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

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

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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: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^2-y^2}$

… ►As a special case of (37.2.20) define for $m,n\in {\mathbb{N}}_0$ the (complex) circular Hermite polynomials (first introduced by Itô (1952, §12)) by ►

37.6.3

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\overline{z}$: complex conjugate, $!$: factorial (as in $n!$), $(a,b)$: open interval, $m$: nonnegative integer, $n$: nonnegative integer and $z$: complex number

Sources:

Itô (1952, §12); Intissar and Intissar (2006, Proposition 2.1(i)($\beta$))

Referenced by:

§37.12(iv), §37.6(ii), §37.6(iii)

Permalink:

http://dlmf.nist.gov/37.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §37.6(i), §37.6, Part 37.II and Ch.37

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37.6.9 $$S_{m,n}(z,\overline{z})={(-1)}^{m+n}{\mathrm{e}}^{{\left|z\right|}^2}{D}_{\overline{z}}^m{D}_z^n{\mathrm{e}}^{-{\left|z\right|}^2}.$$

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Symbols:

$\overline{z}$: complex conjugate, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $(a,b)$: open interval, $m$: nonnegative integer, $n$: nonnegative integer and $z$: complex number

Proved:

Intissar and Intissar (2006, Proposition 2.1)(proved)

Referenced by:

(37.6.10), (37.6.8)

Permalink:

http://dlmf.nist.gov/37.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §37.6(i), §37.6(i), §37.6, Part 37.II and Ch.37

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37.6.10 $$\sum _{m,n=0}^{\mathrm{\infty }}{S}_{m,n}(z,\overline{z})\frac{u^m{v}^n}{m!n!}={\mathrm{e}}^{uz+v\overline{z}-uv}.$$

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Symbols:

$\overline{z}$: complex conjugate, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $(a,b)$: open interval, $m$: nonnegative integer, $n$: nonnegative integer and $z$: complex number

Source:

Ismail (2016, (1.2))

Proof sketch:

Take the double Taylor expansion in $u,v$ of ${\mathrm{e}}^{-(z+u)(\overline{z}+v)}$ and use (37.6.9).

Permalink:

http://dlmf.nist.gov/37.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §37.6(i), §37.6(i), §37.6, Part 37.II and Ch.37

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37.6.15 $$\underset{\alpha \to \mathrm{\infty }}{lim}{\alpha }^{\frac{1}{2}(m+n)}{R}_{m,n}^{\alpha }({\alpha }^{-\frac{1}{2}}z,{\alpha }^{-\frac{1}{2}}\overline{z})=S_{m,n}(z,\overline{z}),$$

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Symbols:

$\overline{z}$: complex conjugate, $(a,b)$: open interval, $m$: nonnegative integer, $n$: nonnegative integer and $z$: complex number

Proof sketch:

Use (18.7.22).

Referenced by:

(37.6.14)

Permalink:

http://dlmf.nist.gov/37.6.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §37.6(iv), §37.6, Part 37.II and Ch.37

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7: 23.12 Asymptotic Approximations

… ►Also, ►

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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8: 37.1 Notation

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$x$, $y$	real variables.
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$j,k,l,m,n$	nonnegative integers.
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9: 37.9 Jacobi Polynomials Associated with Root System $A_2$

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37.9.1 $$\omega (x,y)=-{\left(x^2+y^2+9\right)}^2+8\left(x^3-3x{y}^2\right)+108=-{\left(z\overline{z}+9\right)}^2+4\left(z^3+{\overline{z}}^3\right)+108,$$

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Symbols:

$\overline{z}$: complex conjugate, $(a,b)$: open interval, $z$: complex number, $x$: real variable and $y$: real variable

Source:

Koornwinder (1974c, (3.16))

Permalink:

http://dlmf.nist.gov/37.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.9, Part 37.II and Ch.37

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37.9.3

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Symbols:

$\overline{z}$: complex conjugate, $m$: nonnegative integer, $n$: nonnegative integer, $P$: polynomial of two variables and $z$: complex number

Permalink:

http://dlmf.nist.gov/37.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §37.9, §37.9, Part 37.II and Ch.37

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37.9.6 $$z=z(t)={\mathrm{e}}^{2\pi \mathrm{i}(t_1-t_2)/3}+{\mathrm{e}}^{2\pi \mathrm{i}(t_2-t_3)/3}+{\mathrm{e}}^{2\pi \mathrm{i}(t_3-t_1)/3},$$ $t=(t_1,t_2,t_3)$, $t_1+t_2+t_3=0$,

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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 and $z$: complex number

Permalink:

http://dlmf.nist.gov/37.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §37.9, §37.9, Part 37.II and Ch.37

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37.9.8 $${\int }_{\mathrm{△}}{P}_{m,n}^{\alpha }(z(t),\overline{z(t)})\overline{P_{k,l}^{\alpha }(z(t),\overline{z(t)})}{\left(\sin \left(\pi t_1\right)\sin \left(\pi t_2\right)\sin \left(\pi t_3\right)\right)}^{2\alpha +1}d{t}_{1}d{t}_2=0,$$ $\alpha >-\frac{5}{6}$, $(m,n)\ne (k,l)$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral, $(a,b)$: open interval, $\sin z$: sine function, $k$: nonnegative integer, $l$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $P$: polynomial of two variables, $z$: complex number and $\mathrm{△}$: equilateral triangle

Permalink:

http://dlmf.nist.gov/37.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §37.9, §37.9, Part 37.II and Ch.37

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

ⓘ

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

ⓘ

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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