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11: 4.43 Cubic Equations

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$A={\left(-\frac{4}{3}p\right)}^{1/2},$

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$B={\left(\frac{4}{3}p\right)}^{1/2}.$

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$A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=4q/A^3$, when $4p^3+27q^2\le 0$.

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$A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=-4q/A^3$, when , , and $4p^3+27q^2>0$.

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$B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=-4q/B^3$, when $p>0$.

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12: 12.7 Relations to Other Functions

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12.7.8 $$U(-2,z)=\frac{z^{5/2}}{4\sqrt{2\pi }}\left(2K_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right)+3K_{\frac{3}{4}}\left(\frac{1}{4}{z}^2\right)-K_{\frac{5}{4}}\left(\frac{1}{4}{z}^2\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.11 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

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12.7.9 $$U(-1,z)=\frac{z^{3/2}}{2\sqrt{2\pi }}\left(K_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right)+K_{\frac{3}{4}}\left(\frac{1}{4}{z}^2\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.10 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

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12.7.11 $$U(1,z)=\frac{z^{3/2}}{\sqrt{2\pi }}\left(K_{\frac{3}{4}}\left(\frac{1}{4}{z}^2\right)-K_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.3 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

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12.7.12 $$u_1(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}M(\frac{1}{2}a+\frac{1}{4},\frac{1}{2},\frac{1}{2}{z}^2)={\mathrm{e}}^{\frac{1}{4}{z}^2}M(-\frac{1}{2}a+\frac{1}{4},\frac{1}{2},-\frac{1}{2}{z}^2),$$

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Defines:

$u_1(a,z)$: solution (locally)

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.2 (modification of)

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iv), §12.7 and Ch.12

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12.7.13 $$u_2(a,z)=z{\mathrm{e}}^{-\frac{1}{4}{z}^2}M(\frac{1}{2}a+\frac{3}{4},\frac{3}{2},\frac{1}{2}{z}^2)=z{\mathrm{e}}^{\frac{1}{4}{z}^2}M(-\frac{1}{2}a+\frac{3}{4},\frac{3}{2},-\frac{1}{2}{z}^2).$$

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Defines:

$u_2(a,z)$: solution (locally)

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.4 (modification of)

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iv), §12.7 and Ch.12

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13: 4.17 Special Values and Limits

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

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$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
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$\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$	$\sqrt{2}(\sqrt{3}+1)$	$\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$
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$\pi /4$	$\frac{1}{2}\sqrt{2}$	$\frac{1}{2}\sqrt{2}$	$1$	$\sqrt{2}$	$\sqrt{2}$	$1$
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$5\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$	$\sqrt{2}(\sqrt{3}-1)$	$\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$
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$7\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$-\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$-(2+\sqrt{3})$	$\sqrt{2}(\sqrt{3}-1)$	$-\sqrt{2}(\sqrt{3}+1)$	$-(2-\sqrt{3})$
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$11\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$-\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$-(2-\sqrt{3})$	$\sqrt{2}(\sqrt{3}+1)$	$-\sqrt{2}(\sqrt{3}-1)$	$-(2+\sqrt{3})$
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14: 14.4 Graphics

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Figure 14.4.1: ${𝖯}_{\nu }^0\left(x\right)$, $\nu =0,\frac{1}{2},1,2,4$. Magnify

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Figure 14.4.2: ${𝖰}_{\nu }^0\left(x\right)$, $\nu =0,\frac{1}{2},1,2,4$. Magnify

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Figure 14.4.3: ${𝖯}_{\nu }^{-1/2}\left(x\right)$, $\nu =0,\frac{1}{2},1,2,4$. Magnify

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Figure 14.4.24: ${𝑸}_0^{\mu }\left(x\right)$, $\mu =0,2,4,8$. Magnify

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Figure 14.4.28: ${𝑸}_1^{\mu }\left(x\right)$, $\mu =0,2,4,8$. Magnify

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15: 19.38 Approximations

… ►Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^2$ with can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. …

16: 29.7 Asymptotic Expansions

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29.7.4 $${\tau }_1=\frac{p}{2^6}({(1+k^2)}^2(p^2+3)-4k^2(p^2+5)).$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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29.7.6 $${\tau }_2=\frac{1}{2^{10}}(1+k^2){(1-k^2)}^2(5p^4+34p^2+9),$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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29.7.7 $${\tau }_3=\frac{p}{2^{14}}({(1+k^2)}^4(33p^4+410p^2+405)-24k^2{(1+k^2)}^2(7p^4+90p^2+95)+16k^4(9p^4+130p^2+173)),$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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29.7.8 $${\tau }_4=\frac{1}{2^{16}}({(1+k^2)}^5(63p^6+1260p^4+2943p^2+486)-8k^2{(1+k^2)}^3(49p^6+1010p^4+2493p^2+432)+16k^4(1+k^2)(35p^6+760p^4+2043p^2+378)).$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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17: 3.4 Differentiation

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$B_{-1}^4=-\frac{1}{6}(4-8t-3t^2+4t^3),$

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$B_1^4=\frac{1}{6}(4+8t-3t^2-4t^3),$

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$$B_0^5=-\frac{1}{12}(4+30t-15t^2-12t^3+5t^4),$$

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$$B_3^5=\frac{1}{120}(4-15t^2+5t^4).$$

… ►For additional formulas involving values of ${\nabla }^{2}u$ and ${\nabla }^{4}u$ on square, triangular, and cubic grids, see Collatz (1960, Table VI, pp. 542–546). …

18: 26.2 Basic Definitions

… ►If, for example, a permutation of the integers 1 through 6 is denoted by $256413$, then the cycles are $\left(1,2,5\right)$, $\left(3,6\right)$, and $\left(4\right)$. …The function $\sigma$ also interchanges 3 and 6, and sends 4 to itself. … ►As an example, $\{1,3,4\}$, $\{2,6\}$, $\{5\}$ is a partition of $\{1,2,3,4,5,6\}$. … ►As an example, $\{1,1,1,2,4,4\}$ is a partition of 13. … ►The example $\{1,1,1,2,4,4\}$ has six parts, three of which equal 1. …

19: 27.15 Chinese Remainder Theorem

… ►Choose four relatively prime moduli $m_1,m_2,m_3$, and $m_4$ of five digits each, for example $2^{16}-3$, $2^{16}-1$, $2^{16}+1$, and $2^{16}+3$. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_1$), (mod $m_2$), (mod $m_3$), and (mod $m_4$), respectively. Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers $a_1(mod{m}_1)$, $a_2(mod{m}_2)$, $a_3(mod{m}_3)$, and $a_4(mod{m}_4)$, where each $a_j$ has no more than five digits. …

20: 28.15 Expansions for Small $q$

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28.15.1 $${\lambda }_{\nu }\left(q\right)={\nu }^2+\frac{1}{2({\nu }^2-1)}{q}^2+\frac{5{\nu }^2+7}{32{({\nu }^2-1)}^3({\nu }^2-4)}{q}^4+\frac{9{\nu }^4+58{\nu }^2+29}{64{({\nu }^2-1)}^5({\nu }^2-4)({\nu }^2-9)}{q}^6+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

A&S Ref:

20.3.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

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28.15.2 $$a-{\nu }^2-\frac{q^2}{a-{(\nu +2)}^2-}\frac{q^2}{a-{(\nu +4)}^2-}\mathrm{\cdots }=\frac{q^2}{a-{(\nu -2)}^2-}\frac{q^2}{a-{(\nu -4)}^2-}\mathrm{\cdots }.$$

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Symbols:

$q=h^2$: parameter, $\nu$: complex parameter and $a$: parameter

Referenced by:

item (e)

Permalink:

http://dlmf.nist.gov/28.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

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28.15.3 $${\mathrm{me}}_{\nu }(z,q)={\mathrm{e}}^{\mathrm{i}\nu z}-\frac{q}{4}\left(\frac{1}{\nu +1}{\mathrm{e}}^{\mathrm{i}(\nu +2)z}-\frac{1}{\nu -1}{\mathrm{e}}^{\mathrm{i}(\nu -2)z}\right)+\frac{q^2}{32}\left(\frac{1}{(\nu +1)(\nu +2)}{\mathrm{e}}^{\mathrm{i}(\nu +4)z}+\frac{1}{(\nu -1)(\nu -2)}{\mathrm{e}}^{\mathrm{i}(\nu -4)z}-\frac{2({\nu }^2+1)}{{({\nu }^2-1)}^2}{\mathrm{e}}^{\mathrm{i}\nu z}\right)+\mathrm{\cdots };$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

20.3.17 (only two terms and without normalization)

Permalink:

http://dlmf.nist.gov/28.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(ii), §28.15 and Ch.28

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