【AG真人官方qee9.com】加拿大28是不是骗局局89WEE4Mn

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21: Bibliography U

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Theory of the Whittaker-Hill equation. Proc. Roy. Soc. Edinburgh Sect. A 69, pp. 28–44.

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External Links:

MathReview (R. K. Rathy), ZentralBlatt

Referenced by:

§28.31(i), §28.31(ii), §28.32(ii).

Encodings:

BibTeX

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22: Bibliography Y

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Computation of the derivatives of the Riemann zeta-function in the complex domain. USSR Comput. Math. and Math. Phys. 28 (4), pp. 115–124.

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Notes:

Includes Fortran programs.

External Links:

Document, ZentralBlatt

Referenced by:

§25.18(i), §25.21(iii).

Encodings:

BibTeX

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23: Bibliography K

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Replica field theories, Painlevé transcendents, and exact correlation functions. Phys. Rev. Lett. 89 (25), pp. (250201–1)–(250201–4).

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External Links:

ISSN 0031-9007, Document, MathReview, ZentralBlatt

Referenced by:

§32.16.

Encodings:

BibTeX

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Zeros of exceptional Hermite polynomials. J. Approx. Theory 200, pp. 28–39.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Mario Pérez Riera)

Referenced by:

§18.39(i).

Encodings:

BibTeX

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24: 25.14 Lerch’s Transcendent

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25.14.6 $$\mathrm{\Phi }(z,s,a)=\frac{1}{2}{a}^{-s}+{\int }_0^{\mathrm{\infty }}\frac{z^x}{{(a+x)}^s}dx-2{\int }_0^{\mathrm{\infty }}\frac{\sin \left(x\ln z-s\arctan \left(x/a\right)\right)}{{(a^2+x^2)}^{s/2}({\mathrm{e}}^{2\pi x}-1)}dx,$$ $\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$.

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Symbols:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.11.4), p. 28)

Notes:

For the case see Erdélyi et al. (1953a, (1.11.4), p. 28). In the case $\left|z\right|=1$ one checks that the first integral converges absolutely iff $\mathrm{\Re }s>1$.

Referenced by:

Erratum (V1.1.4) for Equation (25.14.6)

Permalink:

http://dlmf.nist.gov/25.14.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The constraint which originally read “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $|z|=1,\mathrm{\Re }a>0$” has been improved to be “$\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$”.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.14(ii), §25.14 and Ch.25

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25: 31.12 Confluent Forms of Heun’s Equation

… ►Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation. …

26: 25.10 Zeros

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25.10.1 $$Z(t)\equiv \exp \left(\mathrm{i}\vartheta (t)\right)\zeta \left(\frac{1}{2}+\mathrm{i}t\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\equiv$: equals by definition, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $Z(t)$: zeros function and $\vartheta (t)$: function

Keywords:

definition

Source:

Titchmarsh (1986b, (4.17.2), p. 89)

Permalink:

http://dlmf.nist.gov/25.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(i), §25.10 and Ch.25

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25.10.2 $$\vartheta (t)\equiv \mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)-\frac{1}{2}t\ln \pi$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $\vartheta (t)$: function and $\chi (s)$: function

Keywords:

definition

Proof sketch:

Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta (t)\equiv -\frac{1}{2}\mathrm{ph}\chi (\frac{1}{2}+\mathrm{i}t)$, by using (25.9.2), (1.9.18), (1.9.20), $\mathrm{\Gamma }\left(\overline{z}\right)=\overline{\mathrm{\Gamma }\left(z\right)}$, and (4.8.10).

Permalink:

http://dlmf.nist.gov/25.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(i), §25.10 and Ch.25

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25.10.3 $$Z(t)=2\sum _{n=1}^m\frac{\cos \left(\vartheta (t)-t\ln n\right)}{n^{1/2}}+R(t),$$ $m=\lfloor \sqrt{t/(2\pi )}\rfloor$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\lfloor x\rfloor$: floor of $x$ , $\ln z$: principal branch of logarithm function, $m$: nonnegative integer, $n$: nonnegative integer, $Z(t)$: zeros function and $\vartheta (t)$: function

Keywords:

Riemann–Siegel formula

Source:

Titchmarsh (1986b, (4.17.4), p. 89)

Referenced by:

§25.18(i), Erratum (V1.1.4) for Equation (25.10.3)

Permalink:

http://dlmf.nist.gov/25.10.E3

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The constraint $m=\lfloor \sqrt{t/(2\pi )}\rfloor$ was added.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.10(ii), §25.10 and Ch.25

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25.10.4 $$R(t)={(-1)}^{m-1}{\left(\frac{2\pi }{t}\right)}^{1/4}\frac{\cos \left(t-(2m+1)\sqrt{2\pi t}-\frac{1}{8}\pi \right)}{\cos \left(\sqrt{2\pi t}\right)}+O\left(t^{-3/4}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $m$: nonnegative integer

Keywords:

asymptotic approximation

Source:

Titchmarsh (1986b, (4.17.5), p. 89)

Referenced by:

§25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii)

Permalink:

http://dlmf.nist.gov/25.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(ii), §25.10 and Ch.25

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27: Bibliography M

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Asymptotic expansions of ellipsoidal wave functions and their characteristic numbers. Math. Nachr. 31, pp. 89–101.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.11, §29.7(i), §29.7(ii).

Encodings:

BibTeX

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Rational solutions of the second and the fourth Painlevé equations. Funkcial. Ekvac. 28 (1), pp. 1–32.

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External Links:

ISSN 0532-8721, FullText, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.8(iv).

Encodings:

BibTeX

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28: 24.2 Definitions and Generating Functions

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Table 24.2.3: Bernoulli numbers $B_n=N/D$.

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$n$	$N$	$D$	$B_n$
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$28$	$-\mathrm{2\hspace{0.33em}37494\hspace{0.33em}61029}$	$870$	$-\mathrm{2.72982\hspace{0.33em}3107}$	$\mathrm{\times 10⁷}$
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Table 24.2.4: Euler numbers $E_n$.

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$n$	$E_n$
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$28$	$\mathrm{12522\hspace{0.33em}59641\hspace{0.33em}40362\hspace{0.33em}98654\hspace{0.33em}68285}$
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Table 24.2.6: Coefficients $e_{n,k}$ of the Euler polynomials $E_n\left(x\right)={\sum }_{k=0}^n{e}_{n,k}{x}^k$.

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	$k$
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$8$	$0$	$17$	$0$	$-28$	$0$	$14$	$0$	$-4$	$1$
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29: Bibliography P

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Stokes phenomenon demystified. Bull. Inst. Math. Appl. 31 (1-2), pp. 21–28.

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External Links:

ISSN 0905-5628, MathReview Entry, ZentralBlatt

Referenced by:

§2.11(iv).

Encodings:

BibTeX

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Algorithm 28. Algorithm for the computation of Bessel function integrals. J. Comput. Appl. Math. 11 (1), pp. 119–137.

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External Links:

ISSN 0377-0427, Document, ZentralBlatt

Referenced by:

§10.77(ix).

Encodings:

BibTeX

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30: 23.9 Laurent and Other Power Series

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$c_3={\displaystyle \frac{1}{28}}{g}_3,$

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