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11: 23.6 Relations to Other Functions

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23.6.8 $${\eta }_1=-\frac{{\pi }^2}{12{\omega }_1}\frac{{\theta }_1^{\prime \prime \prime }(0,q)}{{\theta }_1^{\prime }(0,q)}.$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\pi$: 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:

§23.22(i), §23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

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23.6.9 $$\sigma \left(z\right)=2{\omega }_1\exp \left(\frac{{\eta }_1{z}^2}{2{\omega }_1}\right)\frac{{\theta }_1(\pi z/(2{\omega }_1),q)}{\pi {\theta }_1^{\prime }(0,q)},$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $𝕃$: lattice, $q$: nome, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

A&S Ref:

18.10.8

Referenced by:

§23.22(i), §23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

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23.6.10 $$\sigma \left({\omega }_1\right)=2{\omega }_1\frac{\exp \left(\frac{1}{2}{\eta }_1{\omega }_1\right){\theta }_2(0,q)}{\pi {\theta }_1^{\prime }(0,q)},$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $𝕃$: lattice, $q$: nome, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Referenced by:

§23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

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23.6.11 $$\sigma \left({\omega }_2\right)=-2{\omega }_1\frac{\exp \left(\frac{1}{2}{\eta }_1({\omega }_1{\tau }^2+{\omega }_3-{\omega }_2)\right){\theta }_3(0,q)}{\pi q^{1/4}{\theta }_1^{\prime }(0,q)},$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $𝕃$: lattice, $q$: nome, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $\tau$: complex variable and ${\eta }_j$: complex number

Referenced by:

Erratum (V1.1.1) for Equation (23.6.11)

Permalink:

http://dlmf.nist.gov/23.6.E11

Encodings:

pMML, png, TeX

Correction (effective with 1.1.1):

The factor $2{\omega }_1\mathrm{i}\exp \left(\frac{1}{2}{\eta }_1{\omega }_1{\tau }^2\right)$ has been corrected to be $-2{\omega }_1\exp \left(\frac{1}{2}{\eta }_1({\omega }_1{\tau }^2+{\omega }_3-{\omega }_2)\right)$.

See also:

Annotations for §23.6(i), §23.6 and Ch.23

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23.6.13 $$\zeta \left(u\right)=\frac{{\eta }_1}{{\omega }_1}u+\frac{\pi }{2{\omega }_1}\frac{d}{dz}\ln {\theta }_1(z,q),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\ln z$: principal branch of logarithm function, $𝕃$: lattice, $q$: nome, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, ${\eta }_j$: complex number and $u$: change of variable

Referenced by:

§23.22(i), §23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

…

12: 25.14 Lerch’s Transcendent

… ►If $s$ is not an integer then ; if $s$ is a positive integer then $a\ne 0,-1,-2,\mathrm{\dots }$; if $s$ is a non-positive integer then $a$ can be any complex number. …

13: 20.9 Relations to Other Functions

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20.9.3 $$R_F(\frac{{\theta }_2^2(z,q)}{{\theta }_2^2(0,q)},\frac{{\theta }_3^2(z,q)}{{\theta }_3^2(0,q)},\frac{{\theta }_4^2(z,q)}{{\theta }_4^2(0,q)})=\frac{{\theta }_1^{\prime }(0,q)}{{\theta }_1(z,q)}z,$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, ${\theta }_j(z,q)$: theta function, $z$: complex and $q$: nome

Referenced by:

§19.25(v), §20.9(i)

Permalink:

http://dlmf.nist.gov/20.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.9(i), §20.9 and Ch.20

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14: 21.5 Modular Transformations

… ►where $\kappa (𝜶,𝜷,𝚪)$ is a complex number that depends on $𝜶$, $𝜷$, and $𝚪$. …

15: 3.1 Arithmetics and Error Measures

… ►If $x^{\ast }$ is an approximation to a real or complex number $x$, then the absolute error is …

16: 22.20 Methods of Computation

… ►Given real or complex numbers $a_0,b_0$, with $b_0/a_0$ not real and negative, define …

17: 3.9 Acceleration of Convergence

… ►All sequences (series) in this section are sequences (series) of real or complex numbers. …

18: 25.15 Dirichlet $L$-functions

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25.15.2 $$L(s,\chi )=\prod _p{\left(1-\frac{\chi (p)}{p^s}\right)}^{-1},$$ $\mathrm{\Re }s>1$,

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\mathrm{\Re }$: real part, $p$: prime number, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

infinite product, product representation

Source:

Apostol (1976, p. 231)

Permalink:

http://dlmf.nist.gov/25.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

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25.15.4 $$L(s,\chi )=L(s,{\chi }_0)\prod _{p|k}\left(1-\frac{{\chi }_0(p)}{p^s}\right),$$

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $k$: nonnegative integer, $p$: prime number, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

product representation

Source:

Apostol (1976, p. 262)

Referenced by:

§25.15(i)

Permalink:

http://dlmf.nist.gov/25.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

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19: 3.2 Linear Algebra

… ►If $𝐀$ is an $n\times n$ matrix, then a real or complex number $\lambda$ is called an eigenvalue of $𝐀$, and a nonzero vector $𝐱$ a corresponding (right) eigenvector, if …

20: 19.22 Quadratic Transformations

… ►However, if $x$ and $y$ are complex conjugates and $z$ and $p$ are real, then the right-hand sides of all transformations in §§19.22(i) and 19.22(iii)—except (19.22.3) and (19.22.22)—are free of complex numbers and $p_{\pm }^2-p_{\mp }^2=\pm |p^2-x^2|\ne 0$. …