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1: 4.17 Special Values and Limits

… ►

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

► ►►►►►

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
…
$\pi /6$	$\frac{1}{2}$	$\frac{1}{2}\sqrt{3}$	$\frac{1}{3}\sqrt{3}$	$2$	$\frac{2}{3}\sqrt{3}$	$\sqrt{3}$
…
$\pi /3$	$\frac{1}{2}\sqrt{3}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{2}{3}\sqrt{3}$	$2$	$\frac{1}{3}\sqrt{3}$
…
$2\pi /3$	$\frac{1}{2}\sqrt{3}$	$-\frac{1}{2}$	$-\sqrt{3}$	$\frac{2}{3}\sqrt{3}$	$-2$	$-\frac{1}{3}\sqrt{3}$
…

►

… ►

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

2: 24.2 Definitions and Generating Functions

… ►

Table 24.2.5: Coefficients $b_{n,k}$ of the Bernoulli polynomials $B_n\left(x\right)={\sum }_{k=0}^n{b}_{n,k}{x}^k$.

► ►►►►►

	$k$
…
$5$	$0$	$-\frac{1}{6}$	$0$	$\frac{5}{3}$	$-\frac{5}{2}$	$1$
$6$	$\frac{1}{42}$	$0$	$-\frac{1}{2}$	$0$	$\frac{5}{2}$	$-3$	$1$
…
$14$	$\frac{7}{6}$	$0$	$-\frac{691}{30}$	$0$	$\frac{455}{6}$	$0$	$-\frac{1001}{10}$	$0$	$\frac{143}{2}$	$0$	$-\frac{1001}{30}$	$0$	$\frac{91}{6}$	$-7$	$1$
…

►

►

Table 24.2.6: Coefficients $e_{n,k}$ of the Euler polynomials $E_n\left(x\right)={\sum }_{k=0}^n{e}_{n,k}{x}^k$.

► ►►►►

	$k$
…
$5$	$-\frac{1}{2}$	$0$	$\frac{5}{2}$	$0$	$-\frac{5}{2}$	$1$
…
$13$	$-\frac{5461}{2}$	$0$	$\frac{26949}{2}$	$0$	$-\frac{22165}{2}$	$0$	$\frac{7293}{2}$	$0$	$-\frac{1287}{2}$	$0$	$\frac{143}{2}$	$0$	$-\frac{13}{2}$	$1$
…

►

3: 12.7 Relations to Other Functions

… ►

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

ⓘ

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

… ►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). … ►

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

ⓘ

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

►

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

ⓘ

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

… ►

12.7.14 $$U(a,z)=2^{-\frac{1}{4}-\frac{1}{2}a}{\mathrm{e}}^{-\frac{1}{4}{z}^2}U(\frac{1}{2}a+\frac{1}{4},\frac{1}{2},\frac{1}{2}{z}^2)=2^{-\frac{3}{4}-\frac{1}{2}a}z{\mathrm{e}}^{-\frac{1}{4}{z}^2}U(\frac{1}{2}a+\frac{3}{4},\frac{3}{2},\frac{1}{2}{z}^2)=2^{-\frac{1}{2}a}{z}^{-\frac{1}{2}}{W}_{-\frac{1}{2}a,\pm \frac{1}{4}}\left(\frac{1}{2}{z}^2\right).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.12.2 (modification of)

Referenced by:

§12.12, §12.20, §12.7(iv), §12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.7.E14

Encodings:

pMML, png, TeX

See also:

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

…

4: 26.3 Lattice Paths: Binomial Coefficients

… ► $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ is the number of ways of choosing $n$ objects from a collection of $m$ distinct objects without regard to order. $\left(\genfrac{}{}{0.0pt}{}{m+n}{n}\right)$ is the number of lattice paths from $(0,0)$ to $(m,n)$. …The number of lattice paths from $(0,0)$ to $(m,n)$, $m\le n$, that stay on or above the line $y=x$ is $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)-\left(\genfrac{}{}{0.0pt}{}{m+n}{m-1}\right).$ … ►For numerical values of $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ and $\left(\genfrac{}{}{0.0pt}{}{m+n}{n}\right)$ see Tables 26.3.1 and 26.3.2. ►

Table 26.3.1: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$.

► ►►

$m$	$n$
	…

►

…

5: 13.15 Recurrence Relations and Derivatives

… ►

13.15.3 $$(\kappa -\mu -\frac{1}{2})M_{\kappa -\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)+(1+2\mu )\sqrt{z}{M}_{\kappa ,\mu }\left(z\right)-(\kappa +\mu +\frac{1}{2})M_{\kappa +\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)=0,$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

… ►

13.15.12 $$(\kappa -\mu -\frac{1}{2})\sqrt{z}{W}_{\kappa -\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)+2\mu W_{\kappa ,\mu }\left(z\right)-(\kappa +\mu -\frac{1}{2})\sqrt{z}{W}_{\kappa -\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)=0,$$

ⓘ

Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

… ►

13.15.16 $$\frac{d^n}{{dz}^n}\left({\mathrm{e}}^{\frac{1}{2}z}{z}^{-\mu -\frac{1}{2}}{M}_{\kappa ,\mu }\left(z\right)\right)=\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_n}{{\left(1+2\mu \right)}_n}{\mathrm{e}}^{\frac{1}{2}z}{z}^{-\mu -\frac{1}{2}(n+1)}{M}_{\kappa -\frac{1}{2}n,\mu +\frac{1}{2}n}\left(z\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(ii), §13.15 and Ch.13

… ►

13.15.19 $$\frac{d^n}{{dz}^n}\left({\mathrm{e}}^{-\frac{1}{2}z}{z}^{-\mu -\frac{1}{2}}{M}_{\kappa ,\mu }\left(z\right)\right)={(-1)}^n\frac{{\left(\frac{1}{2}+\mu +\kappa \right)}_n}{{\left(1+2\mu \right)}_n}{\mathrm{e}}^{-\frac{1}{2}z}{z}^{-\mu -\frac{1}{2}(n+1)}{M}_{\kappa +\frac{1}{2}n,\mu +\frac{1}{2}n}\left(z\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(ii), §13.15 and Ch.13

… ►

13.15.21 $$\frac{d^n}{{dz}^n}\left({\mathrm{e}}^{\frac{1}{2}z}{z}^{-\mu -\frac{1}{2}}{W}_{\kappa ,\mu }\left(z\right)\right)={(-1)}^n{\left(\frac{1}{2}+\mu -\kappa \right)}_n{\mathrm{e}}^{\frac{1}{2}z}{z}^{-\mu -\frac{1}{2}(n+1)}{W}_{\kappa -\frac{1}{2}n,\mu +\frac{1}{2}n}\left(z\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(ii), §13.15 and Ch.13

…

6: 12.13 Sums

… ►

12.13.2 $$U(a,x+y)={\mathrm{e}}^{-\frac{1}{2}xy-\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{-a-\frac{1}{2}}{m}\right)y^{m}U(a+m,x),$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

►

12.13.3 $$V(a,x+y)={\mathrm{e}}^{\frac{1}{2}xy+\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{a-\frac{1}{2}}{m}\right)y^{m}V(a-m,x),$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $V(a,z)$: parabolic cylinder function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

►

12.13.4 $$V(a,x+y)={\mathrm{e}}^{-\frac{1}{2}xy-\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\frac{y^m}{m!}V(a+m,x).$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $V(a,z)$: parabolic cylinder function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Referenced by:

§12.13(i)

Permalink:

http://dlmf.nist.gov/12.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

►

12.13.5 $$U(a,x\cos t+y\sin t)={\mathrm{e}}^{\frac{1}{4}{(x\sin t-y\cos t)}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{-a-\frac{1}{2}}{m}\right){(\tan t)}^{m}U(m+a,x)U(-m-\frac{1}{2},y),$$ $\mathrm{\Re }a\le -\frac{1}{2},0\le t\le \frac{1}{4}\pi$.

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $\tan z$: tangent function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Referenced by:

§12.13(i)

Permalink:

http://dlmf.nist.gov/12.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

►

12.13.6 $$n!U(n+\frac{1}{2},z)={\mathrm{i}}^n{\mathrm{e}}^{-\frac{1}{2}{z}^2}\mathrm{erfc}(z/\sqrt{2})U(-n-\frac{1}{2},\mathrm{i}z)+\sum _{m=1}^{\lfloor \frac{1}{2}n+\frac{1}{2}\rfloor }U(2m-n-\frac{1}{2},z),$$ $n=0,1,2,\mathrm{\dots }.$

ⓘ

Symbols:

$\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§12.13(i)

Permalink:

http://dlmf.nist.gov/12.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

…

7: 12.4 Power-Series Expansions

… ►

12.4.3 $$u_1(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(1+(a+\frac{1}{2})\frac{z^2}{2!}+(a+\frac{1}{2})(a+\frac{5}{2})\frac{z^4}{4!}+\mathrm{\cdots }\right),$$

ⓘ

Defines:

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

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.1 (modification of)

Referenced by:

§12.14(v), §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

►

12.4.4 $$u_2(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(z+(a+\frac{3}{2})\frac{z^3}{3!}+(a+\frac{3}{2})(a+\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

ⓘ

Defines:

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

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.3 (modification of)

Permalink:

http://dlmf.nist.gov/12.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

… ►

12.4.5 $$u_1(a,z)={\mathrm{e}}^{\frac{1}{4}{z}^2}\left(1+(a-\frac{1}{2})\frac{z^2}{2!}+(a-\frac{1}{2})(a-\frac{5}{2})\frac{z^4}{4!}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $a$: real or complex parameter and $u_1(a,z)$: solution

A&S Ref:

19.2.2 (modification of)

Permalink:

http://dlmf.nist.gov/12.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

►

12.4.6 $$u_2(a,z)={\mathrm{e}}^{\frac{1}{4}{z}^2}\left(z+(a-\frac{3}{2})\frac{z^3}{3!}+(a-\frac{3}{2})(a-\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $a$: real or complex parameter and $u_2(a,z)$: solution

A&S Ref:

19.2.4 (modification of)

Referenced by:

§12.14(v), §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

…

8: 10.57 Uniform Asymptotic Expansions for Large Order

… ►Asymptotic expansions for ${𝗃}_n\left((n+\frac{1}{2})z\right)$, ${𝗒}_n\left((n+\frac{1}{2})z\right)$, ${𝗁}_n^{(1)}\left((n+\frac{1}{2})z\right)$, ${𝗁}_n^{(2)}\left((n+\frac{1}{2})z\right)$, ${𝗂}_n^{(1)}\left((n+\frac{1}{2})z\right)$, and ${𝗄}_n\left((n+\frac{1}{2})z\right)$ as $n\to \mathrm{\infty }$ that are uniform with respect to $z$ can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). Subsequently, for ${𝗂}_n^{(2)}\left((n+\frac{1}{2})z\right)$ the connection formula (10.47.11) is available. ►For the corresponding expansion for ${𝗃}_n^{\prime }\left((n+\frac{1}{2})z\right)$ use ►

10.57.1 $${𝗃}_n^{\prime }\left((n+\frac{1}{2})z\right)=\frac{{\pi }^{\frac{1}{2}}}{{((2n+1)z)}^{\frac{1}{2}}}{J}_{n+\frac{1}{2}}^{\prime }\left((n+\frac{1}{2})z\right)-\frac{{\pi }^{\frac{1}{2}}}{{((2n+1)z)}^{\frac{3}{2}}}{J}_{n+\frac{1}{2}}\left((n+\frac{1}{2})z\right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

Referenced by:

§10.57

Permalink:

http://dlmf.nist.gov/10.57.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.57 and Ch.10

…

9: 28.6 Expansions for Small $q$

… ►

28.6.1 $$a_0\left(q\right)=-\frac{1}{2}{q}^2+\frac{7}{128}{q}^4-\frac{29}{2304}{q}^6+\frac{68687}{\mathrm{188\hspace{0.33em}74368}}{q}^8+\mathrm{\cdots },$$

ⓘ

Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation and $q=h^2$: parameter

A&S Ref:

20.2.25

Referenced by:

§28.6(i)

Permalink:

http://dlmf.nist.gov/28.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

►

28.6.2 $$a_1\left(q\right)=1+q-\frac{1}{8}{q}^2-\frac{1}{64}{q}^3-\frac{1}{1536}{q}^4+\frac{11}{36864}{q}^5+\frac{49}{\mathrm{5\hspace{0.33em}89824}}{q}^6+\frac{55}{\mathrm{94\hspace{0.33em}37184}}{q}^7-\frac{83}{\mathrm{353\hspace{0.33em}89440}}{q}^8+\mathrm{\cdots },$$

ⓘ

Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation and $q=h^2$: parameter

Permalink:

http://dlmf.nist.gov/28.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

►

28.6.3 $$b_1\left(q\right)=1-q-\frac{1}{8}{q}^2+\frac{1}{64}{q}^3-\frac{1}{1536}{q}^4-\frac{11}{36864}{q}^5+\frac{49}{\mathrm{5\hspace{0.33em}89824}}{q}^6-\frac{55}{\mathrm{94\hspace{0.33em}37184}}{q}^7-\frac{83}{\mathrm{353\hspace{0.33em}89440}}{q}^8+\mathrm{\cdots },$$

ⓘ

Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation and $q=h^2$: parameter

Permalink:

http://dlmf.nist.gov/28.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

… ►

28.6.6 $$a_3\left(q\right)=9+\frac{1}{16}{q}^2+\frac{1}{64}{q}^3+\frac{13}{20480}{q}^4-\frac{5}{16384}{q}^5-\frac{1961}{\mathrm{235\hspace{0.33em}92960}}{q}^6-\frac{609}{\mathrm{1048\hspace{0.33em}57600}}{q}^7+\mathrm{\cdots },$$

ⓘ

Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation and $q=h^2$: parameter

Permalink:

http://dlmf.nist.gov/28.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

►

28.6.7 $$b_3\left(q\right)=9+\frac{1}{16}{q}^2-\frac{1}{64}{q}^3+\frac{13}{20480}{q}^4+\frac{5}{16384}{q}^5-\frac{1961}{\mathrm{235\hspace{0.33em}92960}}{q}^6+\frac{609}{\mathrm{1048\hspace{0.33em}57600}}{q}^7+\mathrm{\cdots },$$

ⓘ

Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation and $q=h^2$: parameter

Permalink:

http://dlmf.nist.gov/28.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

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10: 12.9 Asymptotic Expansions for Large Variable

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12.9.1 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ ,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

A&S Ref:

19.8.1 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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12.9.2 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ .

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

A&S Ref:

19.8.2 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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12.9.3 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s}\pm \mathrm{i}\frac{\sqrt{2\pi }}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\mathrm{e}}^{\mp \mathrm{i}\pi a}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ $\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{5}{4}\pi -\delta$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/12.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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12.9.4 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s}\pm \frac{\mathrm{i}}{\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}{\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ $-\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{4}\pi -\delta$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Referenced by:

§12.9(i)

Permalink:

http://dlmf.nist.gov/12.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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