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1: 21.7 Riemann Surfaces

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21.7.8 $$\zeta ={\sum _{j=1}^g{\omega }_j\frac{\partial }{\partial z_j}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)|}_{𝐳=\mathrm{𝟎}}.$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $g$: positive integer, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector and $\omega$: differential

Permalink:

http://dlmf.nist.gov/21.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(ii), §21.7 and Ch.21

►Then the prime form on the corresponding compact Riemann surface $\mathrm{\Gamma }$ is defined by ►

21.7.9 $$E(P_1,P_2)=\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left({\int }_{P_1}^{P_2}𝝎\right|𝛀)/\left(\sqrt{\zeta (P_1)}\sqrt{\zeta (P_2)}\right),$$

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Defines:

$E(P_1,P_2)$: prime form (locally)

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\int$: integral, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector, $P_1,P_2$: points and $𝝎$: vector of differentials

Permalink:

http://dlmf.nist.gov/21.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(ii), §21.7 and Ch.21

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21.7.10 $$\theta \left(𝐳+{\int }_{P_1}^{P_3}𝝎\right|𝛀)\theta \left(𝐳+{\int }_{P_2}^{P_4}𝝎\right|𝛀)E(P_3,P_2)E(P_1,P_4)+\theta \left(𝐳+{\int }_{P_2}^{P_3}𝝎\right|𝛀)\theta \left(𝐳+{\int }_{P_1}^{P_4}𝝎\right|𝛀)E(P_3,P_1)E(P_4,P_2)=\theta \left(𝐳\right|𝛀)\theta \left(𝐳+{\int }_{P_1}^{P_3}𝝎+{\int }_{P_2}^{P_4}𝝎\right|𝛀)E(P_1,P_2)E(P_3,P_4),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\int$: integral, $𝛀$: a Riemann matrix, $E(P_1,P_2)$: prime form, $P_1,P_2$: points and $𝝎$: vector of differentials

Referenced by:

§21.7(ii)

Permalink:

http://dlmf.nist.gov/21.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(ii), §21.7 and Ch.21

…

2: 27.12 Asymptotic Formulas: Primes

… ►A Mersenne prime is a prime of the form $2^p-1$. …

3: 1.12 Continued Fractions

… ►A contraction of a continued fraction $C$ is a continued fraction $C^{\prime }$ whose convergents $\{C_n^{\prime }\}$ form a subsequence of the convergents $\{C_n\}$ of $C$. …

4: 25.2 Definition and Expansions

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25.2.12 $$\zeta \left(s\right)=\frac{{(2\pi )}^s{\mathrm{e}}^{-s-(\gamma s/2)}}{2(s-1)\mathrm{\Gamma }\left(\frac{1}{2}s+1\right)}\prod _{\rho }\left(1-\frac{s}{\rho }\right){\mathrm{e}}^{s/\rho },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $s$: complex variable and $\rho$: zeros

Keywords:

infinite product, primes

Source:

Titchmarsh (1986b, (2.12.6), (2.12.8), p. 30–31)

A&S Ref:

23.2.10 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iv), §25.2 and Ch.25

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5: 27.2 Functions

… ►An equivalent form states that the $n$th prime $p_n$ (when the primes are listed in increasing order) is asymptotic to $n\ln n$ as $n\to \mathrm{\infty }$: …

6: 19.26 Addition Theorems

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$$({\xi }^{\prime },{\eta }^{\prime },{\zeta }^{\prime })=(x+\mu ,y+\mu ,z+\mu ),$$

… ►(Note that $\xi {\zeta }^{\prime }+{\eta }^{\prime }\zeta -\xi {\eta }^{\prime }={\xi }^{\prime }\zeta +\eta {\zeta }^{\prime }-{\xi }^{\prime }\eta$.) Equivalent forms of (19.26.2) are given by …Equivalent forms of (19.26.11) are given by … ►Equivalent forms are given by (19.22.22). …

7: 32.10 Special Function Solutions

… ►The solution (32.10.34) is an essentially transcendental function of both constants of integration since ${\text{P}}_{\text{VI}}$ with $\alpha =\beta =\gamma =0$ and $\delta =\frac{1}{2}$ does not admit an algebraic first integral of the form $P(z,w,w^{\prime },C)=0$, with $C$ a constant. …

8: 9.10 Integrals

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9.10.1 $${\int }_z^{\mathrm{\infty }}\mathrm{Ai}\left(t\right)dt=\pi \left(\mathrm{Ai}\left(z\right){\mathrm{Gi}}^{\prime }\left(z\right)-{\mathrm{Ai}}^{\prime }\left(z\right)\mathrm{Gi}\left(z\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.12.4) and its differentiated form with the first of (9.10.11)

Referenced by:

(9.10.2)

Permalink:

http://dlmf.nist.gov/9.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(i), §9.10 and Ch.9

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9.10.2 $${\int }_{-\mathrm{\infty }}^z\mathrm{Ai}\left(t\right)dt=\pi \left(\mathrm{Ai}\left(z\right){\mathrm{Hi}}^{\prime }\left(z\right)-{\mathrm{Ai}}^{\prime }\left(z\right)\mathrm{Hi}\left(z\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.12.11) and its differentiated form with (9.10.1), then apply (9.2.7) and (9.10.11)

Permalink:

http://dlmf.nist.gov/9.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(i), §9.10 and Ch.9

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9.10.3 $${\int }_{-\mathrm{\infty }}^z\mathrm{Bi}\left(t\right)dt={\int }_0^z\mathrm{Bi}\left(t\right)dt=\pi \left({\mathrm{Bi}}^{\prime }\left(z\right)\mathrm{Gi}\left(z\right)-\mathrm{Bi}\left(z\right){\mathrm{Gi}}^{\prime }\left(z\right)\right)=\pi \left(\mathrm{Bi}\left(z\right){\mathrm{Hi}}^{\prime }\left(z\right)-{\mathrm{Bi}}^{\prime }\left(z\right)\mathrm{Hi}\left(z\right)\right).$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.12.5) and its differentiated form with (9.2.7).

Permalink:

http://dlmf.nist.gov/9.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(i), §9.10 and Ch.9

…

9: 2.4 Contour Integrals

… ►If $p^{\prime }(t_0)\ne 0$, then $\mu =1$, $\lambda$ is a positive integer, and the two resulting asymptotic expansions are identical. …However, if $p^{\prime }(t_0)=0$, then $\mu \ge 2$ and different branches of some of the fractional powers of $p_0$ are used for the coefficients $b_s$; again see §2.3(iii). … ►Zeros of $p^{\prime }(t)$ are called saddle points (or cols) owing to the shape of the surface $|p(t)|$, $t\in ℂ$, in their vicinity. … ►In the commonest case the interior minimum $t_0$ of $\mathrm{\Re }\left(zp(t)\right)$ is a simple zero of $p^{\prime }(t)$. The final expansion then has the form …

10: 22.4 Periods, Poles, and Zeros

… ►Again, one member of each congruent set of zeros appears in the second row; all others are generated by translations of the form $2mK+2n\mathrm{i}{K}^{\prime }$, where $m,n\in \mathbb{Z}$. …