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11: 23.14 Integrals

… ►

23.14.1 $$\int \mathrm{\wp }\left(z\right)dz=-\zeta \left(z\right),$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $dx$: differential of $x$ , $\int$: integral, $𝕃$: lattice and $z$: complex

A&S Ref:

18.6.1

Permalink:

http://dlmf.nist.gov/23.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.14 and Ch.23

►

23.14.2 $$\int {\mathrm{\wp }}^2\left(z\right)dz=\frac{1}{6}{\mathrm{\wp }}^{\prime }\left(z\right)+\frac{1}{12}{g}_{2}z,$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $dx$: differential of $x$ , $\int$: integral, $𝕃$: lattice and $z$: complex

A&S Ref:

18.7.1

Permalink:

http://dlmf.nist.gov/23.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.14 and Ch.23

►

23.14.3 $$\int {\mathrm{\wp }}^3\left(z\right)dz=\frac{1}{120}{\mathrm{\wp }}^{\prime \prime \prime }\left(z\right)-\frac{3}{20}{g}_2\zeta \left(z\right)+\frac{1}{10}{g}_{3}z.$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $dx$: differential of $x$ , $\int$: integral, $𝕃$: lattice and $z$: complex

A&S Ref:

18.7.2

Permalink:

http://dlmf.nist.gov/23.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.14 and Ch.23

…

12: 23.22 Methods of Computation

… ►Suppose that the invariants $g_2=c$, $g_3=d$, are given, for example in the differential equation (23.3.10) or via coefficients of an elliptic curve (§23.20(ii)). … ►

In the general case, given by $cd\ne 0$, we compute the roots $\alpha$, $\beta$, $\gamma$, say, of the cubic equation $4t^3-ct-d=0$; see §1.11(iii). These roots are necessarily distinct and represent $e_1$, $e_2$, $e_3$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is $c^3-27d^2>0$, then $e_1$, $e_2$, $e_3$ can be identified via (23.5.1), and $k^2$, $k_{}^{\prime }{}_{}{}^2$ obtained from (23.6.16).

If , or $c$ and $d$ are not both real, then we label $\alpha$, $\beta$, $\gamma$ so that the triangle with vertices $\alpha$, $\beta$, $\gamma$ is positively oriented and $[\alpha ,\gamma ]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha$, $\beta$, $\gamma$ are collinear, then we label them so that $\beta$ is on the line segment $(\alpha ,\gamma )$. In consequence, $k^2=(\beta -\gamma )/(\alpha -\gamma )$, $k_{}^{\prime }{}_{}{}^2=(\alpha -\beta )/(\alpha -\gamma )$ satisfy $\mathrm{\Im }{k}^2\ge 0\ge \mathrm{\Im }k_{}^{\prime }{}_{}{}^2$ (with strict inequality unless $\alpha$, $\beta$, $\gamma$ are collinear); also $|k^2|$, $|k_{}^{\prime }{}_{}{}^2|\le 1$.

Finally, on taking the principal square roots of $k^2$ and $k_{}^{\prime }{}_{}{}^2$ we obtain values for $k$ and $k^{\prime }$ that lie in the 1st and 4th quadrants, respectively, and $2{\omega }_1$, $2{\omega }_3$ are given by

23.22.1 $$2{\omega }_{1}M(1,k^{\prime })=-2\mathrm{i}{\omega }_{3}M(1,k)=\frac{\pi }{3}\sqrt{\frac{c(2+k^{2}k_{}^{\prime }{}_{}{}^2)(k_{}^{\prime }{}_{}{}^2-k^2)}{d(1-k^{2}k_{}^{\prime }{}_{}{}^2)}},$$

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

Proof sketch:

Combine (23.6.16), (23.6.17), (22.20.6), (23.3.5)–(23.3.7), and use analytical continuation.

Referenced by:

§23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 2 possible choices of the square root.

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If $d=0$, then

23.22.2 $$2{\omega }_1=-2\mathrm{i}{\omega }_3=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{4}\right)\right)}^2}{2\sqrt{\pi }{c}^{1/4}}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§22.20(iv), §23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

There are 4 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 4 rotations of a square lattice. The lemniscatic case occurs when $c>0$ and ${\omega }_1>0$.

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If $c=0$, then

23.22.3 $$2{\omega }_1=2{\mathrm{e}}^{-\pi \mathrm{i}/3}{\omega }_3=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{3}\right)\right)}^3}{2\pi d^{1/6}}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

There are 6 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 6 rotations of a lattice of equilateral triangles. The equianharmonic case occurs when $d>0$ and ${\omega }_1>0$.

… ►Assume $c=g_2=-4(3-2\mathrm{i})$ and $d=g_3=4(4-2\mathrm{i})$. …

13: 32.2 Differential Equations

… ►be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $dw/dz$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $ℂ$. … ►in which $a(z)$, $b(z)$, $c(z)$, $d(z)$, and $\varphi (z)$ are locally analytic functions. … ►where ${\mu }_1$, ${\mu }_2$, ${\mu }_3$ are constants, $f_1$, $f_2$, $f_3$ are functions of $z$, with … ►where ${\mu }_1$, ${\mu }_2$, ${\mu }_3$, ${\mu }_4$ are constants, $f_1$, $f_2$, $f_3$, $f_4$ are functions of $z$, with … ►then as $ϵ\to 0$, $W(\zeta ;a,b,c,d)$ satisfies ${\text{P}}_{\text{III}}$ with $z=\zeta$, $\alpha =a$, $\beta =b$, $\gamma =c$, $\delta =d$. …

14: 9.12 Scorer Functions

… ►

9.12.21 $$\mathrm{Gi}\left(z\right)=-\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\exp \left(-\frac{1}{3}{t}^3-\frac{1}{2}zt\right)\cos \left(\frac{1}{2}\sqrt{3}zt+\frac{2}{3}\pi \right)dt.$$

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Symbols:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral and $z$: complex variable

Source:

Lee (1980, (2.20))

Permalink:

http://dlmf.nist.gov/9.12.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vii), §9.12 and Ch.9

►If $\zeta =\frac{2}{3}{z}^{3/2}$ or $\frac{2}{3}{x}^{3/2}$, and $K_{1/3}$ is the modified Bessel function (§10.25(ii)), then ►

9.12.22 $$\mathrm{Hi}\left(-z\right)=\frac{4z^2}{3^{3/2}{\pi }^2}{\int }_0^{\mathrm{\infty }}\frac{K_{1/3}\left(t\right)}{{\zeta }^2+t^2}dt,$$ ,

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Symbols:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\zeta$: variable

Source:

Gordon (1970, Appendix A)

Permalink:

http://dlmf.nist.gov/9.12.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vii), §9.12 and Ch.9

… ►

9.12.24 $$\mathrm{Hi}\left(z\right)=\frac{3^{-2/3}}{2{\pi }^2\mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{1}{3}+\frac{1}{3}t\right)\mathrm{\Gamma }\left(-t\right){(3^{1/3}{\mathrm{e}}^{\pi \mathrm{i}}z)}^{t}dt,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

Keywords:

Mellin transform

Source:

Exton (1983, (2.1), p. 100)

Permalink:

http://dlmf.nist.gov/9.12.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vii), §9.12(vii), §9.12 and Ch.9

►where the integration contour separates the poles of $\mathrm{\Gamma }\left(\frac{1}{3}+\frac{1}{3}t\right)$ from those of $\mathrm{\Gamma }\left(-t\right)$. …

15: 23.2 Definitions and Periodic Properties

… ►If ${\omega }_1$ and ${\omega }_3$ are nonzero real or complex numbers such that $\mathrm{\Im }\left({\omega }_3/{\omega }_1\right)>0$, then the set of points $2m{\omega }_1+2n{\omega }_3$, with $m,n\in \mathbb{Z}$, constitutes a lattice $𝕃$ with $2{\omega }_1$ and $2{\omega }_3$ lattice generators. … ►then $2{\omega }_2$, $2{\omega }_3$ are generators, as are $2{\omega }_2$, $2{\omega }_1$. …where $a,b,c,d$ are integers, then $2{\chi }_1$, $2{\chi }_3$ are generators of $𝕃$ iff … ►For $j=1,2,3$, the function $\sigma \left(z\right)$ satisfies …More generally, if $j=1,2,3$, $k=1,2,3$, $j\ne k$, and $m,n\in \mathbb{Z}$, then …

16: 26.10 Integer Partitions: Other Restrictions

… ► $p({𝒟}^{\prime }3,n)$ denotes the number of partitions of $n$ into parts with difference at least 3, except that multiples of 3 must differ by at least 6. …The set $\{2,3,4,\mathrm{\dots }\}$ is denoted by $T$. … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►Note that $p({𝒟}^{\prime }3,n)\le p(𝒟3,n)$, with strict inequality for $n\ge 9$. It is known that for $k>3$, $p(𝒟k,n)\ge p(\in A_{1,k+3},n)$, with strict inequality for $n$ sufficiently large, provided that $k=2^m-1,m=3,4,5$, or $k\ge 32$; see Yee (2004). …

17: 34.11 Higher-Order $3nj$ Symbols

§34.11 Higher-Order $3nj$ Symbols

…

18: 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). … ►In formulas (10.20.10)–(10.20.13) replace ${\zeta }^{\frac{1}{2}}$, ${\zeta }^{-\frac{1}{2}}$, ${\zeta }^{-3j/2}$, and ${(1-z^2)}^{-\frac{1}{2}}$ by $-\mathrm{i}{(-\zeta )}^{\frac{1}{2}}$, $\mathrm{i}{(-\zeta )}^{-\frac{1}{2}}$, ${\mathrm{i}}^{3j}{(-\zeta )}^{-3j/2}$, and $\mathrm{i}{(z^2-1)}^{-\frac{1}{2}}$, respectively. ►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 . … ►The equations of the curved boundaries $D_1{E}_1$ and $D_2{E}_2$ in the $\zeta$-plane are given parametrically by …

19: 26.13 Permutations: Cycle Notation

… ►is $\left(1,3,2,5,7\right)\left(4\right)\left(6,8\right)$ in cycle notation. …In consequence, (26.13.2) can also be written as $\left(1,3,2,5,7\right)\left(6,8\right)$. … ►The derangement number, $d(n)$, is the number of elements of ${𝔖}_n$ with no fixed points: … ►For the example (26.13.2), this decomposition is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(1,3\right)\left(2,3\right)\left(2,5\right)\left(5,7\right)\left(6,8\right).$ … ►Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(2,3\right)\left(1,2\right)\left(4,5\right)\left(3,4\right)\left(2,3\right)\left(3,4\right)\left(4,5\right)\left(6,7\right)\left(5,6\right)\left(7,8\right)\left(6,7\right)$: $inv(\left(1,3,2,5,7\right)\left(6,8\right))=11$.

20: 9.2 Differential Equation

… ►

9.2.3 $$\mathrm{Ai}\left(0\right)=\frac{1}{3^{2/3}\mathrm{\Gamma }\left(\frac{2}{3}\right)}=\mathrm{0.35502\hspace{0.33em}80538}\mathrm{\dots },$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function and $\mathrm{\Gamma }\left(z\right)$: gamma function

Source:

Olver (1997b, (1.03), p. 392)

A&S Ref:

10.4.4 (with more digits)

Permalink:

http://dlmf.nist.gov/9.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(ii), §9.2 and Ch.9

►

9.2.4 $${\mathrm{Ai}}^{\prime }\left(0\right)=-\frac{1}{3^{1/3}\mathrm{\Gamma }\left(\frac{1}{3}\right)}=-\mathrm{0.25881\hspace{0.33em}94037}\mathrm{\dots },$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function and $\mathrm{\Gamma }\left(z\right)$: gamma function

Source:

Olver (1997b, (1.03), p. 392)

A&S Ref:

10.4.5 (with more digits)

Permalink:

http://dlmf.nist.gov/9.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(ii), §9.2 and Ch.9

►

9.2.5 $$\mathrm{Bi}\left(0\right)=\frac{1}{3^{1/6}\mathrm{\Gamma }\left(\frac{2}{3}\right)}=\mathrm{0.61492\hspace{0.33em}66274}\mathrm{\dots },$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function and $\mathrm{\Gamma }\left(z\right)$: gamma function

Source:

Olver (1997b, (1.11), p. 393)

A&S Ref:

10.4.4 (in different form)

Referenced by:

(9.12.15)

Permalink:

http://dlmf.nist.gov/9.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(ii), §9.2 and Ch.9

►

9.2.6 $${\mathrm{Bi}}^{\prime }\left(0\right)=\frac{3^{1/6}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}=\mathrm{0.44828\hspace{0.33em}83573}\mathrm{\dots }.$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function and $\mathrm{\Gamma }\left(z\right)$: gamma function

Source:

Olver (1997b, (1.11), p. 393)

A&S Ref:

10.4.5 (in different form)

Referenced by:

(9.12.15)

Permalink:

http://dlmf.nist.gov/9.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(ii), §9.2 and Ch.9

… ► $W=(1/w)dw/dz$, where $w$ is any nontrivial solution of (9.2.1). …