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11: 32.9 Other Elementary Solutions

… ►where $P_{n^2+1}(\zeta )$ and $Q_{n^2}(\zeta )$ are polynomials of degrees $n^2+1$ and $n^2$, respectively, with no common zeros. … ►These are rational solutions in $\zeta =z^{1/2}$ of the form …where $P_{n^2-n+1}(\zeta )$ and $Q_{n^2-n}(\zeta )$ are polynomials of degrees $n^2-n+1$ and $n^2-n$, respectively, with no common zeros. … ► ${\text{P}}_{\text{V}}$, with $\alpha =\beta =0$ and ${\gamma }^2+2\delta =0$, has solutions $w(z)=C\exp \left(\pm \sqrt{-2\delta }z\right)$, with $C$ an arbitrary constant. … ►Dubrovin and Mazzocco (2000) classifies all algebraic solutions for the special case of ${\text{P}}_{\text{VI}}$ with $\beta =\gamma =0$, $\delta =\frac{1}{2}$. …

12: 22.6 Elementary Identities

… ►

22.6.4 $$-k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^2(z,k)=k^2({\mathrm{cd}}^2(z,k)-1)=k_{}^{\prime }{}_{}{}^2(1-{\mathrm{nd}}^2(z,k)).$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.9.2

Permalink:

http://dlmf.nist.gov/22.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(i), §22.6 and Ch.22

… ►

22.6.7 $$\mathrm{dn}(2z,k)=\frac{{\mathrm{dn}}^2(z,k)-k^2{\mathrm{sn}}^2(z,k){\mathrm{cn}}^2(z,k)}{1-k^2{\mathrm{sn}}^4(z,k)}=\frac{{\mathrm{dn}}^4(z,k)+k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sn}}^4(z,k)}{1-k^2{\mathrm{sn}}^4(z,k)}.$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.18.3

Referenced by:

§22.6(ii), Erratum (V1.0.7) for Equation (22.6.7)

Permalink:

http://dlmf.nist.gov/22.6.E7

Encodings:

pMML, png, TeX

Errata (effective with 1.0.7):

Originally the term $k^2{\mathrm{sn}}^2(z,k){\mathrm{cn}}^2(z,k)$ was given incorrectly as $k^2{\mathrm{sn}}^2(z,k){\mathrm{dn}}^2(z,k)$.

Reported 2014-02-28 by Svante Janson

See also:

Annotations for §22.6(ii), §22.6 and Ch.22

►

22.6.8 $$\mathrm{cd}(2z,k)=\frac{{\mathrm{cd}}^2(z,k)-k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^2(z,k){\mathrm{nd}}^2(z,k)}{1+k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^4(z,k)},$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(ii), §22.6 and Ch.22

… ►

22.6.10 $$\mathrm{nd}(2z,k)=\frac{{\mathrm{nd}}^2(z,k)+k^2{\mathrm{sd}}^2(z,k){\mathrm{cd}}^2(z,k)}{1+k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^4(z,k)},$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(ii), §22.6 and Ch.22

… ►

22.6.20 $${\mathrm{cn}}^2(\frac{1}{2}z,k)=\frac{-k_{}^{\prime }{}_{}{}^2+\mathrm{dn}(z,k)+k^2\mathrm{cn}(z,k)}{k^2(1+\mathrm{cn}(z,k))}=\frac{k_{}^{\prime }{}_{}{}^2(1-\mathrm{dn}(z,k))}{k^2(\mathrm{dn}(z,k)-\mathrm{cn}(z,k))}=\frac{k_{}^{\prime }{}_{}{}^2(1+\mathrm{cn}(z,k))}{k_{}^{\prime }{}_{}{}^2+\mathrm{dn}(z,k)-k^2\mathrm{cn}(z,k)},$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.19.2

Permalink:

http://dlmf.nist.gov/22.6.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(iii), §22.6 and Ch.22

…

13: Staff

… ►

Frank W. J. Olver, University of Maryland and NIST, Chaps. 1, 2, 4, 9, 10

… ►

Ian J. Thompson, Lawrence Livermore National Laboratory, Chap. 33

… ►

Roderick S. C. Wong, City University of Hong Kong, Chaps. 1, 2, 18

… ►

Ian J. Thompson, Lawrence Livermore National Laboratory, for Chap. 33

… ►

Roderick S. C. Wong, City University of Hong Kong, for Chaps. 2, 18

…

14: 18.39 Applications in the Physical Sciences

… ►The spectrum is mixed, as in §1.18(viii), the positive energy, non-$L^2$, scattering states are the subject of Chapter 33. … ►Namely for fixed $l$ the infinite set labeled by $p$ describe only the $L^2$ bound states for that single $l$, omitting the continuum briefly mentioned below, and which is the subject of Chapter 33, and so an unusual example of the mixed spectra of §1.18(viii). … ►This is also the normalization and notation of Chapter 33 for $Z=1$ , and the notation of Weinberg (2013, Chapter 2). … ►The positive energy (scattering) eigenfunctions for the above Coulomb problem, with potential $V(r)=-Z{e}^2/r$ are discussed in Chapter 33 for each integer $l$. … ►The bound state $L^2$ eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the $\delta$-function normalized (non-$L^2$) in Chapter 33, where the solutions appear as Whittaker functions. …

15: 10.67 Asymptotic Expansions for Large Argument

… ►

10.67.9 $${\mathrm{ber}}^{2}x+{\mathrm{bei}}^{2}x\sim \frac{{\mathrm{e}}^{x\sqrt{2}}}{2\pi x}\left(1+\frac{1}{4\sqrt{2}}\frac{1}{x}+\frac{1}{64}\frac{1}{x^2}-\frac{33}{256\sqrt{2}}\frac{1}{x^3}-\frac{1797}{8192}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

ⓘ

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 and $x$: real variable

A&S Ref:

9.10.27

Permalink:

http://dlmf.nist.gov/10.67.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(ii), §10.67 and Ch.10

►

10.67.10 $$\mathrm{ber}x{\mathrm{bei}}^{\prime }x-{\mathrm{ber}}^{\prime }x\mathrm{bei}x\sim \frac{{\mathrm{e}}^{x\sqrt{2}}}{2\pi x}\left(\frac{1}{\sqrt{2}}+\frac{1}{8}\frac{1}{x}+\frac{9}{64\sqrt{2}}\frac{1}{x^2}+\frac{39}{512}\frac{1}{x^3}+\frac{75}{8192\sqrt{2}}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

ⓘ

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 and $x$: real variable

A&S Ref:

9.10.28

Permalink:

http://dlmf.nist.gov/10.67.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(ii), §10.67 and Ch.10

… ►

10.67.12 $${\left({\mathrm{ber}}^{\prime }x\right)}^2+{\left({\mathrm{bei}}^{\prime }x\right)}^2\sim \frac{{\mathrm{e}}^{x\sqrt{2}}}{2\pi x}\left(1-\frac{3}{4\sqrt{2}}\frac{1}{x}+\frac{9}{64}\frac{1}{x^2}+\frac{75}{256\sqrt{2}}\frac{1}{x^3}+\frac{2475}{8192}\frac{1}{x^4}+\mathrm{\cdots }\right).$$

ⓘ

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 and $x$: real variable

A&S Ref:

9.10.30

Permalink:

http://dlmf.nist.gov/10.67.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(ii), §10.67 and Ch.10

►

10.67.13 $${\ker }^{2}x+{\mathrm{kei}}^{2}x\sim \frac{\pi }{2x}{\mathrm{e}}^{-x\sqrt{2}}\left(1-\frac{1}{4\sqrt{2}}\frac{1}{x}+\frac{1}{64}\frac{1}{x^2}+\frac{33}{256\sqrt{2}}\frac{1}{x^3}-\frac{1797}{8192}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

ⓘ

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 and $x$: real variable

A&S Ref:

9.10.31

Permalink:

http://dlmf.nist.gov/10.67.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(ii), §10.67 and Ch.10

… ►

10.67.16 $${\left({\ker }^{\prime }x\right)}^2+{\left({\mathrm{kei}}^{\prime }x\right)}^2\sim \frac{\pi }{2x}{\mathrm{e}}^{-x\sqrt{2}}\left(1+\frac{3}{4\sqrt{2}}\frac{1}{x}+\frac{9}{64}\frac{1}{x^2}-\frac{75}{256\sqrt{2}}\frac{1}{x^3}+\frac{2475}{8192}\frac{1}{x^4}+\mathrm{\cdots }\right).$$

ⓘ

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 and $x$: real variable

A&S Ref:

9.10.34

Permalink:

http://dlmf.nist.gov/10.67.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(ii), §10.67 and Ch.10

16: 4.17 Special Values and Limits

… ►

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

► ►►►►►►

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

►

… ►

4.17.3 $$\underset{z\to 0}{lim}\frac{1-\cos z}{z^2}=\frac{1}{2}.$$

ⓘ

Symbols:

$\cos z$: cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

17: 34.5 Basic Properties: $\mathit{6}j$ Symbol

… ►If any lower argument in a $\mathit{6}j$ symbol is $0$, $\frac{1}{2}$, or $1$, then the $\mathit{6}j$ symbol has a simple algebraic form. … ►

34.5.5

ⓘ

Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $J$: sum

Permalink:

http://dlmf.nist.gov/34.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.6

ⓘ

Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $J$: sum

Permalink:

http://dlmf.nist.gov/34.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.7

ⓘ

Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $J$: sum

Permalink:

http://dlmf.nist.gov/34.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(i), §34.5 and Ch.34

… ►

34.5.13 $$E(j)={\left((j^2-{(j_2-j_3)}^2)({(j_2+j_3+1)}^2-j^2)(j^2-{(l_2-l_3)}^2)({(l_2+l_3+1)}^2-j^2)\right)}^{\frac{1}{2}}.$$

ⓘ

Defines:

$E(j)$ (locally)

Symbols:

$j,j_r$: non-negative integers or non-negative integers plus one half. and $l,l_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(iii), §34.5 and Ch.34

…

18: 28.26 Asymptotic Approximations for Large $q$

… ►

28.26.1 $${\mathrm{Mc}}_m^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fc}}_m(z,h)-\mathrm{i}{\mathrm{Gc}}_m(z,h)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

… ►

28.26.3 $$\varphi =2h\sinh z-\left(m+\frac{1}{2}\right)\arctan \left(\sinh z\right).$$

ⓘ

Symbols:

$\sinh z$: hyperbolic sine function, $\arctan z$: arctangent function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.8 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

►Then as $h\to +\mathrm{\infty }$ with fixed $z$ in $\mathrm{\Re }z>0$ and fixed $s=2m+1$, ►

28.26.4 $${\mathrm{Fc}}_m(z,h)\sim 1+\frac{s}{8h{\cosh }^{2}z}+\frac{1}{2^{11}{h}^2}\left(\frac{s^4+86s^2+105}{{\cosh }^{4}z}-\frac{s^4+22s^2+57}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(-\frac{s^5+14s^3+33s}{{\cosh }^{2}z}-\frac{2s^5+124s^3+1122s}{{\cosh }^{4}z}+\frac{3s^5+290s^3+1627s}{{\cosh }^{6}z}\right)+\mathrm{\cdots },$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.9 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

►

28.26.5 $${\mathrm{Gc}}_m(z,h)\sim \frac{\sinh z}{{\cosh }^{2}z}\left(\frac{s^2+3}{2^{5}h}+\frac{1}{2^9{h}^2}\left(s^3+3s+\frac{4s^3+44s}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(5s^4+34s^2+9-\frac{s^6-47s^4+667s^2+2835}{12{\cosh }^{2}z}+\frac{s^6+505s^4+12139s^2+10395}{12{\cosh }^{4}z}\right)\right)+\mathrm{\cdots }.$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.10 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

…

19: 24.2 Definitions and Generating Functions

… ►

$B_{2n+1}=0$ ,

►

${(-1)}^{n+1}{B}_{2n}>0$ , $n=1,2,\mathrm{\dots }$.

… ►

$E_{2n+1}=0$ ,

►

${(-1)}^n{E}_{2n}>0$ .

… ►

24.2.9 $$E_n=2^n{E}_n\left(\frac{1}{2}\right)=\text{integer},$$

ⓘ

Symbols:

$E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials and $n$: integer

A&S Ref:

23.1.2

Permalink:

http://dlmf.nist.gov/24.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.2(ii), §24.2 and Ch.24

…

20: 23.21 Physical Applications

… ►In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form $(1-x^2)(1-k^2{x}^2)$. The Weierstrass function $\mathrm{\wp }$ plays a similar role for cubic potentials in canonical form $g_3+g_{2}x-4x^3$. … ►where $x,y,z$ are the corresponding Cartesian coordinates and $e_1$, $e_2$, $e_3$ are constants. The Laplacian operator ${\nabla }^2$ (§1.5(ii)) is given by … ►Another form is obtained by identifying $e_1$, $e_2$, $e_3$ as lattice roots (§23.3(i)), and setting …