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1: 23.8 Trigonometric Series and Products

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23.8.1 $$\mathrm{\wp }\left(z\right)+\frac{{\eta }_1}{{\omega }_1}-\frac{{\pi }^2}{4{\omega }_1^2}{\csc }^2\left(\frac{\pi z}{2{\omega }_1}\right)=-\frac{2{\pi }^2}{{\omega }_1^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n{q}^{2n}}{1-q^{2n}}\cos \left(\frac{n\pi z}{{\omega }_1}\right),$$

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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, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $𝕃$: lattice, $q$: nome, $n$: integer, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.8(i), §23.8 and Ch.23

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23.8.3 $$\mathrm{\wp }\left(z\right)=-\frac{{\eta }_1}{{\omega }_1}+\frac{{\pi }^2}{4{\omega }_1^2}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\csc }^2\left(\frac{\pi (z+2n{\omega }_3)}{2{\omega }_1}\right),$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $𝕃$: lattice, $n$: integer, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.8(ii), §23.8 and Ch.23

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23.8.4 $$\zeta \left(z\right)=\frac{{\eta }_{1}z}{{\omega }_1}+\frac{\pi }{2{\omega }_1}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\cot \left(\frac{\pi (z+2n{\omega }_3)}{2{\omega }_1}\right),$$

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $𝕃$: lattice, $n$: integer, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Referenced by:

§23.8(ii)

Permalink:

http://dlmf.nist.gov/23.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.8(ii), §23.8 and Ch.23

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23.8.5 $${\eta }_1=\frac{{\pi }^2}{2{\omega }_1}\left(\frac{1}{6}+\sum _{n=1}^{\mathrm{\infty }}{\csc }^2\left(\frac{n\pi {\omega }_3}{{\omega }_1}\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $n$: integer, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.8(ii), §23.8 and Ch.23

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2: 1.9 Calculus of a Complex Variable

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§1.9(i) Complex Numbers

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Polar Representation

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Modulus and Phase

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Complex Conjugate

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Powers

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3: 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. … ►

23.2.11 $$\zeta \left(z+2{\omega }_j\right)=\zeta \left(z\right)+2{\eta }_j,$$

ⓘ

Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $𝕃$: lattice, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

A&S Ref:

18.2.19

Permalink:

http://dlmf.nist.gov/23.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §23.2(iii), §23.2 and Ch.23

►where … ►

23.2.13 $${\eta }_1+{\eta }_2+{\eta }_3=0,$$

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Symbols:

${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.2.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §23.2(iii), §23.2 and Ch.23

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23.2.15 $$\sigma \left(z+2{\omega }_j\right)=-{\mathrm{e}}^{2{\eta }_j(z+{\omega }_j)}\sigma \left(z\right),$$

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Symbols:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\mathrm{e}$: base of natural logarithm, $𝕃$: lattice, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Referenced by:

§23.2(iii)

Permalink:

http://dlmf.nist.gov/23.2.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §23.2(iii), §23.2 and Ch.23

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4: 20.12 Mathematical Applications

… ►The space of complex tori $ℂ/(\mathbb{Z}+\tau \mathbb{Z})$ (that is, the set of complex numbers $z$ in which two of these numbers $z_1$ and $z_2$ are regarded as equivalent if there exist integers $m,n$ such that $z_1-z_2=m+\tau n$) is mapped into the projective space $P^3$ via the identification $z\to ({\theta }_1\left(2z\right|\tau ),{\theta }_2\left(2z\right|\tau ),{\theta }_3\left(2z\right|\tau ),{\theta }_4\left(2z\right|\tau ))$. …

5: 1.1 Special Notation

… ►In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{\ast }$, $a$ being a complex number or a matrix; the Hermitian conjugate of $𝐀$ is usually being denoted ${𝐀}^{†}$.

6: 23.12 Asymptotic Approximations

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

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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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7: 23.5 Special Lattices

… ►The parallelogram $0$, $2{\omega }_1$, $2{\omega }_1+2{\omega }_3$, $2{\omega }_3$ is a square, and ►

23.5.2 $${\eta }_1=\mathrm{i}{\eta }_3=\pi /(4{\omega }_1),$$

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Symbols:

$\pi$: 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 and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.5(iii), §23.5 and Ch.23

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23.5.6 $${\eta }_1={\mathrm{e}}^{\pi \mathrm{i}/3}{\eta }_3=\frac{\pi }{2\sqrt{3}{\omega }_1},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.5(v), §23.5 and Ch.23

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8: 19.25 Relations to Other Functions

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19.25.36 $$\mathrm{\wp }\left(z\right)-e_j\in ℂ\setminus (-\mathrm{\infty },0],$$ $j=1,2,3.$

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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, $e_j$: Weierstrass lattice roots, $ℂ$: complex plane, $\in$: element of, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $j$: $(=1,2,3)$ and $z$: complex number

Referenced by:

(19.25.39)

Permalink:

http://dlmf.nist.gov/19.25.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

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19.25.38 $${\omega }_j=R_F(0,e_j-e_k,e_j-e_{\mathrm{\ell }}),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $e_j$: Weierstrass lattice roots, $j$: $(=1,2,3)$ , $z$: complex number, $k$: $(=1,2,3)$ , $\mathrm{\ell }$: $(=1,2,3)$ and ${\omega }_j$: nonzero complex number

Proof sketch:

Take $z=-\omega$ in (19.25.35).

Referenced by:

§19.25(vi), §19.25(vi)

Permalink:

http://dlmf.nist.gov/19.25.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

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19.25.39 $$\zeta \left({\omega }_j\right)+{\omega }_j{e}_j=2R_G(0,e_j-e_k,e_j-e_{\mathrm{\ell }}),$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $e_j$: Weierstrass lattice roots, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $j$: $(=1,2,3)$ , $z$: complex number, $k$: $(=1,2,3)$ , $\mathrm{\ell }$: $(=1,2,3)$ , ${\omega }_j$: nonzero complex number and ${\eta }_j$: complex number

Proof sketch:

Take $z=-\omega$ in (19.25.36).

Referenced by:

§19.25(vi), §19.25(vi), Erratum (V1.1.0) for Subsection 19.25(vi)

Permalink:

http://dlmf.nist.gov/19.25.E39

Encodings:

pMML, png, TeX

Correction (effective with 1.1.0):

This equation has been corrected such that the left-hand side ${\eta }_j$ has been replaced by $\zeta \left({\omega }_j\right)$.

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

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19.25.41 $${\sigma }_j(z)=\exp \left(-{\eta }_{j}z\right)\sigma \left(z+{\omega }_j\right)/\sigma \left({\omega }_j\right),$$ $j=1,2,3,$

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Defines:

${\sigma }_j(z)$: function (locally)

Symbols:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\exp z$: exponential function, $j$: $(=1,2,3)$ , $z$: complex number, ${\omega }_j$: nonzero complex number and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/19.25.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

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9: 23.10 Addition Theorems and Other Identities

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23.10.12 $$n\zeta \left(nz\right)=-n(n-1)({\eta }_1+{\eta }_3)+\sum _{j=0}^{n-1}\sum _{\mathrm{\ell }=0}^{n-1}\zeta \left(z+\frac{2j}{n}{\omega }_1+\frac{2\mathrm{\ell }}{n}{\omega }_3\right),$$

ⓘ

Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $𝕃$: lattice, $n$: integer, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Referenced by:

§23.10(iii)

Permalink:

http://dlmf.nist.gov/23.10.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §23.10(iii), §23.10 and Ch.23

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23.10.13 $$\sigma \left(nz\right)=A_n{\mathrm{e}}^{-n(n-1)({\eta }_1+{\eta }_3)z}\prod _{j=0}^{n-1}\prod _{\mathrm{\ell }=0}^{n-1}\sigma \left(z+\frac{2j}{n}{\omega }_1+\frac{2\mathrm{\ell }}{n}{\omega }_3\right),$$

ⓘ

Symbols:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\mathrm{e}$: base of natural logarithm, $𝕃$: lattice, $n$: integer, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $A_n$: coefficients and ${\eta }_j$: complex number

Referenced by:

§23.10(iii)

Permalink:

http://dlmf.nist.gov/23.10.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §23.10(iii), §23.10 and Ch.23

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23.10.15 $$A_n={\left(\frac{{\pi }^2{G}^2}{{\omega }_1}\right)}^{n^2-1}\frac{q^{n(n-1)/2}}{{\mathrm{i}}^{n-1}}\exp \left(-\frac{(n-1){\eta }_1}{3{\omega }_1}\left((2n-1)({\omega }_1^2+{\omega }_3^2)+3(n-1){\omega }_1{\omega }_3\right)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $q$: nome, $n$: integer, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $A_n$: coefficients, $G$ and ${\eta }_j$: complex number

Referenced by:

§23.10(iii)

Permalink:

http://dlmf.nist.gov/23.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §23.10(iii), §23.10 and Ch.23

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10: 31.2 Differential Equations

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31.2.11 $$d^{2}W/{d\xi }^2+\left(H+b_0\mathrm{\wp }\left(\xi \right)+b_1\mathrm{\wp }\left(\xi +{\omega }_1\right)+b_2\mathrm{\wp }\left(\xi +{\omega }_2\right)+b_3\mathrm{\wp }\left(\xi +{\omega }_3\right)\right)W=0,$$

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Defines:

$W(\xi )$: solution (locally)

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${\omega }_j$: nonzero complex number, $𝕃$: lattice, $b_j$: coefficients and $H$

Permalink:

http://dlmf.nist.gov/31.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

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