periodic Euler functions

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11: Bibliography B

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Periods for Calabi-Yau and Landau-Ginzburg vacua. Nuclear Phys. B 419 (2), pp. 352–403.

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External Links:

ISSN 0550-3213, Document, MathReview (Bruce Hunt), ZentralBlatt

Referenced by:

§16.23(i).

Encodings:

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).

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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Stability of repulsive Bose-Einstein condensates in a periodic potential. Phys. Rev. E (3) 63 (036612), pp. 1–11.

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External Links:

Document

Referenced by:

§29.19(i).

Encodings:

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Euler functions $E_{\alpha }(z)$ with complex $\alpha$ and applications. In Approximation, probability, and related fields (Santa Barbara, CA, 1993), G. Anastassiou and S. T. Rachev (Eds.), pp. 127–150.

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External Links:

MathReview (R. N. Kalia), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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12: 29.3 Definitions and Basic Properties

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§29.3(i) Eigenvalues

►For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2K$ or $4K$. … ►

§29.3(iv) Lamé Functions

… ►They are called Lamé functions with real periods and of order $\nu$ , or more simply, Lamé functions. … ►

§29.3(v) Normalization

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13: 31.2 Differential Equations

… ►This equation has regular singularities at $0,1,a,\mathrm{\infty }$, with corresponding exponents $\{0,1-\gamma \}$, $\{0,1-\delta \}$, $\{0,1-ϵ\}$, $\{\alpha ,\beta \}$, respectively (§2.7(i)). … ►

§31.2(iv) Doubly-Periodic Forms

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Jacobi’s Elliptic Form

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Weierstrass’s Form

… ► $w(z)=z^{1-\gamma }{w}_1(z)$ satisfies (31.2.1) if $w_1$ is a solution of (31.2.1) with transformed parameters $q_1=q+(a\delta +ϵ)(1-\gamma )$; ${\alpha }_1=\alpha +1-\gamma$, ${\beta }_1=\beta +1-\gamma$, ${\gamma }_1=2-\gamma$. …

14: 28.32 Mathematical Applications

§28.32 Mathematical Applications

… ► … ► … ►where $c$ is a parameter, , , and . …The first is the $2\pi$-periodicity of the solutions; the second can be their asymptotic form. …

15: 27.8 Dirichlet Characters

§27.8 Dirichlet Characters

►If $k$ $(>1)$ is a given integer, then a function $\chi \left(n\right)$ is called a Dirichlet character (mod $k$) if it is completely multiplicative, periodic with period $k$, and vanishes when $\left(n,k\right)>1$. … ►For any character $\chi (modk)$, $\chi \left(n\right)\ne 0$ if and only if $\left(n,k\right)=1$, in which case the Euler–Fermat theorem (27.2.8) implies ${\left(\chi \left(n\right)\right)}^{\varphi \left(k\right)}=1$. There are exactly $\varphi \left(k\right)$ different characters (mod $k$), which can be labeled as ${\chi }_1,\mathrm{\dots },{\chi }_{\varphi \left(k\right)}$. …If $\left(n,k\right)=1$, then the characters satisfy the orthogonality relation …

16: 3.5 Quadrature

… ►If in addition $f$ is periodic, $f\in C^k(ℝ)$, and the integral is taken over a period, then … ►The given quantities ${\gamma }_n$ follow from (18.2.5), (18.2.7), Table 18.3.1, and the relation ${\gamma }_n=h_n/k_n^2$. … ►In the case of Chebyshev weight functions $w(x)={(1-x)}^{\alpha }{(1+x)}^{\beta }$ on $[-1,1]$, with $\left|\alpha \right|=\left|\beta \right|=\frac{1}{2}$, the nodes $x_k$, weights $w_k$, and error constant ${\gamma }_n$, are as follows: … ►

Example

… ►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. …

17: Errata

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In previous versions of the DLMF, in §8.18(ii), the notation $\tilde{\mathrm{\Gamma }}(z)$ was used for the scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$. Now in §8.18(ii), we adopt the notation which was introduced in Version 1.1.7 (October 15, 2022) and correspondingly, Equation (8.18.13) has been removed. In place of Equation (8.18.13), it is now mentioned to see (5.11.3).

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The generalized hypergeometric function of matrix argument ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$, was linked inadvertently as its single variable counterpart ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$. Furthermore, the Jacobi function of matrix argument $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and the Laguerre function of matrix argument $L_{\nu }^{(\gamma )}\left(𝐓\right)$, were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and $L_{\nu }^{(\gamma )}\left(𝐓\right)$. In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

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The table of extrema for the Euler gamma function $\mathrm{\Gamma }$ had several entries in the $x_n$ column that were wrong in the last 2 or 3 digits. These have been corrected and 10 extra decimal places have been included.

$n$	$x_n$	$\mathrm{\Gamma }\left(x_n\right)$
0	$\mathrm{1.46163\hspace{0.33em}21449\hspace{0.33em}68362\hspace{0.33em}34126}$	$\mathrm{0.88560\hspace{0.33em}31944\hspace{0.33em}10888\hspace{0.33em}70028}$
1	$-\mathrm{0.50408\hspace{0.33em}30082\hspace{0.33em}64455\hspace{0.33em}40926}$	$-\mathrm{3.54464\hspace{0.33em}36111\hspace{0.33em}55005\hspace{0.33em}08912}$
2	$-\mathrm{1.57349\hspace{0.33em}84731\hspace{0.33em}62390\hspace{0.33em}45878}$	$\mathrm{2.30240\hspace{0.33em}72583\hspace{0.33em}39680\hspace{0.33em}13582}$
3	$-\mathrm{2.61072\hspace{0.33em}08684\hspace{0.33em}44144\hspace{0.33em}65000}$	$-\mathrm{0.88813\hspace{0.33em}63584\hspace{0.33em}01241\hspace{0.33em}92010}$
4	$-\mathrm{3.63529\hspace{0.33em}33664\hspace{0.33em}36901\hspace{0.33em}09784}$	$\mathrm{0.24512\hspace{0.33em}75398\hspace{0.33em}34366\hspace{0.33em}25044}$
5	$-\mathrm{4.65323\hspace{0.33em}77617\hspace{0.33em}43142\hspace{0.33em}44171}$	$-\mathrm{0.05277\hspace{0.33em}96395\hspace{0.33em}87319\hspace{0.33em}40076}$
6	$-\mathrm{5.66716\hspace{0.33em}24415\hspace{0.33em}56885\hspace{0.33em}53585}$	$\mathrm{0.00932\hspace{0.33em}45944\hspace{0.33em}82614\hspace{0.33em}85052}$
7	$-\mathrm{6.67841\hspace{0.33em}82130\hspace{0.33em}73426\hspace{0.33em}74283}$	$-\mathrm{0.00139\hspace{0.33em}73966\hspace{0.33em}08949\hspace{0.33em}76730}$
8	$-\mathrm{7.68778\hspace{0.33em}83250\hspace{0.33em}31626\hspace{0.33em}03744}$	$\mathrm{0.00018\hspace{0.33em}18784\hspace{0.33em}44909\hspace{0.33em}40419}$
9	$-\mathrm{8.69576\hspace{0.33em}41638\hspace{0.33em}16401\hspace{0.33em}26649}$	$-\mathrm{0.00002\hspace{0.33em}09252\hspace{0.33em}90446\hspace{0.33em}52667}$
10	$-\mathrm{9.70267\hspace{0.33em}25400\hspace{0.33em}01863\hspace{0.33em}73608}$	$\mathrm{0.00000\hspace{0.33em}21574\hspace{0.33em}16104\hspace{0.33em}52285}$

Reported 2018-02-17 by David Smith.

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Originally all six integrands in these equations were incorrect because their numerators contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$. The new equations are:

25.11.6 $$\zeta (s,a)=\frac{1}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $s\ne 1$, $\mathrm{\Re }s>-1$, $a>0$

Reported 2016-05-08 by Clemens Heuberger.

25.11.19 $${\zeta }^{\prime }(s,a)=-\frac{\ln a}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{1-s}}{{(s-1)}^2}+\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2)\ln \left(x+a\right)}{{(x+a)}^{s+2}}dx-\frac{(2s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $\mathrm{\Re }s>-1$, $s\ne 1$, $a>0$

Reported 2016-06-27 by Gergő Nemes.

25.11.20 $${(-1)}^k{\zeta }^{(k)}(s,a)=\frac{{(\ln a)}^k}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum _{r=0}^{k-1}\frac{{(\ln a)}^r}{r!{(s-1)}^{k-r+1}}-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^k}{{(x+a)}^{s+2}}dx+\frac{k(2s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^{k-1}}{{(x+a)}^{s+2}}dx-\frac{k(k-1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^{k-2}}{{(x+a)}^{s+2}}dx,$$ $\mathrm{\Re }s>-1$, $s\ne 1$, $a>0$

Reported 2016-06-27 by Gergő Nemes.

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To increase the regions of validity the logarithms of the gamma function that appears on their left-hand sides have all been changed to $\mathrm{Ln}\mathrm{\Gamma }(\cdot )$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }(\cdot )$ was used, where $\ln$ is the principal branch of the logarithm. These changes were recommended by Philippe Spindel on 2015-02-06.

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18: Bibliography H

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Two-dimensional periodic waves in shallow water. II. Asymmetric waves. J. Fluid Mech. 285, pp. 95–122.

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External Links:

ISSN 0022-1120, Document, MathReview

Referenced by:

Figure 21.9.1, Figure 21.9.1, Figure 21.9.2, Figure 21.9.2, §21.9, Sidebar 21.SB1, Sidebar 21.SB2.

Encodings:

BibTeX

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Two-dimensional periodic waves in shallow water. J. Fluid Mech. 209, pp. 567–589.

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External Links:

ISSN 0022-1120, Document, MathReview

Referenced by:

§21.9, Sidebar 21.SB2.

Encodings:

BibTeX

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Note on Dr. Vacca’s series for $\gamma$ . Quart. J. Math. 43, pp. 215–216.

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External Links:

ZentralBlatt

Referenced by:

(25.2.4), (25.2.5), §25.2(ii).

Encodings:

BibTeX

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A Boole-type Formula involving Conjugate Euler Polynomials. In Charlemagne and his Heritage. 1200 Years of Civilization and Science in Europe, Vol. 2 (Aachen, 1995), P.L. Butzer, H. Th. Jongen, and W. Oberschelp (Eds.), pp. 361–375.

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External Links:

ISBN 2-503-50674-7, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Roots of the Euler polynomials. Pacific J. Math. 64 (1), pp. 181–191.

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External Links:

MathReview (M. Marden), ZentralBlatt

Referenced by:

§24.12(ii).

Encodings:

BibTeX

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19: 1.14 Integral Transforms

… ►The Fourier transform of a real- or complex-valued function $f(t)$ is defined by … ►Moreover, if $ℒf\left(s\right)=O\left(s^{-K}\right)$ in some half-plane $\mathrm{\Re }s\ge \gamma$ and $K>1$, then (1.14.20) holds for $\sigma >\gamma$. … ►

Periodic Functions

… ►The Mellin transform of a real- or complex-valued function $f(x)$ is defined by … ►The Stieltjes transform of a real-valued function $f(t)$ is defined by …

20: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Functions $f,g\in L^2(X,d\alpha )$ for which $⟨f-g,f-g⟩=0$ are identified with each other. … ►Case 3: Periodic Boundary Conditions: $\varphi (0)=\varphi (\pi )$ and ${\varphi }^{\prime }(0)={\varphi }^{\prime }(\pi )$. … ►Eigenfunctions corresponding to the continuous spectrum are non-$L^2$ functions. … ►More generally, continuous spectra may occur in sets of disjoint finite intervals $[{\lambda }_a,{\lambda }_b]\in (0,\mathrm{\infty })$, often called bands, when $q(x)$ is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). … ►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 $1/{\mathrm{\Gamma }}_{\mathrm{res}}$. …