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41: 4.14 Definitions and Periodicity

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4.14.8 $$\sin \left(z+2k\pi \right)=\sin z,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $k$: integer and $z$: complex variable

A&S Ref:

4.3.7

Permalink:

http://dlmf.nist.gov/4.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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4.14.9 $$\cos \left(z+2k\pi \right)=\cos z,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $k$: integer and $z$: complex variable

A&S Ref:

4.3.8

Permalink:

http://dlmf.nist.gov/4.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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4.14.10 $$\tan \left(z+k\pi \right)=\tan z.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tan z$: tangent function, $k$: integer and $z$: complex variable

A&S Ref:

4.3.9

Permalink:

http://dlmf.nist.gov/4.14.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

42: 5.4 Special Values and Extrema

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5.4.3 $$|\mathrm{\Gamma }\left(\mathrm{i}y\right)|={\left(\frac{\pi }{y\sinh \left(\pi y\right)}\right)}^{1/2},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.1.29

Referenced by:

§10.24, §5.4(i)

Permalink:

http://dlmf.nist.gov/5.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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5.4.6 $$\mathrm{\Gamma }\left(\frac{1}{2}\right)={\pi }^{1/2}=\mathrm{1.77245\hspace{0.33em}38509\hspace{0.33em}05516\hspace{0.33em}02729}\mathrm{\dots },$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $\pi$: the ratio of the circumference of a circle to its diameter

A&S Ref:

6.1.8

Notes:

For more digits see OEIS Sequence A002161; see also Sloane (2003).

Referenced by:

§4.10(ii), §5.4(i)

Permalink:

http://dlmf.nist.gov/5.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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5.4.16 $$\mathrm{\Im }\psi \left(\mathrm{i}y\right)=\frac{1}{2y}+\frac{\pi }{2}\coth \left(\pi y\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\coth z$: hyperbolic cotangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.11

Permalink:

http://dlmf.nist.gov/5.4.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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5.4.17 $$\mathrm{\Im }\psi \left(\frac{1}{2}+\mathrm{i}y\right)=\frac{\pi }{2}\tanh \left(\pi y\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\tanh z$: hyperbolic tangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.12

Permalink:

http://dlmf.nist.gov/5.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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5.4.18 $$\mathrm{\Im }\psi \left(1+\mathrm{i}y\right)=-\frac{1}{2y}+\frac{\pi }{2}\coth \left(\pi y\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\coth z$: hyperbolic cotangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.13

Referenced by:

§5.4(ii)

Permalink:

http://dlmf.nist.gov/5.4.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(ii), §5.4 and Ch.5

…

43: 25.13 Periodic Zeta Function

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25.13.1 $$F(x,s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}nx}}{n^s},$$

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Defines:

$F(x,s)$: periodic zeta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, (9), p. 257)

Permalink:

http://dlmf.nist.gov/25.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

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25.13.2 $$F(x,s)=\frac{\mathrm{\Gamma }\left(1-s\right)}{{(2\pi )}^{1-s}}\left({\mathrm{e}}^{\pi \mathrm{i}(1-s)/2}\zeta (1-s,x)+{\mathrm{e}}^{\pi \mathrm{i}(s-1)/2}\zeta (1-s,1-x)\right),$$ , $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, Exercise 2, p. 273)

Permalink:

http://dlmf.nist.gov/25.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

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25.13.3 $$\zeta (1-s,x)=\frac{\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}F(x,s)+{\mathrm{e}}^{\pi \mathrm{i}s/2}F(-x,s)\right),$$ $\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $x=1$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, (10), p. 257)

Notes:

This formula is equivalent to (25.11.9).

Referenced by:

Erratum (V1.1.4) for Equation (25.13.3)

Permalink:

http://dlmf.nist.gov/25.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

44: 11.10 Anger–Weber Functions

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11.10.1 $${𝐉}_{\nu }\left(z\right)=\frac{1}{\pi }{\int }_0^{\pi }\cos \left(\nu \theta -z\sin \theta \right)d\theta ,$$

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Defines:

${𝐉}_{\nu }\left(z\right)$: Anger function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order

A&S Ref:

12.3.1

Referenced by:

§11.10(v), §11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(i), §11.10 and Ch.11

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11.10.6 $$f(\nu ,z)=\frac{(z-\nu )}{\pi z^2}\sin \left(\pi \nu \right),$$ $w={𝐉}_{\nu }\left(z\right)$,

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(ii), §11.10 and Ch.11

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11.10.7 $$f(\nu ,z)=-\frac{1}{\pi z^2}(z+\nu +(z-\nu )\cos \left(\pi \nu \right)),$$ $w={𝐄}_{\nu }\left(z\right)$.

ⓘ

Symbols:

${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(ii), §11.10 and Ch.11

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11.10.9 $${𝐄}_{\nu }\left(z\right)=\sin \left(\frac{1}{2}\pi \nu \right)S_1(\nu ,z)-\cos \left(\frac{1}{2}\pi \nu \right)S_2(\nu ,z),$$

ⓘ

Symbols:

${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order

Referenced by:

(11.11.6)

Permalink:

http://dlmf.nist.gov/11.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(iii), §11.10 and Ch.11

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11.10.26 ${𝐄}_0\left(z\right)=-{𝐇}_0\left(z\right),$ ${𝐄}_1\left(z\right)={\displaystyle \frac{2}{\pi }}-{𝐇}_1\left(z\right).$

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E26

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §11.10(vii), §11.10 and Ch.11

…

45: 5.5 Functional Relations

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5.5.3 $$\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(1-z\right)=\pi /\sin \left(\pi z\right),$$ $z\ne 0,\pm 1,\mathrm{\dots }$,

ⓘ

Symbols:

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

A&S Ref:

6.1.17 (without the condition on $z$.)

Referenced by:

§10.19(i), §15.4(ii), §15.8(ii), (25.9.2), §5.21, (9.10.18), (9.12.15)

Permalink:

http://dlmf.nist.gov/5.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(ii), §5.5 and Ch.5

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5.5.4 $$\psi \left(z\right)-\psi \left(1-z\right)=-\pi /\tan \left(\pi z\right),$$ $z\ne 0,\pm 1,\mathrm{\dots }$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

6.3.7 (without the condition on $z$.)

Permalink:

http://dlmf.nist.gov/5.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(ii), §5.5 and Ch.5

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5.5.5 $$\mathrm{\Gamma }\left(2z\right)={\pi }^{-1/2}{2}^{2z-1}\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(z+\frac{1}{2}\right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

A&S Ref:

6.1.18

Referenced by:

§14.19(ii), §15.4(i), §15.4(iii), (25.9.2), §5.5(iii)

Permalink:

http://dlmf.nist.gov/5.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

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5.5.6 $$\mathrm{\Gamma }\left(nz\right)={(2\pi )}^{(1-n)/2}{n}^{nz-(1/2)}\prod _{k=0}^{n-1}\mathrm{\Gamma }\left(z+\frac{k}{n}\right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.20

Referenced by:

§15.4(iii), §5.5(iii)

Permalink:

http://dlmf.nist.gov/5.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

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5.5.7 $$\prod _{k=1}^{n-1}\mathrm{\Gamma }\left(\frac{k}{n}\right)={(2\pi )}^{(n-1)/2}{n}^{-1/2}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

…

46: 9.9 Zeros

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9.9.6 $$a_k=-T\left(\frac{3}{8}\pi (4k-1)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $a_k$: $k$th zero of Airy $\mathrm{Ai}$ , $k$: nonnegative integer and $T$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E6

Encodings:

pMML, png, TeX

See also:

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

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9.9.7 $${\mathrm{Ai}}^{\prime }\left(a_k\right)={(-1)}^{k-1}V\left(\frac{3}{8}\pi (4k-1)\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $a_k$: $k$th zero of Airy $\mathrm{Ai}$ , $k$: nonnegative integer and $V$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E7

Encodings:

pMML, png, TeX

See also:

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

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9.9.8 $$a_k^{\prime }=-U\left(\frac{3}{8}\pi (4k-3)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $a_k^{\prime }$: $k$th zero of Airy ${\mathrm{Ai}}^{\prime }$ , $k$: nonnegative integer and $U$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E8

Encodings:

pMML, png, TeX

See also:

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

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9.9.9 $$\mathrm{Ai}\left(a_k^{\prime }\right)={(-1)}^{k-1}W\left(\frac{3}{8}\pi (4k-3)\right).$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $a_k^{\prime }$: $k$th zero of Airy ${\mathrm{Ai}}^{\prime }$ , $k$: nonnegative integer and $W$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E9

Encodings:

pMML, png, TeX

See also:

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

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9.9.10 $$b_k=-T\left(\frac{3}{8}\pi (4k-3)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $b_k$: $k$th zero of Airy $\mathrm{Bi}$ , $k$: nonnegative integer and $T$: expansion

Source:

Olver (1954, (A 21))

Permalink:

http://dlmf.nist.gov/9.9.E10

Encodings:

pMML, png, TeX

See also:

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

…

47: 10.5 Wronskians and Cross-Products

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10.5.1 $$𝒲\left\{J_{\nu }\left(z\right),J_{-\nu }\left(z\right)\right\}=J_{\nu +1}\left(z\right)J_{-\nu }\left(z\right)+J_{\nu }\left(z\right)J_{-\nu -1}\left(z\right)=-2\sin \left(\nu \pi \right)/(\pi z),$$

ⓘ

Symbols:

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

A&S Ref:

9.1.15

Permalink:

http://dlmf.nist.gov/10.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

►

10.5.2 $$𝒲\left\{J_{\nu }\left(z\right),Y_{\nu }\left(z\right)\right\}=J_{\nu +1}\left(z\right)Y_{\nu }\left(z\right)-J_{\nu }\left(z\right)Y_{\nu +1}\left(z\right)=2/(\pi z),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.16

Permalink:

http://dlmf.nist.gov/10.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

►

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

ⓘ

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

►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

48: 10.54 Integral Representations

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10.54.1 $${𝗃}_n\left(z\right)=\frac{z^n}{2^{n+1}n!}{\int }_0^{\pi }\cos \left(z\cos \theta \right){(\sin \theta )}^{2n+1}d\theta .$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.13

Referenced by:

§10.54

Permalink:

http://dlmf.nist.gov/10.54.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

►

10.54.2 $${𝗃}_n\left(z\right)=\frac{{(-\mathrm{i})}^n}{2}{\int }_0^{\pi }{\mathrm{e}}^{\mathrm{i}z\cos \theta }{P}_n\left(\cos \theta \right)\sin \theta d\theta .$$

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: 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, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.14

Permalink:

http://dlmf.nist.gov/10.54.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

►

10.54.3 $${𝗄}_n\left(z\right)=\frac{\pi }{2}{\int }_1^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{P}_n\left(t\right)dt,$$

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.54.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

►

10.54.4 $${𝗃}_n\left(z\right)=\frac{{(-\mathrm{i})}^{n+1}}{2\pi }{\int }_{\mathrm{i}\mathrm{\infty }}^{(-1+,1+)}{\mathrm{e}}^{\mathrm{i}zt}{Q}_n\left(t\right)dt,$$

ⓘ

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, $(a,b)$: open interval, $\mathrm{ph}$: phase, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.54.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

…

49: 24.12 Zeros

… ►

24.12.2

ⓘ

Symbols:

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

Referenced by:

§24.12(i)

Permalink:

http://dlmf.nist.gov/24.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.12(i), §24.12 and Ch.24

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24.12.3 $$x_1^{(n)}-\frac{3}{4}\sim \frac{1}{2^{n+1}\pi },$$ $n\to \mathrm{\infty }$,

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: integer and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.12(i), §24.12 and Ch.24

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24.12.6 $$R(n)\sim 2n/(\pi \mathrm{e}),$$ $n\to \mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $n$: integer and $R(n)$: number of zeros

Permalink:

http://dlmf.nist.gov/24.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.12(i), §24.12 and Ch.24

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24.12.9 $n=3,7,11,\mathrm{\dots }$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: integer and $y_j^{(n)}$: zeros

Permalink:

http://dlmf.nist.gov/24.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.12(ii), §24.12 and Ch.24

►

24.12.10 $n=5,9,13,\mathrm{\dots }$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: integer and $y_j^{(n)}$: zeros

Permalink:

http://dlmf.nist.gov/24.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.12(ii), §24.12 and Ch.24

…

50: 9.2 Differential Equation

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9.2.2 $$w=\mathrm{Ai}\left(z\right),\mathrm{Bi}\left(z\right),\mathrm{Ai}\left(z{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\right).$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy 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 $w$: ODE solution

Source:

Olver (1997b, pp. 392–393, 413–414)

Permalink:

http://dlmf.nist.gov/9.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(i), §9.2 and Ch.9

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9.2.7 $$𝒲\left\{\mathrm{Ai}\left(z\right),\mathrm{Bi}\left(z\right)\right\}=\frac{1}{\pi },$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

Source:

Olver (1997b, (1.19), p. 393)

A&S Ref:

10.4.10

Referenced by:

(9.10.2), (9.10.3), (9.11.2), (9.8.13), (9.8.17), (9.9.3), (9.9.4)

Permalink:

http://dlmf.nist.gov/9.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(iv), §9.2 and Ch.9

►

9.2.8 $$𝒲\left\{\mathrm{Ai}\left(z\right),\mathrm{Ai}\left(z{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\right)\right\}=\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}/6}}{2\pi },$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Proof sketch:

Derivable from Olver (1997b, p. 416).

A&S Ref:

10.4.11 10.4.12

Permalink:

http://dlmf.nist.gov/9.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(iv), §9.2 and Ch.9

►

9.2.9 $$𝒲\left\{\mathrm{Ai}\left(z{\mathrm{e}}^{-2\pi \mathrm{i}/3}\right),\mathrm{Ai}\left(z{\mathrm{e}}^{2\pi \mathrm{i}/3}\right)\right\}=\frac{1}{2\pi \mathrm{i}}.$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Proof sketch:

Derivable from Olver (1997b, p. 416).

A&S Ref:

10.4.13

Permalink:

http://dlmf.nist.gov/9.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(iv), §9.2 and Ch.9

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9.2.10 $$\mathrm{Bi}\left(z\right)={\mathrm{e}}^{-\pi \mathrm{i}/6}\mathrm{Ai}\left(z{\mathrm{e}}^{-2\pi \mathrm{i}/3}\right)+{\mathrm{e}}^{\pi \mathrm{i}/6}\mathrm{Ai}\left(z{\mathrm{e}}^{2\pi \mathrm{i}/3}\right).$$

ⓘ

Symbols:

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

Proof sketch:

Derivable from Olver (1997b, p. 414).

A&S Ref:

10.4.6

Referenced by:

(9.10.19), (9.5.5)

Permalink:

http://dlmf.nist.gov/9.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(v), §9.2 and Ch.9

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