periodic zeta function

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11: 22.2 Definitions

§22.2 Definitions

… ►With … ►As a function of $z$, with fixed $k$, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … … ►The Jacobian functions are related in the following way. …

12: Bibliography B

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A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

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BibTeX

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On the Hurwitz zeta-function. Rocky Mountain J. Math. 2 (1), pp. 151–157.

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External Links:

MathReview (K. Thanigasalam), ZentralBlatt

Referenced by:

(25.11.26).

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BibTeX

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Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§24.16(iii), §24.8(ii).

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BibTeX

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The Riemann-Siegel expansion for the zeta function: High orders and remainders. Proc. Roy. Soc. London Ser. A 450, pp. 439–462.

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External Links:

ISSN 0962-8444, FullText, MathReview (Christian Grosche), ZentralBlatt

Referenced by:

§25.10(ii).

Encodings:

BibTeX

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Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems. J. Phys. A 34 (40), pp. 8199–8220.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§29.19(i).

Encodings:

BibTeX

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13: 32.10 Special Function Solutions

§32.10 Special Function Solutions

… ►with $\zeta =\sqrt{{\epsilon }_1{\epsilon }_2}z$, $\nu =\frac{1}{2}\alpha {\epsilon }_1$, and $C_1$, $C_2$ arbitrary constants. … ►with $\zeta ={\epsilon }_{3}z$, $\kappa =\frac{1}{2}(a-b+1)$, $\mu =\frac{1}{2}(a+b)$, and $C_1$, $C_2$ arbitrary constants. … ►where the fundamental periods $2{\varphi }_1$ and $2{\varphi }_2$ are linearly independent functions satisfying the hypergeometric equation … ►

14: 2.10 Sums and Sequences

… ►As in §24.2, let $B_n$ and $B_n\left(x\right)$ denote the $n$th Bernoulli number and polynomial, respectively, and ${\tilde{B}}_n\left(x\right)$ the $n$th Bernoulli periodic function $B_n\left(x-\lfloor x\rfloor \right)$. … ►Then … ►From §24.12(i), (24.2.2), and (24.4.27), ${\tilde{B}}_{2m}\left(x\right)-B_{2m}$ is of constant sign ${(-1)}^m$. … ►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

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15: 4.2 Definitions

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§4.2(iii) The Exponential Function

… ►It has period $2\pi \mathrm{i}$: … ►If $\zeta \ne 0$ then … ►

§4.2(iv) Powers

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16: 28.32 Mathematical Applications

… ► ►This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting $\zeta =\mathrm{i}\xi$, $z=\eta$ in (28.32.3)). … ►Let $u(\zeta )$ be a solution of Mathieu’s equation (28.2.1) and $K(z,\zeta )$ be a solution of …approaches the same value when $\zeta$ tends to the endpoints of $ℒ$. … ►The first is the $2\pi$-periodicity of the solutions; the second can be their asymptotic form. …

17: 3.5 Quadrature

… ►If in addition $f$ is periodic, $f\in C^k(ℝ)$, and the integral is taken over a period, then … ►The complex Gauss nodes ${\zeta }_k$ have positive real part for all $s>0$. … ►The integral (3.5.39) has the form (3.5.35) if we set $\zeta =tp$, $c=t\sigma$, and $f(\zeta )=t^{-1}{\zeta }^{s}G(\zeta /t)$. … ►With the transformation $\zeta ={\lambda }^{2}t$, (3.5.42) becomes …The integrand can be extended as a periodic $C^{\mathrm{\infty }}$ function on $ℝ$ with period $2\pi$ and as noted in §3.5(i), the trapezoidal rule is exceptionally efficient in this case. …

18: 22.16 Related Functions

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§22.16(i) Jacobi’s Amplitude ($\mathrm{am}$) Function

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Quasi-Periodicity

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§22.16(ii) Jacobi’s Epsilon Function

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Quasi-Addition and Quasi-Periodic Formulas

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§22.16(iii) Jacobi’s Zeta Function

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19: 3.4 Differentiation

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§3.4(ii) Analytic Functions

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3.4.17 $$\frac{1}{k!}{f}^{(k)}(x_0)=\frac{1}{2\pi \mathrm{i}}{\int }_C\frac{f(\zeta )}{{(\zeta -x_0)}^{k+1}}d\zeta ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral and $C$: simple closed contour

Permalink:

http://dlmf.nist.gov/3.4.E17

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Annotations for §3.4(ii), §3.4 and Ch.3

… ►As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands. … ► … ►

3.4.28 $${\nabla }^2{u}_{0,0}=\frac{1}{h^2}(u_{1,0}+u_{0,1}+u_{-1,0}+u_{0,-1}-4u_{0,0})+O\left(h^2\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding and $u$: function

A&S Ref:

25.3.30

Permalink:

http://dlmf.nist.gov/3.4.E28

Encodings:

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See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

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20: 23.6 Relations to Other Functions

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§23.6(i) Theta Functions

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§23.6(ii) Jacobian Elliptic Functions

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§23.6(iii) General Elliptic Functions

►For representations of general elliptic functions (§23.2(iii)) in terms of $\sigma \left(z\right)$ and $\mathrm{\wp }\left(z\right)$ see Lawden (1989, §§8.9, 8.10), and for expansions in terms of $\zeta \left(z\right)$ see Lawden (1989, §8.11). … ►