Riemann theta functions

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21: 25.19 Tables

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Abramowitz and Stegun (1964) tabulates: $\zeta \left(n\right)$, $n=2,3,4,\mathrm{\dots }$, 20D (p. 811); ${\mathrm{Li}}_2\left(1-x\right)$, $x=0(.01)0.5$, 9D (p. 1005); $f(\theta )$, $\theta ={15}^{\circ }(1^{\circ }){30}^{\circ }(2^{\circ }){90}^{\circ }(5^{\circ }){180}^{\circ }$, $f(\theta )+\theta \ln \theta$, $\theta =0(1^{\circ }){15}^{\circ }$, 6D (p. 1006). Here $f(\theta )$ denotes Clausen’s integral, given by the right-hand side of (25.12.9).

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Cloutman (1989) tabulates $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

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Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta \left(s\right)$ for both real and complex $s$. §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta (s,a)$, and §22.17 lists tables for some Dirichlet $L$-functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

22: 31.10 Integral Equations and Representations

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31.10.9

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Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $\cos z$: cosine function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $𝒦(z,t)$: kernel, $\theta$: angle, $\varphi$: angle and $\sigma$: separation constant

Referenced by:

§31.10(i)

Permalink:

http://dlmf.nist.gov/31.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

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31.10.22

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Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $\cos z$: cosine function, $\sin z$: sine function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $𝒦(z;s,t)$: kernel, $r$: radius, $\theta$: angle, $\varphi$: angle, $b$ and $c$

Permalink:

http://dlmf.nist.gov/31.10.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

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23: 25.5 Integral Representations

§25.5 Integral Representations

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§25.5(i) In Terms of Elementary Functions

… ►For ${\theta }_3$ see §20.2(i). For similar representations involving other theta functions see Erdélyi et al. (1954a, p. 339). … ►

§25.5(iii) Contour Integrals

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24: Software Index

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	Open Source													With Book						Commercial
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20 Theta Functions
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …

25: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Eigenfunctions corresponding to the continuous spectrum are non-$L^2$ functions. … ►For example, replacing $2q\cos (2z)$ of (28.2.1) by $\lambda \cos (2\pi \alpha n+\theta )$, $n\in \mathbb{Z}$ gives an almost Mathieu equation which for appropriate $\alpha$ has such properties. … ►Note that the integral in (1.18.66) is not singular if approached separately from above, or below, the real axis: in fact analytic continuation from the upper half of the complex plane, across the cut, and onto higher Riemann Sheets can access complex poles with singularities at discrete energies ${\lambda }_{\mathrm{res}}-\mathrm{i}{\mathrm{\Gamma }}_{\mathrm{res}}/2$ corresponding to quantum resonances, or decaying quantum states with lifetimes proportional to $1/{\mathrm{\Gamma }}_{\mathrm{res}}$. …This is accomplished by the variable change $x\to x{\mathrm{e}}^{\mathrm{i}\theta }$, in $ℒ$, which rotates the continuous spectrum ${𝝈}_c\to {𝝈}_c{\mathrm{e}}^{-2\mathrm{i}\theta }$ and the branch cut of (1.18.66) into the lower half complex plain by the angle $-2\theta$, with respect to the unmoved branch point at $\lambda =0$; thus, providing access to resonances on the higher Riemann sheet should $\theta$ be large enough to expose them. … …

26: Errata

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The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

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The original constraint, $\mathrm{\Re }s>0$, was removed because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$, and therefore $\zeta \left(s\right)-{(s-1)}^{-1}$ is an entire function.

Suggested by John Harper.

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22.19.2 $$\sin \left(\frac{1}{2}\theta (t)\right)=\sin \left(\frac{1}{2}\alpha \right)\mathrm{sn}(t+K,\sin \left(\frac{1}{2}\alpha \right))$$

Originally the first argument to the function $\mathrm{sn}$ was given incorrectly as $t$. The correct argument is $t+K$.

Reported 2014-03-05 by Svante Janson.

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22.19.3 $$\theta (t)=2\mathrm{am}(t\sqrt{E/2},\sqrt{2/E})$$

Originally the first argument to the function $\mathrm{am}$ was given incorrectly as $t$. The correct argument is $t\sqrt{E/2}$.

Reported 2014-03-05 by Svante Janson.

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21.3.4 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶+{𝐦}_1}{𝜷+{𝐦}_2}\right]\left(𝐳\right|𝛀)={\mathrm{e}}^{2\pi \mathrm{i}𝜶\cdot {𝐦}_2}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$$

Originally the vector ${𝐦}_2$ on the right-hand side was given incorrectly as ${𝐦}_1$.

Reported 2012-08-27 by Klaas Vantournhout.

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27: 2.10 Sums and Sequences

… ►Then …${\vartheta }_n$ being some number in the interval $(0,1)$. … ►where $\gamma$ is Euler’s constant (§5.2(ii)) and ${\zeta }^{\prime }$ is the derivative of the Riemann zeta function (§25.2(i)). … ►

§2.10(iii) Asymptotic Expansions of Entire Functions

… ►Hence by the Riemann–Lebesgue lemma (§1.8(i)) …

28: 1.15 Summability Methods

… ► $A(r,\theta )$ is a harmonic function in polar coordinates (1.9.27), and … ►at every point $\theta$ where both limits exist. If $f(\theta )$ is also continuous, then the convergence is uniform for all $\theta$. ►For real-valued $f(\theta )$, if …is the Fourier series of $f(\theta )$, then the series …

29: 27.11 Asymptotic Formulas: Partial Sums

… ►The behavior of a number-theoretic function $f(n)$ for large $n$ is often difficult to determine because the function values can fluctuate considerably as $n$ increases. …where $F(x)$ is a known function of $x$, and $O\left(g(x)\right)$ represents the error, a function of smaller order than $F(x)$ for all $x$ in some prescribed range. …Dirichlet’s divisor problem (unsolved as of 2022) is to determine the least number ${\theta }_0$ such that the error term in (27.11.2) is $O\left(x^{\theta }\right)$ for all $\theta >{\theta }_0$. Huxley (2003) proves that ${\theta }_0\le \frac{131}{416}$. ►Equations (27.11.3)–(27.11.11) list further asymptotic formulas related to some of the functions listed in §27.2. …

30: 25.12 Polylogarithms

… ►When $z={\mathrm{e}}^{\mathrm{i}\theta }$, $0\le \theta \le 2\pi$, (25.12.1) becomes … ►The special case $z=1$ is the Riemann zeta function: $\zeta \left(s\right)={\mathrm{Li}}_s\left(1\right)$. … ►Further properties include …and … ►In terms of polylogarithms …