with respect to summation

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1: 18.3 Definitions

… ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials $T_n\left(x\right)$, $n=0,1,\mathrm{\dots },N$, are orthogonal on the discrete point set comprising the zeros $x_{N+1,n},n=1,2,\mathrm{\dots },N+1$, of $T_{N+1}\left(x\right)$: …

2: 3.11 Approximation Techniques

… ►When $n>0$ and $0\le j\le n$, $0\le k\le n$, … ►Now suppose that $X_{k\mathrm{\ell }}=0$ when $k\ne \mathrm{\ell }$, that is, the functions ${\varphi }_k(x)$ are orthogonal with respect to weighted summation on the discrete set $x_1,x_2,\mathrm{\dots },x_J$. … …

3: 25.11 Hurwitz Zeta Function

… ►The Riemann zeta function is a special case: … ►For other series expansions similar to (25.11.10) see Coffey (2008). … ►In (25.11.18)–(25.11.24) primes on $\zeta$ denote derivatives with respect to $s$. … ►When $a=1$, (25.11.35) reduces to (25.2.3). … ►uniformly with respect to bounded nonnegative values of $\alpha$. …

4: 1.6 Vectors and Vector-Valued Functions

… ►

Einstein Summation Convention

►Much vector algebra involves summation over suffices of products of vector components. In almost all cases of repeated suffices, we can suppress the summation notation entirely, if it is understood that an implicit sum is to be taken over any repeated suffix. … ► … ►If ${𝚽}_1$ and ${𝚽}_2$ are both orientation preserving or both orientation reversing parametrizations of $S$ defined on open sets $D_1$ and $D_2$ respectively, then …

5: 1.17 Integral and Series Representations of the Dirac Delta

… ►subject to certain conditions on the function $\varphi (x)$. … ►The last condition is satisfied, for example, when $\varphi (x)=O\left({\mathrm{e}}^{\alpha x^2}\right)$ as $x\to \pm \mathrm{\infty }$, where $\alpha$ is a real constant. … ►Integral representation (1.17.12_1), (1.17.12_2) is the equivalent of the transform pairs, (1.14.9) (1.14.11), (1.14.10) (1.14.12), respectively. … ►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non-$L^2$ improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. … ►Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)): …

6: 23.18 Modular Transformations

… ►according as the elements of $𝒜$ in (23.15.3) have the respective forms …Here e and o are generic symbols for even and odd integers, respectively. … ►

23.18.6 $$\epsilon (𝒜)=\exp \left(\pi \mathrm{i}\left(\frac{a+d}{12c}+s(-d,c)\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $𝒜$: bilinear transformation, $a$: integer, $c$: integer, $d$: integer and $\epsilon (𝒜)$: function

Permalink:

http://dlmf.nist.gov/23.18.E6

Encodings:

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See also:

Annotations for §23.18, §23.18 and Ch.23

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23.18.7 $$s(d,c)=\sum _{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\lfloor \frac{dr}{c}\rfloor -\frac{1}{2}\right),$$ $c>0$.

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Symbols:

$\lfloor x\rfloor$: floor of $x$ , $\left(m,n\right)$: greatest common divisor (gcd), $c$: integer and $d$: integer

Referenced by:

§23.18, Erratum (V1.0.11) for Equation (23.18.7)

Permalink:

http://dlmf.nist.gov/23.18.E7

Encodings:

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Errata (effective with 1.0.11):

Originally the sum ${\sum }_{r=1}^{c-1}$ was written with an additional constraint on the summation, that $\left(r,c\right)=1$. This additional condition was incorrect and has been removed.

Reported 2015-10-05 by Howard Cohl and Tanay Wakhare

See also:

Annotations for §23.18, §23.18 and Ch.23

…

7: 1.14 Integral Transforms

… ►If $f(t)$ is absolutely integrable on $(-\mathrm{\infty },\mathrm{\infty })$, then $F(x)$ is continuous, $F(x)\to 0$ as $x\to \pm \mathrm{\infty }$, and … ►

Poisson’s Summation Formula

… ►The Fourier cosine transform and Fourier sine transform are defined respectively by … ►If also ${lim}_{t\to 0+}f(t)/t$ exists, then … ►Sufficient conditions for the integral to converge are that $s$ is a positive real number, and $f(t)=O\left(t^{-\delta }\right)$ as $t\to \mathrm{\infty }$, where $\delta >0$. …

8: Errata

… ►

The following additions were made in Chapter 1:

• New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

• The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).

• In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).

• In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

• New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

• New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

• Another form of Parseval’s formula, (1.14.7_5).

• We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

• Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

• An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

… ►

The Weierstrass lattice roots $e_j,$ were linked inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots $e_j$, and lattice invariants $g_2$, $g_3$, now link to their respective definitions (see §§23.2(i), 23.3(i)).

Reported by Felix Ospald.

… ►

The titles have been changed to $\mu =0$ , $\nu =0,1$ , and Addendum to §14.5(ii): $\mu =0$, $\nu =2$ , respectively, in order to be more descriptive of their contents.

… ►

8.12.18 $$\begin{array}{c}Q(a,z)\\ P(a,z)\end{array}\}\sim \frac{z^{a-\frac{1}{2}}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a\right)}\left(d(\pm \chi )\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\chi )}{z^{k/2}}\mp \sum _{k=1}^{\mathrm{\infty }}\frac{B_k(\chi )}{z^{k/2}}\right)$$

The original $\pm$ in front of the second summation was replaced by $\mp$ to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.

Reported 2017-01-28 by Richard Paris.

… ►

26.7.6 $$B\left(n+1\right)=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B\left(k\right)$$

Originally this equation appeared with $B\left(n\right)$ in the summation, instead of $B\left(k\right)$.

Reported 2010-11-07 by Layne Watson.

…

9: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►in each case uniformly with respect to $\lambda$ in the sector $|\mathrm{ph}\lambda |\le 2\pi -\delta$ (). … ►For the asymptotic behavior of $c_k(\eta )$ as $k\to \mathrm{\infty }$ see Dunster et al. (1998) and Olde Daalhuis (1998c). … ►Higher coefficients $A_k(\chi )$, $B_k(\chi )$, up to $k=8$, are given in Paris (2002b). … ►For asymptotic expansions, as $a\to \mathrm{\infty }$, of the inverse function $x=x(a,q)$ that satisfies the equation …These expansions involve the inverse error function $\mathrm{inverfc}\left(x\right)$ (§7.17), and are uniform with respect to $q\in [0,1]$. …

10: 11.10 Anger–Weber Functions

… ►

11.10.5 $$\frac{d^{2}w}{{dz}^2}+\frac{1}{z}\frac{dw}{dz}+\left(1-\frac{{\nu }^2}{z^2}\right)w=f(\nu ,z),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.10.E5

Encodings:

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See also:

Annotations for §11.10(ii), §11.10 and Ch.11

… ►

§11.10(vi) Relations to Other Functions

… ► … ►

11.10.27 $${\frac{\partial }{\partial \nu }{𝐉}_{\nu }\left(z\right)|}_{\nu =0}=\frac{1}{2}\pi {𝐇}_0\left(z\right),$$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E27

Encodings:

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See also:

Annotations for §11.10(vii), §11.10 and Ch.11

… ►where the prime on the second summation symbols means that the first term is to be halved. …