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21: 36.8 Convergent Series Expansions

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36.8.3 $$\frac{3^{2/3}}{4{\pi }^2}{\mathrm{\Psi }}^{(\mathrm{H})}\left(3^{1/3}𝐱\right)=\mathrm{Ai}\left(x\right)\mathrm{Ai}\left(y\right)\sum _{n=0}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{c_n(x)c_n(y)}{n!}+\mathrm{Ai}\left(x\right){\mathrm{Ai}}^{\prime }\left(y\right)\sum _{n=2}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{c_n(x)d_n(y)}{n!}+{\mathrm{Ai}}^{\prime }\left(x\right)\mathrm{Ai}\left(y\right)\sum _{n=2}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{d_n(x)c_n(y)}{n!}+{\mathrm{Ai}}^{\prime }\left(x\right){\mathrm{Ai}}^{\prime }\left(y\right)\sum _{n=1}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{d_n(x)d_n(y)}{n!},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $z$: real parameter, $m$: integer, $x$: real parameter, $c_n(t)$: coefficients and $d_n(t)$: coefficients

Referenced by:

§36.8

Permalink:

http://dlmf.nist.gov/36.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.8 and Ch.36

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36.8.4 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)=2{\pi }^2{\left(\frac{2}{3}\right)}^{2/3}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-\mathrm{i}{(2/3)}^{2/3}z\right)}^n}{n!}\mathrm{\Re }\left(f_n(\frac{x+\mathrm{i}y}{{12}^{1/3}},\frac{x-\mathrm{i}y}{{12}^{1/3}})\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter, $m$: integer, $x$: real parameter and $f_n$: function

Referenced by:

§36.8

Permalink:

http://dlmf.nist.gov/36.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.8 and Ch.36

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$d_0(t)=0,$

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$c_{n+1}(t)=c_n^{\prime }(t)+t{d}_n(t),$

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$d_{n+1}(t)=c_n(t)+d_n^{\prime }(t).$

22: 9.11 Products

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9.11.1 $$\frac{d^{3}w}{{dz}^3}-4z\frac{dw}{dz}-2w=0,$$ $w=w_1{w}_2$,

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $w$: function, $w_1$: function and $w_2$: function

Permalink:

http://dlmf.nist.gov/9.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(i), §9.11 and Ch.9

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

9.11.10 $$\int z{w}_1^{\prime }{w}_2^{\prime }dz=\frac{3}{10}(-w_1{w}_2+z{w}_1{w}_2^{\prime }+z{w}_1^{\prime }{w}_2)+\frac{1}{5}(z^2{w}_1^{\prime }{w}_2^{\prime }-z^3{w}_1{w}_2).$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: function

Source:

Albright (1977, (A.21))

Permalink:

http://dlmf.nist.gov/9.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(iv), §9.11 and Ch.9

►For $\int z^n{w}_1{w}_{2}dz$, $\int z^n{w}_1{w}_2^{\prime }dz$, $\int z^n{w}_1^{\prime }{w}_2^{\prime }dz$, where $n$ is any positive integer, see Albright (1977). … ►For further definite integrals see Prudnikov et al. (1990, §1.8.2), Laurenzi (1993), Reid (1995, 1997a, 1997b), and Vallée and Soares (2010, Chapters 3, 4).

23: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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§17.9(i) ${}_2{}^{}\varphi _1^{}\to {}_2{}^{}\varphi _2^{}$, ${}_3{}^{}\varphi _1^{}$, or ${}_3{}^{}\varphi _2^{}$

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§17.9(ii) ${}_3{}^{}\varphi _2^{}\to {}_3{}^{}\varphi _2^{}$

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Transformations of ${}_3{}^{}\varphi _2^{}$-Series

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Sears’ Balanced ${}_4{}^{}\varphi _3^{}$ Transformations

►With $def=abc{q}^{1-n}$ …

24: 31.2 Differential Equations

… ►where $2{\omega }_1$ and $2{\omega }_3$ with $\mathrm{\Im }\left({\omega }_3/{\omega }_1\right)>0$ are generators of the lattice $𝕃$ for $\mathrm{\wp }\left(z|𝕃\right)$. … ►Lastly, $w(z)={(z-a)}^{1-ϵ}{w}_3(z)$ satisfies (31.2.1) if $w_3$ is a solution of (31.2.1) with transformed parameters $q_3=q+\gamma (1-ϵ)$; ${\alpha }_3=\alpha +1-ϵ$, ${\beta }_3=\beta +1-ϵ$, ${ϵ}_3=2-ϵ$. By composing these three steps, there result $2^3=8$ possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). … ►If $\tilde{z}=\tilde{z}(z)$ is one of the $3!=6$ homographies that map $\mathrm{\infty }$ to $\mathrm{\infty }$, then $w(z)=\tilde{w}(\tilde{z})$ satisfies (31.2.1) if $\tilde{w}(\tilde{z})$ is a solution of (31.2.1) with $z$ replaced by $\tilde{z}$ and appropriately transformed parameters. …If $\tilde{z}=\tilde{z}(z)$ is one of the $4!-3!=18$ homographies that do not map $\mathrm{\infty }$ to $\mathrm{\infty }$, then an appropriate prefactor must be included on the right-hand side. …

25: 29.2 Differential Equations

… ►Next, let $e_1,e_2,e_3$ be any real constants that satisfy $e_1>e_2>e_3$ and … ►

$(e_2-e_3)/(e_1-e_3)=k^2.$

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29.2.9 $$\frac{d^{2}w}{{d\eta }^2}+(g-\nu (\nu +1)\mathrm{\wp }\left(\eta \right))w=0,$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\nu$: real parameter, $g$ and $\eta$

Permalink:

http://dlmf.nist.gov/29.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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29.2.10 $$\frac{d^{2}w}{{d\zeta }^2}+\frac{1}{2}\left(\frac{1}{\zeta -e_1}+\frac{1}{\zeta -e_2}+\frac{1}{\zeta -e_3}\right)\frac{dw}{d\zeta }+\frac{g-\nu (\nu +1)\zeta }{4(\zeta -e_1)(\zeta -e_2)(\zeta -e_3)}w=0,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\nu$: real parameter, $e_j$: constants, $g$ and $\zeta$

Referenced by:

§29.2(ii)

Permalink:

http://dlmf.nist.gov/29.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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$g_3=4e_1{e}_2{e}_3.$

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26: 19.34 Mutual Inductance of Coaxial Circles

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19.34.1 $$\frac{c^{2}M}{2\pi }=ab{\int }_0^{2\pi }{(h^2+a^2+b^2-2ab\cos \theta )}^{-1/2}\cos \theta d\theta =2ab{\int }_{-1}^1\frac{tdt}{\sqrt{(1+t)(1-t)(a_3-2abt)}}=2abI({𝐞}_5),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $I(𝐦)$: integral, $M$: mutual inductance and $h$: distance

Referenced by:

§19.34

Permalink:

http://dlmf.nist.gov/19.34.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

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$a_3=h^2+a^2+b^2,$

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19.34.3 $$2abI({𝐞}_5)=a_{3}I(\mathrm{𝟎})-I({𝐞}_3)=a_{3}I(\mathrm{𝟎})-r_{+}^2{r}_{-}^{2}I(-{𝐞}_3)=2ab(I(\mathrm{𝟎})-r_{-}^{2}I({𝐞}_1-{𝐞}_3)),$$

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Symbols:

$I(𝐦)$: integral and $r_{\pm }$

Permalink:

http://dlmf.nist.gov/19.34.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

… ►Application of (19.29.4) and (19.29.7) with $\alpha =1$, $a_{\beta }+b_{\beta }t=1-t$, $\delta =3$, and $a_{\gamma }+b_{\gamma }t=1$ yields ►

19.34.5 $$\frac{3c^2}{8\pi ab}M=3R_F(0,r_{+}^2,r_{-}^2)-2r_{-}^2{R}_D(0,r_{+}^2,r_{-}^2),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $M$: mutual inductance and $r_{\pm }$

Permalink:

http://dlmf.nist.gov/19.34.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

…

27: 23.6 Relations to Other Functions

… ►In this subsection $2{\omega }_1$, $2{\omega }_3$ are any pair of generators of the lattice $𝕃$, and the lattice roots $e_1$, $e_2$, $e_3$ are given by (23.3.9). … ►Again, in Equations (23.6.16)–(23.6.26), $2{\omega }_1,2{\omega }_3$ are any pair of generators of the lattice $𝕃$ and $e_1,e_2,e_3$ are given by (23.3.9). … ►Let $z$ be on the perimeter of the rectangle with vertices $0,2{\omega }_1,2{\omega }_1+2{\omega }_3,2{\omega }_3$. … ►Let $z$ be a point of $ℂ$ different from $e_1,e_2,e_3$, and define $w$ by …where the integral is taken along any path from $z$ to $\mathrm{\infty }$ that does not pass through any of $e_1,e_2,e_3$. …

28: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

► ►►►►►►►

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
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$\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$	$\sqrt{2}(\sqrt{3}+1)$	$\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$
$\pi /6$	$\frac{1}{2}$	$\frac{1}{2}\sqrt{3}$	$\frac{1}{3}\sqrt{3}$	$2$	$\frac{2}{3}\sqrt{3}$	$\sqrt{3}$
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$\pi /3$	$\frac{1}{2}\sqrt{3}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{2}{3}\sqrt{3}$	$2$	$\frac{1}{3}\sqrt{3}$
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$2\pi /3$	$\frac{1}{2}\sqrt{3}$	$-\frac{1}{2}$	$-\sqrt{3}$	$\frac{2}{3}\sqrt{3}$	$-2$	$-\frac{1}{3}\sqrt{3}$
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$5\pi /6$	$\frac{1}{2}$	$-\frac{1}{2}\sqrt{3}$	$-\frac{1}{3}\sqrt{3}$	$2$	$-\frac{2}{3}\sqrt{3}$	$-\sqrt{3}$
…

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29: 4.24 Inverse Trigonometric Functions: Further Properties

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4.24.1 $$\arcsin z=z+\frac{1}{2}\frac{z^3}{3}+\frac{1\cdot 3}{2\cdot 4}\frac{z^5}{5}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{z^7}{7}+\mathrm{\cdots },$$ $|z|\le 1$.

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Symbols:

$\arcsin z$: arcsine function and $z$: complex variable

A&S Ref:

4.4.40 (where the constraint is .)

Permalink:

http://dlmf.nist.gov/4.24.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(i), §4.24 and Ch.4

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4.24.3 $$\arctan z=z-\frac{z^3}{3}+\frac{z^5}{5}-\frac{z^7}{7}+\mathrm{\cdots },$$ $\left|z\right|\le 1$, $z\ne \pm \mathrm{i}$.

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Symbols:

$\mathrm{i}$: imaginary unit, $\arctan z$: arctangent function and $z$: complex variable

A&S Ref:

4.4.42

Referenced by:

§3.10(ii), §4.45(i), §4.45(i)

Permalink:

http://dlmf.nist.gov/4.24.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(i), §4.24 and Ch.4

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4.24.5 $$\arctan z=\frac{z}{z^2+1}\left(1+\frac{2}{3}\frac{z^2}{1+z^2}+\frac{2\cdot 4}{3\cdot 5}{\left(\frac{z^2}{1+z^2}\right)}^2+\mathrm{\cdots }\right),$$ $\mathrm{\Re }\left(z^2\right)>-\frac{1}{2}$,

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Symbols:

$\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.4.42 (has an error in the conditions on $z$.)

Permalink:

http://dlmf.nist.gov/4.24.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(i), §4.24 and Ch.4

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4.24.7 $$\frac{d}{dz}\arcsin z={(1-z^2)}^{-1/2},$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\arcsin z$: arcsine function and $z$: complex variable

A&S Ref:

4.4.52

Permalink:

http://dlmf.nist.gov/4.24.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(ii), §4.24 and Ch.4

►

4.24.8 $$\frac{d}{dz}\arccos z=-{(1-z^2)}^{-1/2},$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\arccos z$: arccosine function and $z$: complex variable

A&S Ref:

4.4.53

Permalink:

http://dlmf.nist.gov/4.24.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(ii), §4.24 and Ch.4

…

30: 9.19 Approximations

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Moshier (1989, §6.14) provides minimax rational approximations for calculating $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$. They are in terms of the variable $\zeta$, where $\zeta =\frac{2}{3}{x}^{3/2}$ when $x$ is positive, $\zeta =\frac{2}{3}{(-x)}^{3/2}$ when $x$ is negative, and $\zeta =0$ when $x=0$. The approximations apply when , that is, when or . The precision in the coefficients is 21S.

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Németh (1992, Chapter 8) covers $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$, and integrals ${\int }_0^x\mathrm{Ai}\left(t\right)dt$, ${\int }_0^x\mathrm{Bi}\left(t\right)dt$, ${\int }_0^x{\int }_0^v\mathrm{Ai}\left(t\right)dtdv$, ${\int }_0^x{\int }_0^v\mathrm{Bi}\left(t\right)dtdv$ (see also (9.10.20) and (9.10.21)). The Chebyshev coefficients are given to 15D. Chebyshev coefficients are also given for expansions of the second and higher (real) zeros of $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$, again to 15D.

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