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11: 15.19 Methods of Computation

… ►The Gauss series (15.2.1) converges for

|z|<1

. … ►Large values of

|a|

|b|

, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. … ►The representation (15.6.1) can be used to compute the hypergeometric function in the sector

|\operatorname{ph}\left(1-z\right)|<\pi

. … ►Initial values for moderate values of

|a|

and

|b|

can be obtained by the methods of §15.19(i), and for large values of

|a|

|b|

, or

|c|

via the asymptotic expansions of §§15.12(ii) and 15.12(iii). …

12: 4.5 Inequalities

… ►

4.5.3

|\ln\left(1-x\right)|<\tfrac{3}{2}x,

0<x\leq 0.5828\dots

ⓘ

Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 4.1.35
Referenced by:: §4.5(i)
Permalink:: http://dlmf.nist.gov/4.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.5(i), §4.5 and Ch.4

… ►

4.5.6

|\ln\left(1+z\right)|\leq-\ln\left(1-|z|\right),

|z|<1

ⓘ

Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 4.1.38
Referenced by:: §4.5(i)
Permalink:: http://dlmf.nist.gov/4.5.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §4.5(i), §4.5 and Ch.4

… ►

4.5.15

\tfrac{1}{4}|z|<|e^{z}-1|<\tfrac{7}{4}|z|,

0<|z|<1

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable
A&S Ref:: 4.2.38
Referenced by:: §4.5(ii)
Permalink:: http://dlmf.nist.gov/4.5.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §4.5(ii), §4.5 and Ch.4

►

4.5.16

|e^{z}-1|\leq e^{|z|}-1\leq|z|e^{|z|},

z\in\mathbb{C}

ⓘ

Symbols:: $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable
A&S Ref:: 4.2.39
Referenced by:: §4.5(ii)
Permalink:: http://dlmf.nist.gov/4.5.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §4.5(ii), §4.5 and Ch.4

…

13: 15.15 Sums

… ►Here

z_{0}

(

{\neq 0}

) is an arbitrary complex constant and the expansion converges when

|z-z_{0}|>\max(|z_{0}|,|z_{0}-1|)

. …

14: 1.2 Elementary Algebra

… ►

§1.2(v) Matrices, Vectors, Scalar Products, and Norms

… ►

Vector Norms

… ►the

l^{1}

norm … ►

Norms of Square Matrices

►Let

\left\|{\mathbf{x}}\right\|=\left\|{\mathbf{x}}\right\|_{2}

the

l^{2}

norm, and

\mathbf{E}_{n}

the space of all

n

-dimensional vectors. …

15: 5.3 Graphics

… ►

See accompanying text — Figure 5.3.4: $|\Gamma\left(x+iy\right)|$ . Magnify 3D Help

►

16: 21.2 Definitions

… ►This

g

-tuple Fourier series converges absolutely and uniformly on compact sets of the

\mathbf{z}

and

\boldsymbol{{\Omega}}

spaces; hence

\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)

is an analytic function of (each element of)

\mathbf{z}

and (each element of)

\boldsymbol{{\Omega}}

\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)

is also referred to as a theta function with

g

components, a

g

-dimensional theta function or as a genus

g

theta function. ►For numerical purposes we use the scaled Riemann theta function

\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)

, defined by (Deconinck et al. (2004)), …

\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)

is a bounded nonanalytic function of

\mathbf{z}

. … ►

21.2.8

\theta\left(z\middle|\Omega\right)=\theta_{3}\left(\pi z\middle|\Omega\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function and $\pi$ : the ratio of the circumference of a circle to its diameter
Permalink:: http://dlmf.nist.gov/21.2.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §21.2(iii), §21.2 and Ch.21

…

17: 20.10 Integrals

… ►

20.10.1

\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle|ix^{2}\right)\,\mathrm{d}x=2^% {s}(1-2^{-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),

\Re s>1

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Referenced by:: §20.10(i), Erratum (V1.1.5) for §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E1
Encodings:: pMML, png, TeX
Modification (effective with 1.1.5):: Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>1$ because $\theta_{2}\left(0\middle|ix^{2}\right)=x^{-1}\theta_{4}\left(0\middle|ix^{-2}% \right)\sim x^{-1}$ as $x\to 0+$ , which follows from (20.7.31) and (20.4.5).
See also:: Annotations for §20.10(i), §20.10 and Ch.20

►

20.10.2

\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|ix^{2}\right)-1)\,\mathrm{d}% x=\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),

\Re s>1

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Referenced by:: §20.10(i), Erratum (V1.1.5) for §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E2
Encodings:: pMML, png, TeX
Modification (effective with 1.1.5):: Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>1$ because $\theta_{3}\left(0\middle|ix^{2}\right)=x^{-1}\theta_{3}\left(0\middle|ix^{-2}% \right)\sim x^{-1}$ as $x\to 0+$ , which follows from (20.7.32) and (20.4.4).
See also:: Annotations for §20.10(i), §20.10 and Ch.20

►

20.10.3

\int_{0}^{\infty}x^{s-1}(1-\theta_{4}\left(0\middle|ix^{2}\right))\,\mathrm{d}% x=(1-2^{1-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),

\Re s>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Referenced by:: §20.10(i), Erratum (V1.1.5) for §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E3
Encodings:: pMML, png, TeX
Modification (effective with 1.1.5):: Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>0$ because $1-\theta_{4}\left(0\middle|ix^{2}\right)=1-x^{-1}\theta_{2}\left(0\middle|ix^{% -2}\right)\sim 1$ as $x\to 0+$ , which follows from (20.7.33) and (20.4.3).
See also:: Annotations for §20.10(i), §20.10 and Ch.20

… ►Let

s

\ell

, and

\beta

be constants such that

\Re s>0

\ell>0

, and

\left|\Re\beta\right|+\left|\Im\beta\right|\leq\ell

. … ►

20.10.4

\int_{0}^{\infty}e^{-st}\theta_{1}\left(\frac{\beta\pi}{2\ell}\middle|\frac{i% \pi t}{\ell^{2}}\right)\,\mathrm{d}t=\int_{0}^{\infty}e^{-st}\theta_{2}\left(% \frac{(1+\beta)\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\,\mathrm{d}t=% -\frac{\ell}{\sqrt{s}}\sinh\left(\beta\sqrt{s}\right)\operatorname{sech}\left(% \ell\sqrt{s}\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $s$ : parameter, $\beta$ : parameter and $\ell$ : parameter
Permalink:: http://dlmf.nist.gov/20.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §20.10(ii), §20.10 and Ch.20

…

18: 20.7 Identities

… ►

20.7.16

\theta_{1}\left(2z\middle|2\tau\right)=A\theta_{1}\left(z\middle|\tau\right)% \theta_{2}\left(z\middle|\tau\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $A$
Permalink:: http://dlmf.nist.gov/20.7.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(vi), §20.7 and Ch.20

… ►

20.7.19

\theta_{4}\left(2z\middle|2\tau\right)=A\theta_{3}\left(z\middle|\tau\right)% \theta_{4}\left(z\middle|\tau\right).

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $z$ : complex, $\tau$ : lattice parameter and $A$
Permalink:: http://dlmf.nist.gov/20.7.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(vi), §20.7 and Ch.20

… ►

20.7.21

\theta_{1}\left(4z\middle|4\tau\right)=B\theta_{1}\left(z\middle|\tau\right)% \theta_{1}\left(\tfrac{1}{4}\pi-z\middle|\tau\right)\*\theta_{1}\left(\tfrac{1% }{4}\pi+z\middle|\tau\right)\theta_{2}\left(z\middle|\tau\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex, $\tau$ : lattice parameter and $B$
Permalink:: http://dlmf.nist.gov/20.7.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(vi), §20.7 and Ch.20

… ►

20.7.28

\theta_{3}\left(z\middle|\tau+1\right)=\theta_{4}\left(z\middle|\tau\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $z$ : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.7.E28
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(viii), §20.7 and Ch.20

►

20.7.29

\theta_{4}\left(z\middle|\tau+1\right)=\theta_{3}\left(z\middle|\tau\right).

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $z$ : complex and $\tau$ : lattice parameter
Permalink:: http://dlmf.nist.gov/20.7.E29
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(viii), §20.7 and Ch.20

…

19: 21.3 Symmetry and Quasi-Periodicity

… ►

21.3.1

\theta\left(-\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=\theta\left(% \mathbf{z}\middle|\boldsymbol{{\Omega}}\right),

ⓘ

Symbols:: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function and $\boldsymbol{{\Omega}}$ : a Riemann matrix
Permalink:: http://dlmf.nist.gov/21.3.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §21.3(i), §21.3 and Ch.21

►

21.3.2

\theta\left(\mathbf{z}+\mathbf{m}_{1}\middle|\boldsymbol{{\Omega}}\right)=% \theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right),

ⓘ

Symbols:: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function and $\boldsymbol{{\Omega}}$ : a Riemann matrix
Permalink:: http://dlmf.nist.gov/21.3.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §21.3(i), §21.3 and Ch.21

►when

\mathbf{m}_{1}\in{\mathbb{Z}}^{g}.

Thus

\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)

is periodic, with period

1

, in each element of

\mathbf{z}

. … ►

21.3.3

\theta\left(\mathbf{z}+\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}% \middle|\boldsymbol{{\Omega}}\right)=e^{-2\pi i\left(\frac{1}{2}\mathbf{m}_{2}% \cdot\boldsymbol{{\Omega}}\cdot\mathbf{m}_{2}+\mathbf{m}_{2}\cdot\mathbf{z}% \right)}\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right),

ⓘ

Symbols:: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $\boldsymbol{{\Omega}}$ : a Riemann matrix
Permalink:: http://dlmf.nist.gov/21.3.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §21.3(i), §21.3 and Ch.21

… ►

21.3.6

\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(-\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=(-1)^{4\boldsymbol{{% \alpha}}\cdot\boldsymbol{{\beta}}}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{% \alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right).

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.3.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §21.3(ii), §21.3 and Ch.21

…

20: 10.14 Inequalities; Monotonicity

… ►

|J_{\nu}\left(x\right)|\leq 1,

\nu\geq 0,x\in\mathbb{R}

►

|J_{\nu}\left(x\right)|\leq 2^{-\frac{1}{2}},

\nu\geq 1,x\in\mathbb{R}

… ►

10.14.3

|J_{n}\left(z\right)|\leq e^{|\Im z|},

n\in\mathbb{Z}

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $\mathbb{Z}$ : set of all integers, $n$ : integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §10.14 and Ch.10

►

10.14.4

|J_{\nu}\left(z\right)|\leq\frac{|\tfrac{1}{2}z|^{\nu}e^{|\Im z|}}{\Gamma\left% (\nu+1\right)},

\nu\geq-\frac{1}{2}

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.62 (corrected)
Permalink:: http://dlmf.nist.gov/10.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §10.14 and Ch.10

… ►

10.14.8

|J_{n}\left(nz\right)|\leq\frac{\left|z^{n}\exp\left(n(1-z^{2})^{\frac{1}{2}}% \right)\right|}{\left|1+(1-z^{2})^{\frac{1}{2}}\right|^{n}},

n=0,1,2,\dots

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\exp\NVar{z}$ : exponential function, $n$ : integer and $z$ : complex variable
A&S Ref:: 9.1.63
Permalink:: http://dlmf.nist.gov/10.14.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §10.14, §10.14 and Ch.10

…