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11: 10.20 Uniform Asymptotic Expansions for Large Order

… ►In the following formulas for the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$, $u_k$, $v_k$ are the constants defined in §9.7(i), and $U_k(p)$, $V_k(p)$ are the polynomials in $p$ of degree $3k$ defined in §10.41(ii). … ►Note: Another way of arranging the above formulas for the coefficients $A_k(\zeta ),B_k(\zeta ),C_k(\zeta )$, and $D_k(\zeta )$ would be by analogy with (12.10.42) and (12.10.46). … ►Each of the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$, $k=0,1,2,\mathrm{\dots }$, is real and infinitely differentiable on the interval . … ►For numerical tables of $\zeta =\zeta (z)$, ${(4\zeta /(1-z^2))}^{\frac{1}{4}}$ and $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$ see Olver (1962, pp. 28–42). … ►The equations of the curved boundaries $D_1{E}_1$ and $D_2{E}_2$ in the $\zeta$-plane are given parametrically by …

12: 29.15 Fourier Series and Chebyshev Series

… ►

29.15.34 $${[D_1,D_3,\mathrm{\dots },D_{2n+1}]}^{\mathrm{T}},$$

ⓘ

Symbols:

$n$: nonnegative integer and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

►

29.15.35 $$\left(1-\frac{1}{2}{k}^2\right)\sum _{p=0}^n{D}_{2p+1}^2+\frac{1}{2}{k}^2\left(\frac{1}{2}{D}_1^2-\sum _{p=0}^{n-1}{D}_{2p+1}{D}_{2p+3}\right)=1,$$

ⓘ

Symbols:

$n$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

… ►

Polynomial ${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)$

►When $\nu =2n+3$, $m=0,1,\mathrm{\dots },n$, the Fourier series (29.6.53) terminates: … ►

29.15.39 $${[D_2,D_4,\mathrm{\dots },D_{2n+2}]}^{\mathrm{T}},$$

ⓘ

Symbols:

$n$: nonnegative integer and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

…

13: 29.6 Fourier Series

… ►

29.6.39 $$({\beta }_0-H)D_1+{\alpha }_0{D}_3=0,$$

ⓘ

Symbols:

${\alpha }_p$ , ${\beta }_p$ , $D_{2p+1}$: coefficents and $H$

Permalink:

http://dlmf.nist.gov/29.6.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iii), §29.6 and Ch.29

►

29.6.40 $${\gamma }_p{D}_{2p-1}+({\beta }_p-H)D_{2p+1}+{\alpha }_p{D}_{2p+3}=0,$$ $p\ge 1$,

ⓘ

Symbols:

$p$: nonnegative integer, ${\alpha }_p$ , ${\beta }_p$ , ${\gamma }_p$ , $D_{2p+1}$: coefficents and $H$

Permalink:

http://dlmf.nist.gov/29.6.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iii), §29.6 and Ch.29

… ►

29.6.42 $$\left(1-\frac{1}{2}{k}^2\right)\sum _{p=0}^{\mathrm{\infty }}{D}_{2p+1}^2+\frac{1}{2}{k}^2\left(\frac{1}{2}{D}_1^2-\sum _{p=0}^{\mathrm{\infty }}{D}_{2p+1}{D}_{2p+3}\right)=1,$$

ⓘ

Symbols:

$p$: nonnegative integer, $k$: real parameter and $D_{2p+1}$: coefficents

Permalink:

http://dlmf.nist.gov/29.6.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iii), §29.6 and Ch.29

… ►

29.6.54 $$({\beta }_0-H)D_2+{\alpha }_0{D}_4=0,$$

ⓘ

Symbols:

${\alpha }_p$ , ${\beta }_p$ , $H$ and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.6.E54

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iv), §29.6 and Ch.29

… ►

29.6.59 $$\underset{p\to \mathrm{\infty }}{lim}\frac{D_{2p+2}}{D_{2p}}=\frac{k^2}{{(1+k^{\prime })}^2},$$ $\nu \ne 2n+3$, or $\nu =2n+3$ and $m>n$,

ⓘ

Symbols:

$m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.6.E59

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iv), §29.6 and Ch.29

…

14: 10.13 Other Differential Equations

… ►

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

ⓘ

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

… ►In (10.13.9)–(10.13.11) ${𝒞}_{\nu }\left(z\right)$, ${𝒟}_{\mu }(z)$ are any cylinder functions of orders $\nu ,\mu$, respectively, and $\mathit{\vartheta }=z(d/dz)$. ►

10.13.9 $$z^2{w}^{\prime \prime \prime }+3z{w}^{\prime \prime }+(4z^2+1-4{\nu }^2)w^{\prime }+4zw=0,$$ $w={𝒞}_{\nu }\left(z\right){𝒟}_{\nu }(z)$,

ⓘ

Symbols:

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

A&S Ref:

9.1.58 (modified)

Referenced by:

§10.13, §10.36

Permalink:

http://dlmf.nist.gov/10.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

►

10.13.10 $$z^3{w}^{\prime \prime \prime }+z(4z^2+1-4{\nu }^2)w^{\prime }+(4{\nu }^2-1)w=0,$$ $w=z{𝒞}_{\nu }\left(z\right){𝒟}_{\nu }(z)$,

ⓘ

Symbols:

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

A&S Ref:

9.1.59

Permalink:

http://dlmf.nist.gov/10.13.E10

Encodings:

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See also:

Annotations for §10.13 and Ch.10

►

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

ⓘ

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:

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See also:

Annotations for §10.13 and Ch.10

…

15: 19.27 Asymptotic Approximations and Expansions

… ►

§19.27(iv) $R_D(x,y,z)$

… ►

19.27.7 $$R_D(x,y,z)=\frac{3}{2z^{3/2}}\left(\ln \left(\frac{8z}{a+g}\right)-2\right)\left(1+O\left(\frac{a}{z}\right)\right),$$ $a/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.27.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

►

19.27.8 $$R_D(x,y,z)=\frac{3}{\sqrt{xyz}}-\frac{6}{xy}{R}_G(x,y,0)\left(1+O\left(\frac{z}{g}\right)\right),$$ $z/g\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $g$

Referenced by:

§19.20(iv), §19.29(iii)

Permalink:

http://dlmf.nist.gov/19.27.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

►

19.27.9 $$R_D(x,y,z)=\frac{3}{\sqrt{xz}(\sqrt{y}+\sqrt{z})}\left(1+O\left(\frac{b}{x}\ln \frac{x}{b}\right)\right),$$ $b/x\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\ln z$: principal branch of logarithm function and $b$

Permalink:

http://dlmf.nist.gov/19.27.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

►

19.27.10 $$R_D(x,y,z)=R_D(0,y,z)-\frac{3\sqrt{x}}{hz}\left(1+O\left(\sqrt{\frac{x}{h}}\right)\right),$$ $x/h\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables and $h$

Permalink:

http://dlmf.nist.gov/19.27.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

…

16: 19.20 Special Cases

… ►

§19.20(iv) $R_D(x,y,z)$

►

$$R_D(x,x,x)=x^{-3/2},$$

►

$$R_D(\lambda x,\lambda y,\lambda z)={\lambda }^{-3/2}{R}_D(x,y,z),$$

►

$$R_D(0,y,y)=\frac{3}{4}\pi y^{-3/2},$$

… ►

19.20.22 $${\int }_0^1\frac{t^{2}dt}{\sqrt{1-t^4}}=\frac{1}{3}{R}_D(0,2,1)=\frac{{\left(\mathrm{\Gamma }\left(\frac{3}{4}\right)\right)}^2}{{(2\pi )}^{1/2}}=\mathrm{0.59907\hspace{0.33em}01173\hspace{0.33em}67796\hspace{0.33em}10371}\mathrm{\dots }.$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ and $\int$: integral

Notes:

For more digits see OEIS Sequence A085566; see also Sloane (2003).

Referenced by:

§19.20(iv), §19.21(i), Figure 19.3.6, Figure 19.3.6, Figure 19.3.6

Permalink:

http://dlmf.nist.gov/19.20.E22

Encodings:

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See also:

Annotations for §19.20(iv), §19.20 and Ch.19

…

17: 27.2 Functions

… ►

27.2.9 $$d\left(n\right)=\sum _{d|n}1$$

ⓘ

Symbols:

$d_k\left(n\right)$: divisor function, $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.3 I.A

Permalink:

http://dlmf.nist.gov/27.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. …Note that ${\sigma }_0\left(n\right)=d\left(n\right)$. … ►Table 27.2.2 tabulates the Euler totient function $\varphi \left(n\right)$, the divisor function $d\left(n\right)$ ($={\sigma }_0(n)$), and the sum of the divisors $\sigma (n)$ ($={\sigma }_1(n)$), for $n=1(1)52$. … ►

Table 27.2.2: Functions related to division.

► ►►

$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
…

►

18: 19.4 Derivatives and Differential Equations

… ►Let $D_k=\partial /\partial k$. Then ►

19.4.8 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+(1-3k^2)D_k-k)F(\varphi ,k)=\frac{-k\sin \varphi \cos \varphi }{{(1-k^2{\sin }^2\varphi )}^{3/2}},$$

ⓘ

Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Referenced by:

§19.4(ii)

Permalink:

http://dlmf.nist.gov/19.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

►

19.4.9 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+k_{}^{\prime }{}_{}{}^2{D}_k+k)E(\varphi ,k)=\frac{k\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}.$$

ⓘ

Symbols:

$\cos z$: cosine function, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Permalink:

http://dlmf.nist.gov/19.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

…

19: 12.7 Relations to Other Functions

… ►

12.7.1 $$U(-\frac{1}{2},z)=D_0\left(z\right)={\mathrm{e}}^{-\frac{1}{4}{z}^2},$$

ⓘ

Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/12.7.E1

Encodings:

pMML, png, TeX

See also:

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

►

12.7.2 $$U(-n-\frac{1}{2},z)=D_n\left(z\right)={\mathrm{e}}^{-\frac{1}{4}{z}^2}{\mathrm{𝐻𝑒}}_n\left(z\right)=2^{-n/2}{\mathrm{e}}^{-\frac{1}{4}{z}^2}{H}_n\left(z/\sqrt{2}\right),$$ $n=0,1,2,\mathrm{\dots }$ ,

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $D_{\nu }\left(z\right)$: parabolic cylinder function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

19.13.1 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E2

Encodings:

pMML, png, TeX

See also:

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

… ►

12.7.5 $$U(\frac{1}{2},z)=D_{-1}\left(z\right)=\sqrt{\frac{1}{2}\pi }{\mathrm{e}}^{\frac{1}{4}{z}^2}\mathrm{erfc}\left(z/\sqrt{2}\right),$$

ⓘ

Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Referenced by:

§12.11(ii)

Permalink:

http://dlmf.nist.gov/12.7.E5

Encodings:

pMML, png, TeX

See also:

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

►

12.7.6 $$U(n+\frac{1}{2},z)=D_{-n-1}\left(z\right)=\sqrt{\frac{\pi }{2}}\frac{{(-1)}^n}{n!}{\mathrm{e}}^{-\frac{1}{4}{z}^2}\frac{d^n\left({\mathrm{e}}^{\frac{1}{2}{z}^2}\mathrm{erfc}\left(z/\sqrt{2}\right)\right)}{{dz}^n},$$ $n=0,1,2,\mathrm{\dots }$,

ⓘ

Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $U(a,z)$: parabolic cylinder function, $z$: complex variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/12.7.E6

Encodings:

pMML, png, TeX

See also:

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

… ►For these, the corresponding results for $U(a,z)$ with $a=2$, $\pm 3$, $-\frac{1}{2}$, $-\frac{3}{2}$, $-\frac{5}{2}$, and the corresponding results for $V(a,z)$ with $a=0$, $\pm 1$, $\pm 2$, $\pm 3$, $\frac{1}{2}$, $\frac{3}{2}$, $\frac{5}{2}$, see Miller (1955, pp. 42–43 and 77–79). …

20: 19.29 Reduction of General Elliptic Integrals

… ►For example, 3. …where the arguments of the $R_D$ function are, in order, $(a-b)(u-c)$, $(b-c)(a-u)$, $(a-b)(b-c)$. … ►In the cubic case ($h=3$) the basic integrals are … ►If $h=3$, then the recurrence relation (Carlson (1999, (3.5))) has the special case … ►where …