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41: 32.6 Hamiltonian Structure

… ►The Hamiltonian for ${\text{P}}_{\text{III}}$ is … ►

32.6.17 $$z{q}^{\prime }=2q^{2}p-{\kappa }_{\mathrm{\infty }}z{q}^2-(2{\theta }_0+1)q+{\kappa }_{0}z,$$

ⓘ

Symbols:

$z$: real, $q$: coordinate, $\theta$: parameter, $p$: momentum and ${\kappa }_j$: parameters

Referenced by:

§32.6(iv)

Permalink:

http://dlmf.nist.gov/32.6.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(iv), §32.6 and Ch.32

… ►The Hamiltonian for ${\text{P}}_{\text{III}}$ with $\gamma =0$ is … ►

32.6.33 $$z{q}^{\prime }=2q^{2}p+\theta q-{\kappa }_{0}z,$$

ⓘ

Symbols:

$z$: real, $q$: coordinate, $\theta$: parameter, $p$: momentum and ${\kappa }_j$: parameters

Referenced by:

§32.6(iv)

Permalink:

http://dlmf.nist.gov/32.6.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(iv), §32.6 and Ch.32

►

32.6.34 $$z{p}^{\prime }=-2q{p}^2-\theta p+{\kappa }_{\mathrm{\infty }}z.$$

ⓘ

Symbols:

$z$: real, $q$: coordinate, $\theta$: parameter, $p$: momentum and ${\kappa }_j$: parameters

Referenced by:

§32.6(iv)

Permalink:

http://dlmf.nist.gov/32.6.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(iv), §32.6 and Ch.32

…

42: 36.9 Integral Identities

… ►

36.9.4 $${|{\mathrm{\Psi }}_2\left(x,y\right)|}^2={\int }_0^{\mathrm{\infty }}\left({\mathrm{\Psi }}_1\left(\frac{4u^3+2uy+x}{u^{1/3}}\right)+{\mathrm{\Psi }}_1\left(\frac{4u^3+2uy-x}{u^{1/3}}\right)\right)\frac{du}{u^{1/3}}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\int$: integral, $y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

►

36.9.5 $${|{\mathrm{\Psi }}_2\left(x,y\right)|}^2=2{\int }_0^{\mathrm{\infty }}\cos \left(2xu\right){\mathrm{\Psi }}_1\left(2u^{2/3}(y+2u^2)\right)\frac{du}{u^{1/3}}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

►

36.9.6 $${|{\mathrm{\Psi }}_3\left(x,y,z\right)|}^2=2^{4/5}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{\Psi }}_3\left(2^{4/5}(x+2uy+3u^{2}z+5u^4),0,2^{2/5}(z+10u^2)\right)du.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\int$: integral, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

►

36.9.7 $${|{\mathrm{\Psi }}_3\left(x,y,z\right)|}^2=\frac{2^{7/4}}{5^{1/4}}{\int }_0^{\mathrm{\infty }}\mathrm{\Re }\left({\mathrm{e}}^{2\mathrm{i}u(u^4+z{u}^2+x)}{\mathrm{\Psi }}_2\left(\frac{2^{7/4}}{5^{1/4}}y{u}^{3/4},\sqrt{\frac{2u}{5}}(3z+10u^2)\right)\right)\frac{du}{u^{1/4}}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

►

36.9.8 $${\left|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,z\right)\right|}^2=8{\pi }^2{\left(\frac{2}{9}\right)}^{1/3}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{Ai}\left({\left(\frac{4}{3}\right)}^{1/3}(x+zv+3u^2)\right)\mathrm{Ai}\left({\left(\frac{4}{3}\right)}^{1/3}(y+zu+3v^2)\right)dudv.$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\int$: integral, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

…

43: 31.8 Solutions via Quadratures

… ►For half-odd-integer values of the exponent parameters: … ►

31.8.2 $$w_{\pm }(𝐦;\lambda ;z)=\sqrt{{\mathrm{\Psi }}_{g,N}(\lambda ,z)}\exp \left(\pm \frac{\mathrm{i}\nu (\lambda )}{2}{\int }_{z_0}^z\frac{t^{m_1}{(t-1)}^{m_2}{(t-a)}^{m_3}dt}{{\mathrm{\Psi }}_{g,N}(\lambda ,t)\sqrt{t(t-1)(t-a)}}\right)$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\nu$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\lambda =-4q$ , ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$: polynomial, $g$: degree of $\lambda$ and $N$: degree of $z$

Referenced by:

§31.8

Permalink:

http://dlmf.nist.gov/31.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.8 and Ch.31

… ►The curve $\mathrm{\Gamma }$ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for $m_j\in \mathbb{Z}$. …

44: 31.17 Physical Applications

… ►for the common eigenfunction $\mathrm{\Psi }(𝐱)=\mathrm{\Psi }(x_s,x_t,x_u)$, where $a$ is the coupling parameter of interacting spins. … ►

31.17.2 $$\frac{x_s^2}{z_k}+\frac{x_t^2}{z_k-1}+\frac{x_u^2}{z_k-a}=0,$$ $k=1,2$,

ⓘ

Symbols:

$z$: complex variable, $a$: complex parameter and $x_s$, $x_t$, $x_u$: coordinates

Permalink:

http://dlmf.nist.gov/31.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.17(i), §31.17 and Ch.31

… ►

31.17.4 $$\mathrm{\Psi }(𝐱)={(z_1{z}_2)}^{-s-\frac{1}{4}}{((z_1-1)(z_2-1))}^{-t-\frac{1}{4}}{((z_1-a)(z_2-a))}^{-u-\frac{1}{4}}w(z_1)w(z_2),$$

ⓘ

Symbols:

$z$: complex variable, $a$: complex parameter and $\mathrm{\Psi }(𝐱)$: common eigenfunction

Permalink:

http://dlmf.nist.gov/31.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.17(i), §31.17 and Ch.31

►where $w(z)$ satisfies Heun’s equation (31.2.1) with $a$ as in (31.17.1) and the other parameters given by …

45: 8.18 Asymptotic Expansions of $I_x(a,b)$

… ►

§8.18(i) Large Parameters, Fixed $x$

… ►

§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

… ►

Symmetric Case

… ►

General Case

… ►

Inverse Function

…

46: 36.10 Differential Equations

… ►

36.10.5 $$\frac{{\partial }^4{\mathrm{\Psi }}_3}{{\partial x}^4}-\frac{3}{5}z\frac{{\partial }^2{\mathrm{\Psi }}_3}{{\partial x}^2}-\frac{2\mathrm{i}}{5}y\frac{\partial {\mathrm{\Psi }}_3}{\partial x}+\frac{1}{5}x{\mathrm{\Psi }}_3=0.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(i), §36.10(i), §36.10 and Ch.36

… ►

36.10.13 $$6\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x\partial y}-2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial y}+y{\mathrm{\Psi }}^{(\mathrm{E})}=0,$$

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(iii), §36.10 and Ch.36

… ►

36.10.15 $$3\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{H})}}{{\partial x}^2}+\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial y}-x{\mathrm{\Psi }}^{(\mathrm{H})}=0,$$

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(iii), §36.10 and Ch.36

►

36.10.16 $$3\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{H})}}{{\partial y}^2}+\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x}-y{\mathrm{\Psi }}^{(\mathrm{H})}=0.$$

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(iii), §36.10 and Ch.36

… ►

36.10.18 $$\mathrm{i}\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial z}=\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x\partial y}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(iv), §36.10 and Ch.36

…

47: 28.15 Expansions for Small $q$

… ►

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

ⓘ

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

… ►

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

ⓘ

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

… ►

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

ⓘ

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

…

48: 29.4 Graphics

…

49: 12.1 Special Notation

… ► ►►►

$x,y$	real variables.
…
$a,\nu$	real or complex parameters.
…

…

50: 12.8 Recurrence Relations and Derivatives

… ►

12.8.1 $$zU(a,z)-U(a-1,z)+(a+\frac{1}{2})U(a+1,z)=0,$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.4

Referenced by:

§12.8(i)

Permalink:

http://dlmf.nist.gov/12.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

►

12.8.2 $$U^{\prime }(a,z)+\frac{1}{2}zU(a,z)+(a+\frac{1}{2})U(a+1,z)=0,$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.1

Permalink:

http://dlmf.nist.gov/12.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

►

12.8.3 $$U^{\prime }(a,z)-\frac{1}{2}zU(a,z)+U(a-1,z)=0,$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.2

Permalink:

http://dlmf.nist.gov/12.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

►

12.8.4 $$2U^{\prime }(a,z)+U(a-1,z)+(a+\frac{1}{2})U(a+1,z)=0.$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.3

Referenced by:

§12.8(i)

Permalink:

http://dlmf.nist.gov/12.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

… ►

12.8.5 $$zV(a,z)-V(a+1,z)+(a-\frac{1}{2})V(a-1,z)=0,$$

ⓘ

Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.8

Permalink:

http://dlmf.nist.gov/12.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

…