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31: 28.6 Expansions for Small $q$

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

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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

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

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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

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

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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

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28.6.4 $$a_2\left(q\right)=4+\frac{5}{12}{q}^2-\frac{763}{13824}{q}^4+\frac{\mathrm{10\hspace{0.33em}02401}}{\mathrm{796\hspace{0.33em}26240}}{q}^6-\frac{\mathrm{16690\hspace{0.33em}68401}}{\mathrm{45\hspace{0.33em}86471\hspace{0.33em}42400}}{q}^8+\mathrm{\cdots },$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/28.6.E4

Encodings:

pMML, png, TeX

See also:

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

… ►Leading terms of the of the power series for $m=7,8,9,\mathrm{\dots }$ are: …

32: 10.67 Asymptotic Expansions for Large Argument

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10.67.1 $${\ker }_{\nu }x\sim {\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}+\frac{1}{8}\right)\pi \right),$$

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Symbols:

${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.3, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v), §10.67(ii), §10.68(iii)

Permalink:

http://dlmf.nist.gov/10.67.E1

Encodings:

pMML, png, TeX

See also:

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

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10.67.2 $${\mathrm{kei}}_{\nu }x\sim -{\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\sin \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}+\frac{1}{8}\right)\pi \right).$$

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Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.4, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v)

Permalink:

http://dlmf.nist.gov/10.67.E2

Encodings:

pMML, png, TeX

See also:

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

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10.67.5 $${\ker }_{\nu }^{\prime }x\sim -{\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{b_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}-\frac{1}{8}\right)\pi \right),$$

ⓘ

Symbols:

${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $b_k(\nu )$: polynomial coefficient

A&S Ref:

§9.10 (modified)

Permalink:

http://dlmf.nist.gov/10.67.E5

Encodings:

pMML, png, TeX

See also:

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

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10.67.6 $${\mathrm{kei}}_{\nu }^{\prime }x\sim {\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{b_k(\nu )}{x^k}\sin \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}-\frac{1}{8}\right)\pi \right).$$

ⓘ

Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $b_k(\nu )$: polynomial coefficient

A&S Ref:

§9.10 (modified)

Permalink:

http://dlmf.nist.gov/10.67.E6

Encodings:

pMML, png, TeX

See also:

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

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

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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

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33: 23.17 Elementary Properties

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$\eta \left({\mathrm{e}}^{\pi \mathrm{i}/3}\right)={\displaystyle \frac{3^{1/8}{\left(\mathrm{\Gamma }\left(\frac{1}{3}\right)\right)}^{3/2}}{2\pi }}{\mathrm{e}}^{\pi \mathrm{i}/24}.$

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23.17.4 $$\lambda \left(\tau \right)=16q(1-8q+44q^2+\mathrm{\cdots }),$$

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Symbols:

$\lambda \left(\tau \right)$: elliptic modular function, $q$: nome and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(ii), §23.17 and Ch.23

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23.17.7 $$\lambda \left(\tau \right)=16q\prod _{n=1}^{\mathrm{\infty }}{\left(\frac{1+q^{2n}}{1+q^{2n-1}}\right)}^8,$$

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Symbols:

$\lambda \left(\tau \right)$: elliptic modular function, $q$: nome, $n$: integer and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(iii), §23.17 and Ch.23

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34: 28.16 Asymptotic Expansions for Large $q$

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28.16.1 $${\lambda }_{\nu }\left(h^2\right)\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

ⓘ

Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.16 and Ch.28

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35: Staff

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Richard B. Paris, University of Abertay, Chaps. 8, 11

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Richard B. Paris, University of Abertay Dundee, for Chaps. 8, 11 (deceased)

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36: 26.6 Other Lattice Path Numbers

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Table 26.6.1: Delannoy numbers $D(m,n)$.

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$m$	$n$
	0	1	2	3	4	5	6	7	8	9	10
…
8	1	17	145	833	3649	13073	40081	1 08545	2 65729	5 98417	12 56465
…

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Table 26.6.2: Motzkin numbers $M(n)$.

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$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$
0	1	4	9	8	323	12	15511	16	8 53467
…

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Table 26.6.3: Narayana numbers $N(n,k)$.

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$n$	$k$
	…
8	0	1	28	196	490	490	196	28	1
…

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Table 26.6.4: Schröder numbers $r(n)$.

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$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$
$0$	$1$	$4$	$90$	$8$	$41586$	$12$	$\mathrm{272\hspace{0.33em}97738}$	$16$	$\mathrm{2\hspace{0.33em}09271\hspace{0.33em}56706}$
…

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37: 24.2 Definitions and Generating Functions

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Table 24.2.1: Bernoulli and Euler numbers.

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$n$	$B_n$	$E_n$
…
$8$	$-\frac{1}{30}$	$1385$
…

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Table 24.2.3: Bernoulli numbers $B_n=N/D$.

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$n$	$N$	$D$	$B_n$
…
$8$	$-1$	$30$	$-\mathrm{3.33333\hspace{0.33em}3333}$	$\mathrm{\times 10⁻²}$
…

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Table 24.2.4: Euler numbers $E_n$.

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$n$	$E_n$
…
$8$	$1385$
…

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

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	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
…

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

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	$k$
…
$8$	$0$	$17$	$0$	$-28$	$0$	$14$	$0$	$-4$	$1$
…

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38: 20.4 Values at $z$ = 0

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20.4.9 $$\frac{{\theta }_2^{\prime \prime }(0,q)}{{\theta }_2(0,q)}=-1-8\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{{(1+q^{2n})}^2},$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Permalink:

http://dlmf.nist.gov/20.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(ii), §20.4 and Ch.20

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20.4.10 $$\frac{{\theta }_3^{\prime \prime }(0,q)}{{\theta }_3(0,q)}=-8\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{{(1+q^{2n-1})}^2},$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Permalink:

http://dlmf.nist.gov/20.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(ii), §20.4 and Ch.20

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20.4.11 $$\frac{{\theta }_4^{\prime \prime }(0,q)}{{\theta }_4(0,q)}=8\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{{(1-q^{2n-1})}^2}.$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Permalink:

http://dlmf.nist.gov/20.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(ii), §20.4 and Ch.20

…

39: 23.12 Asymptotic Approximations

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23.12.1 $$\mathrm{\wp }\left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(-\frac{1}{3}+{\csc }^2\left(\frac{\pi z}{2{\omega }_1}\right)+8\left(1-\cos \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.12

Permalink:

http://dlmf.nist.gov/23.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.12 and Ch.23

►

23.12.2 $$\zeta \left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(\frac{z}{3}+\frac{2{\omega }_1}{\pi }\cot \left(\frac{\pi z}{2{\omega }_1}\right)-8\left(z-\frac{{\omega }_1}{\pi }\sin \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.12, Erratum (V1.0.23) for Equation (23.12.2)

Permalink:

http://dlmf.nist.gov/23.12.E2

Encodings:

pMML, png, TeX

Correction (effective with 1.0.23):

The factor of 2, previously omitted from the denominator of the argument of the $\cot$ function has been inserted.

Suggested 2019-05-27 by Blagoje Oblak

See also:

Annotations for §23.12 and Ch.23

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23.12.4 $${\eta }_1=\frac{{\pi }^2}{4{\omega }_1}\left(\frac{1}{3}-8q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $q$: nome, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Referenced by:

§23.12

Permalink:

http://dlmf.nist.gov/23.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.12 and Ch.23

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40: 12.14 The Function $W(a,x)$

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►►

Figure 12.14.1: $k^{-1/2}W(3,x)$, $k^{1/2}W(3,-x)$, $\tilde{F}(3,x)$, $0\le x\le 8$. Magnify

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►►

Figure 12.14.2: $k^{-1/2}{W}^{\prime }(3,x)$, $k^{1/2}{W}^{\prime }(3,-x)$, $\tilde{G}(3,x)$, $0\le x\le 8$. Magnify

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►►

Figure 12.14.3: $k^{-1/2}W(-3,x)$, $k^{1/2}W(-3,-x)$, $\tilde{F}(-3,x)$, $0\le x\le 8$. Magnify

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►►

Figure 12.14.4: $k^{-1/2}{W}^{\prime }(-3,x),k^{1/2}{W}^{\prime }(-3,-x)$, $\tilde{G}(-3,x)$, $0\le x\le 8$. Magnify

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12.14.6 $$\rho =\frac{1}{8}\pi +\frac{1}{2}{\varphi }_2,$$

ⓘ

Defines:

$\rho$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and ${\varphi }_2$

Permalink:

http://dlmf.nist.gov/12.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(iv), §12.14 and Ch.12

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