relation to theta functions

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21: 25.10 Zeros

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§25.10(i) Distribution

… ► ►Calculations relating to the zeros on the critical line make use of the real-valued function …where … ►Sign changes of $Z(t)$ are determined by multiplying (25.9.3) by $\exp \left(\mathrm{i}\vartheta (t)\right)$ to obtain the Riemann–Siegel formula: …

22: 20.5 Infinite Products and Related Results

§20.5 Infinite Products and Related Results

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Jacobi’s Triple Product

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§20.5(iii) Double Products

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23: 36.7 Zeros

… ►Close to the $y$-axis the approximate location of these zeros is given by … ►Deep inside the bifurcation set, that is, inside the three-cusped astroid (36.4.10) and close to the part of the $z$-axis that is far from the origin, the zero contours form an array of rings close to the planes …, $y=0$), the number of rings in the $m$th row, measured from the origin and before the transition to hairpins, is given by …Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral. There are also three sets of zero lines in the plane $z=0$ related by $2\pi /3$ rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates $(x=r\cos \theta ,y=r\sin \theta )$ is given by …

24: 33.11 Asymptotic Expansions for Large $\rho$

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33.11.1 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\sim {\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k{\left(b\right)}_k}{k!{(\pm 2\mathrm{i}\rho )}^k},$$

ⓘ

Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $a$: coefficient and $b$: coefficient

A&S Ref:

14.5.9

Referenced by:

Erratum (V1.0.19) for Equation (33.11.1), Erratum (V1.0.22) for Equation (33.11.1)

Permalink:

http://dlmf.nist.gov/33.11.E1

Encodings:

pMML, png, TeX

Errata (effective with 1.0.22):

Previously this formula was expressed as an equality. Since this formula expresses an asymptotic expansion, it has been corrected by using instead an $\sim$ relation.

Reported 2019-01-29 by Gergő Nemes

Errata (effective with 1.0.19):

Originally the factor in the denominator on the right-hand side was written incorrectly as ${(\mp 2\mathrm{i}\rho )}^k$. This has been corrected to ${(\pm 2\mathrm{i}\rho )}^k$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.11 and Ch.33

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25: 22.20 Methods of Computation

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§22.20(i) Via Theta Functions

►A powerful way of computing the twelve Jacobian elliptic functions for real or complex values of both the argument $z$ and the modulus $k$ is to use the definitions in terms of theta functions given in §22.2, obtaining the theta functions via methods described in §20.14. … ►If either $k$ or $x$ is complex then (22.2.6) gives the definition of $\mathrm{dn}(x,k)$ as a quotient of theta functions. … ►

§22.20(vi) Related Functions

… ►Jacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14. …

26: 14.5 Special Values

§14.5 Special Values

… ►In this subsection and the next two, and $\xi >0$. … ►

§14.5(v) $\mu =0$, $\nu =\pm \frac{1}{2}$

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27: 10.68 Modulus and Phase Functions

§10.68 Modulus and Phase Functions

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§10.68(i) Definitions

… ►where $M_{\nu }\left(x\right)(>0)$, $N_{\nu }\left(x\right)(>0)$, ${\theta }_{\nu }\left(x\right)$, and ${\varphi }_{\nu }\left(x\right)$ are continuous real functions of $x$ and $\nu$, with the branches of ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ chosen to satisfy (10.68.18) and (10.68.21) as $x\to \mathrm{\infty }$. … ►Equations (10.68.8)–(10.68.14) also hold with the symbols $\mathrm{ber}$, $\mathrm{bei}$, $M$, and $\theta$ replaced throughout by $\ker$, $\mathrm{kei}$, $N$, and $\varphi$, respectively. … ►However, care needs to be exercised with the branches of the phases. …

28: 10.9 Integral Representations

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Poisson’s and Related Integrals

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Schläfli’s and Related Integrals

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Mehler–Sonine and Related Integrals

… ►See Paris and Kaminski (2001, p. 116) for related results. … ► …

29: 19.2 Definitions

… ►Let $s^2(t)$ be a cubic or quartic polynomial in $t$ with simple zeros, and let $r(s,t)$ be a rational function of $s$ and $t$ containing at least one odd power of $s$. … ►Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). … ►Lastly, corresponding to Legendre’s incomplete integral of the third kind we have … ►

§19.2(iv) A Related Function: $R_C(x,y)$

… ►In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). …

30: 18.15 Asymptotic Approximations

… ►as $n\to \mathrm{\infty }$, uniformly with respect to $\theta \in [\delta ,\pi -\delta ]$. … ►as $n\to \mathrm{\infty }$, uniformly with respect to $\theta \in [\delta ,\pi -\delta ]$. … ►as $n\to \mathrm{\infty }$ uniformly with respect to $\theta \in [\delta ,\pi -\delta ]$, where … ►as $n\to \mathrm{\infty }$, uniformly with respect to $\theta \in [\delta ,\pi -\delta ]$, where … ►With $\mu =\sqrt{2n+1}$ the expansions in Chapter 12 are for the parabolic cylinder function $U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})$, which is related to the Hermite polynomials via …