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21: 10.36 Other Differential Equations

… ►The quantity ${\lambda }^2$ in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by $-{\lambda }^2$ if at the same time the symbol $𝒞$ in the given solutions is replaced by $𝒵$. … ►

10.36.1 $$z^2(z^2+{\nu }^2)w^{\prime \prime }+z(z^2+3{\nu }^2)w^{\prime }-\left({(z^2+{\nu }^2)}^2+z^2-{\nu }^2\right)w=0,$$ $w={𝒵}_{\nu }^{\prime }\left(z\right)$,

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Symbols:

${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.36.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

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10.36.2 $$z^2{w}^{\prime \prime }+z(1\pm 2z)w^{\prime }+(\pm z-{\nu }^2)w=0,$$ $w={\mathrm{e}}^{\mp z}{𝒵}_{\nu }\left(z\right)$.

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.41

Permalink:

http://dlmf.nist.gov/10.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

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22: 29.4 Graphics

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Figure 29.4.3: $a_{1.5}^m\left(k^2\right)$, $b_{1.5}^{m+1}\left(k^2\right)$ as functions of $k^2$ for $m=0,1,2$. Magnify

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Figure 29.4.8: $a_{2.5}^m\left(k^2\right)$, $b_{2.5}^{m+1}\left(k^2\right)$ as functions of $k^2$ for $m=0,1,2$. Magnify

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Figure 29.4.12: $b_{\nu }^2\left(k^2\right)$ as a function of $\nu$ and $k^2$. Magnify 3D Help

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Figure 29.4.28: ${\mathrm{𝐸𝑠}}_{1.5}^2(x,k^2)$ as a function of $x$ and $k^2$. Magnify 3D Help

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Figure 29.4.32: ${\mathrm{𝐸𝑠}}_{2.5}^2(x,k^2)$ as a function of $x$ and $k^2$. Magnify 3D Help

23: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►For $k=2$ … ► $M_2$ is the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ cycles of length 1, $a_2$ cycles of length 2, $\mathrm{\dots }$, and $a_n$ cycles of length $n$: …$M_3$ is the number of set partitions of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ subsets of size 1, $a_2$ subsets of size 2, $\mathrm{\dots }$, and $a_n$ subsets of size $n$: …For each $n$ all possible values of $a_1,a_2,\mathrm{\dots },a_n$ are covered. … ►where the summation is over all nonnegative integers $n_1,n_2,\mathrm{\dots },n_k$ such that $n_1+n_2+\mathrm{\cdots }+n_k=n$. …

24: 10.13 Other Differential Equations

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10.13.2 $$w^{\prime \prime }+\left(\frac{{\lambda }^2}{4z}-\frac{{\nu }^2-1}{4z^2}\right)w=0,$$ $w=z^{\frac{1}{2}}{𝒞}_{\nu }\left(\lambda z^{\frac{1}{2}}\right)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.50

Permalink:

http://dlmf.nist.gov/10.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

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10.13.3 $$w^{\prime \prime }+{\lambda }^2{z}^{p-2}w=0,$$ $w=z^{\frac{1}{2}}{𝒞}_{1/p}\left(2\lambda z^{\frac{1}{2}p}/p\right)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function and $z$: complex variable

A&S Ref:

9.1.51

Permalink:

http://dlmf.nist.gov/10.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

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10.13.5 $$z^2{w}^{\prime \prime }+(1-2r)z{w}^{\prime }+({\lambda }^2{q}^2{z}^{2q}+r^2-{\nu }^2{q}^2)w=0,$$ $w=z^r{𝒞}_{\nu }\left(\lambda z^q\right)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.53

Permalink:

http://dlmf.nist.gov/10.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

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10.13.7 $$z^2(z^2-{\nu }^2)w^{\prime \prime }+z(z^2-3{\nu }^2)w^{\prime }+({(z^2-{\nu }^2)}^2-(z^2+{\nu }^2))w=0,$$ $w={𝒞}_{\nu }^{\prime }\left(z\right)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.55

Permalink:

http://dlmf.nist.gov/10.13.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

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10.13.11 $$\left({\mathit{\vartheta }}^4-2({\nu }^2+{\mu }^2){\mathit{\vartheta }}^2+{({\nu }^2-{\mu }^2)}^2\right)w+4z^2(\mathit{\vartheta }+1)(\mathit{\vartheta }+2)w=0,$$ $w={𝒞}_{\nu }\left(z\right){𝒟}_{\mu }(z).$

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter, $\mathit{\vartheta }=z(d/dz)$ and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.57

Referenced by:

§10.13, §10.36

Permalink:

http://dlmf.nist.gov/10.13.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

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25: 18.8 Differential Equations

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Table 18.8.1: Classical OP’s: differential equations $A(x)f^{\prime \prime }(x)+B(x)f^{\prime }(x)+C(x)f(x)+{\lambda }_{n}f(x)=0$.

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#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	${\lambda }_n$
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2	$\begin{array}{c}{\left(\sin \frac{1}{2}x\right)}^{\alpha +\frac{1}{2}}{\left(\cos \frac{1}{2}x\right)}^{\beta +\frac{1}{2}}\\ \times P_n^{(\alpha ,\beta )}\left(\cos x\right)\end{array}$	$1$	$0$	${\displaystyle \frac{\frac{1}{4}-{\alpha }^2}{4{\sin }^2\frac{1}{2}x}}+{\displaystyle \frac{\frac{1}{4}-{\beta }^2}{4{\cos }^2\frac{1}{2}x}}$	${\left(n+\frac{1}{2}(\alpha +\beta +1)\right)}^2$
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5	$T_n\left(x\right)$	$1-x^2$	$-x$	$0$	$n^2$
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9	${\mathrm{e}}^{-\frac{1}{2}{x}^2}{x}^{\alpha +\frac{1}{2}}{L}_n^{(\alpha )}\left(x^2\right)$	$1$	$0$	$-x^2+(\frac{1}{4}-{\alpha }^2)x^{-2}$	$4n+2\alpha +2$
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12	$H_n\left(x\right)$	$1$	$-2x$	$0$	$2n$
13	${\mathrm{e}}^{-\frac{1}{2}{x}^2}{H}_n\left(x\right)$	$1$	$0$	$-x^2$	$2n+1$
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26: 22.7 Landen Transformations

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$k_2={\displaystyle \frac{2\sqrt{k}}{1+k}},$

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$k_2^{\prime }={\displaystyle \frac{1-k}{1+k}},$

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22.7.6 $$\mathrm{sn}(z,k)=\frac{(1+k_2^{\prime })\mathrm{sn}(z/(1+k_2^{\prime }),k_2)\mathrm{cn}(z/(1+k_2^{\prime }),k_2)}{\mathrm{dn}(z/(1+k_2^{\prime }),k_2)},$$

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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, $k_2$: change of variable and $k_2^{\prime }$: change of variable

A&S Ref:

16.14.2

Permalink:

http://dlmf.nist.gov/22.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.7(ii), §22.7 and Ch.22

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22.7.7 $$\mathrm{cn}(z,k)=\frac{(1+k_2^{\prime })({\mathrm{dn}}^2(z/(1+k_2^{\prime }),k_2)-k_2^{\prime })}{k_2^2\mathrm{dn}(z/(1+k_2^{\prime }),k_2)},$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus, $k_2$: change of variable and $k_2^{\prime }$: change of variable

A&S Ref:

16.14.3

Permalink:

http://dlmf.nist.gov/22.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.7(ii), §22.7 and Ch.22

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22.7.8 $$\mathrm{dn}(z,k)=\frac{(1-k_2^{\prime })({\mathrm{dn}}^2(z/(1+k_2^{\prime }),k_2)+k_2^{\prime })}{k_2^2\mathrm{dn}(z/(1+k_2^{\prime }),k_2)}.$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus, $k_2$: change of variable and $k_2^{\prime }$: change of variable

A&S Ref:

16.14.4

Permalink:

http://dlmf.nist.gov/22.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.7(ii), §22.7 and Ch.22

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27: 29.21 Tables

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Ince (1940a) tabulates the eigenvalues $a_{\nu }^m\left(k^2\right)$, $b_{\nu }^{m+1}\left(k^2\right)$ (with $a_{\nu }^{2m+1}$ and $b_{\nu }^{2m+1}$ interchanged) for $k^2=0.1,0.5,0.9$, $\nu =-\frac{1}{2},0(1)25$, and $m=0,1,2,3$. Precision is 4D.

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Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^2=0.1(.1)0.9$, $n=1(1)30$. Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.

28: 29.3 Definitions and Basic Properties

… ►They are denoted by $a_{\nu }^{2m}\left(k^2\right)$, $a_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+2}\left(k^2\right)$, where $m=0,1,2,\mathrm{\dots }$; see Table 29.3.1. … ►The quantity $H=2a_{\nu }^{2m+1}\left(k^2\right)-\nu (\nu +1)k^2$ satisfies equation (29.3.10) with … ►The quantity $H=2b_{\nu }^{2m+1}\left(k^2\right)-\nu (\nu +1)k^2$ satisfies equation (29.3.10) with … ►The quantity $H=2b_{\nu }^{2m+2}\left(k^2\right)-\nu (\nu +1)k^2$ satisfies equation (29.3.10) with … ►The eigenfunctions corresponding to the eigenvalues of §29.3(i) are denoted by ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$, ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$. …

29: 10.49 Explicit Formulas

… ►Again, with $a_k(n+\frac{1}{2})$ as in (10.49.1), … ► ${\sum }_{k=0}^n{a}_k(n+\frac{1}{2})z^{n-k}$ is sometimes called the Bessel polynomial of degree $n$. … ►

${𝗃}_2^2\left(z\right)+{𝗒}_2^2\left(z\right)=z^{-2}+3z^{-4}+9z^{-6}.$

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${\left({𝗂}_0^{(1)}\left(z\right)\right)}^2-{\left({𝗂}_0^{(2)}\left(z\right)\right)}^2=-z^{-2},$

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${\left({𝗂}_2^{(1)}\left(z\right)\right)}^2-{\left({𝗂}_2^{(2)}\left(z\right)\right)}^2=-z^{-2}+3z^{-4}-9z^{-6}.$

30: 24.9 Inequalities

… ►Except where otherwise noted, the inequalities in this section hold for $n=1,2,\mathrm{\dots }$. … ►(24.9.3)–(24.9.5) hold for $\frac{1}{2}>x>0$. … ►

24.9.5 $$\frac{4(2n-1)!}{{\pi }^{2n}}\frac{2^{2n}-1}{2^{2n}-2}>{(-1)}^n{E}_{2n-1}\left(x\right)>0.$$

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: integer and $x$: real or complex

A&S Ref:

23.1.14

Referenced by:

§24.9, §24.9

Permalink:

http://dlmf.nist.gov/24.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

►(24.9.6)–(24.9.7) hold for $n=2,3,\mathrm{\dots }$. … ►

24.9.8 $$\frac{2(2n)!}{{(2\pi )}^{2n}}\frac{1}{1-2^{\beta -2n}}\ge {(-1)}^{n+1}{B}_{2n}\ge \frac{2(2n)!}{{(2\pi )}^{2n}}\frac{1}{1-2^{-2n}}$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: integer and $\beta$: quantity

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

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