as Legendre polynomials and integer-degree, zero-order associated Legendre functions of the second kind respectively.

See also:

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

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10.19.13 $$\begin{array}{c}H_{\nu }^{(1)}{}_{}{}^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}{}_{}{}^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim -\frac{2^{\frac{5}{3}}}{{\nu }^{\frac{2}{3}}}{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{R_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{4}{3}}}{{\nu }^{\frac{4}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{S_k(a)}{{\nu }^{2k/3}},$$

ⓘ

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, $\nu$: complex parameter, $R_k(a)$: polynomial coefficient and $S_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E13

Encodings:

pMML, png, TeX

See also:

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

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20: 18.15 Asymptotic Approximations

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18.15.4 $${\theta }_{n,m,\mathrm{\ell }}=\frac{1}{2}(2n+\alpha +\beta +m+1)\theta -\frac{1}{2}(\alpha +\mathrm{\ell }+\frac{1}{2})\pi .$$

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Defines:

${\theta }_{n,m,\mathrm{\ell }}$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§18.15(i), Erratum (V1.0.22) for Subsection 18.15(i)

Permalink:

http://dlmf.nist.gov/18.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(i), §18.15 and Ch.18

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18.15.11 $${\theta }_{n,m}=(n+m+\lambda )\theta -\frac{1}{2}(m+\lambda )\pi .$$

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Defines:

${\theta }_{n,m}$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.15.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(ii), §18.15 and Ch.18

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18.15.13 $${\alpha }_{n,m}=(n-m+\frac{1}{2})\theta +(n-\frac{1}{2}m-\frac{1}{4})\pi .$$

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Defines:

${\alpha }_{n,m}$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§18.15(iii)

Permalink:

http://dlmf.nist.gov/18.15.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(iii), §18.15 and Ch.18

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18.15.15 $${\theta }_n^{(\alpha )}(x)=2{(nx)}^{\frac{1}{2}}-\left(\frac{1}{2}\alpha +\frac{1}{4}\right)\pi .$$

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Defines:

${\theta }_n^{(\alpha )}(x)$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.15.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

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18.15.26 $${\omega }_{n,m}(x)={\mu }^{\frac{1}{2}}x-\frac{1}{2}(m+n)\pi .$$

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Defines:

${\omega }_{n,m}(x)$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.15.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(v), §18.15 and Ch.18

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