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1: 20.10 Integrals

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§20.10(i) Mellin Transforms with respect to the Lattice Parameter

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20.10.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}{\theta }_2\left(0\right|\mathrm{i}{x}^2)dx=2^s(1-2^{-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E1

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_2\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_4\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.31) and (20.4.5).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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20.10.2 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}({\theta }_3\left(0\right|\mathrm{i}{x}^2)-1)dx={\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E2

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_3\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_3\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.32) and (20.4.4).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

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2: 18.5 Explicit Representations

… ►For corresponding formulas for Chebyshev, Legendre, and the Hermite ${\mathrm{𝐻𝑒}}_n$ polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). … ► … ►

$T_5\left(x\right)=16x^5-20x^3+5x,$

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$L_6\left(x\right)=\frac{1}{720}{x}^6-\frac{1}{20}{x}^5+\frac{5}{8}{x}^4-\frac{10}{3}{x}^3+\frac{15}{2}{x}^2-6x+1.$

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3: 25.11 Hurwitz Zeta Function

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Figure 25.11.1: Hurwitz zeta function $\zeta (x,a)$, $a$ = 0. …8, 1, $-20\le x\le 10$. … Magnify

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25.11.9 $$\zeta (1-s,a)=\frac{2\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}\cos \left(\frac{1}{2}\pi s-2n\pi a\right),$$ $\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $a=1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable

Keywords:

infinite series, series representation

Source:

Apostol (1976, (9), (10), p. 257)

Referenced by:

(25.13.3), Erratum (V1.1.4) for Equation (25.11.9)

Permalink:

http://dlmf.nist.gov/25.11.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

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

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.11(iv), §25.11 and Ch.25

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25.11.13 $$\zeta (0,a)=\frac{1}{2}-a.$$

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function and $a$: real or complex parameter

Keywords:

special value

Source:

Apostol (1976, p. 268)

Permalink:

http://dlmf.nist.gov/25.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(v), §25.11 and Ch.25

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§25.11(xii) $a$-Asymptotic Behavior

… ►Similarly, as $a\to \mathrm{\infty }$ in the sector , …

4: Bibliography C

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

Encodings:

BibTeX

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Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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External Links:

Document

Referenced by:

5th item.

Encodings:

BibTeX

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Constraints on the phase of scattering amplitudes due to positivity. Nuclear Phys. B 49, pp. 413–440.

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External Links:

Document

Referenced by:

§18.39(v).

Encodings:

BibTeX

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Constraints on the phases of helicity amplitudes due to positivity. Nuclear Phys. B 77, pp. 141–162.

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External Links:

Document, MathReview

Referenced by:

§18.39(v).

Encodings:

BibTeX

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