finite sum

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11: 34.5 Basic Properties: $\mathit{6}j$ Symbol

… ►Examples are provided by: …

12: 18.30 Associated OP’s

… ►For other cases there may also be, in addition to a possible integral as in (18.30.10), a finite sum of discrete weights on the negative real $x$-axis each multiplied by the polynomial product evaluated at the corresponding values of $x$, as in (18.2.3). …

13: 18.2 General Orthogonal Polynomials

… ►A system $\{p_n(x)\}$ of OP’s satisfying (18.2.1) and (18.2.5) is complete if each $f(x)$ in the Hilbert space $L_w^2((a,b))$ can be approximated in Hilbert norm by finite sums ${\sum }_n{\lambda }_n{p}_n(x)$. …

14: 18.18 Sums

… ►See also (18.38.3) for a finite sum of Jacobi polynomials. …

15: 25.11 Hurwitz Zeta Function

… ►

25.11.36Removed because it is just (25.15.1) combined with (25.15.3).

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function and $k$: nonnegative integer

Referenced by:

§25.11(x), Erratum (V1.0.23) for Equation (25.11.36), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.11.E36

Removal (effective with 1.1.4):

This equation has been removed because it is just (25.15.1) combined with (25.15.3).

Clarification (effective with 1.0.23):

The Dirichlet $L$-function was inserted on the left-hand side. The upper-index of the finitesum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

See also:

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

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16: Errata

… ►

25.15.6 $$G(\chi )\equiv \sum _{r=1}^{k-1}\chi (r){\mathrm{e}}^{2\pi ir/k}.$$

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Reported by Gergő Nemes on 2021-08-23

►

25.15.10

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Reported by Gergő Nemes on 2021-08-23

…

17: 18.17 Integrals

… ►provided that $\mathrm{\ell }+m+n$ is even and the sum of any two of $\mathrm{\ell },m,n$ is not less than the third; otherwise the integral is zero. … ►provided that $\mathrm{\ell }+m+n$ is even and the sum of any two of $\mathrm{\ell },m,n$ is not less than the third; otherwise the integral is zero. …

18: 2.1 Definitions and Elementary Properties

… ►If $c$ is a finite limit point of $𝐗$, then … ►For (2.1.14) $𝐗$ can be the positive real axis or any unbounded sector in $ℂ$ of finite angle. … ►Similarly for finite limit point $c$ in place of $\mathrm{\infty }$. … ►where $c$ is a finite, or infinite, limit point of $𝐗$. … ►It can even happen that a generalized asymptotic expansion converges, but its sum is not the function being represented asymptotically; for an example see §18.15(iii).

19: 31.14 General Fuchsian Equation

… ►The exponents at the finite singularities $a_j$ are $\{0,1-{\gamma }_j\}$ and those at $\mathrm{\infty }$ are $\{\alpha ,\beta \}$, where ►

$\alpha +\beta +1={\displaystyle \sum _{j=1}^N}{\gamma }_j,$

►

$\alpha \beta ={\displaystyle \sum _{j=1}^N}{a}_j{q}_j.$

… ►

${\tilde{q}}_j={\displaystyle \frac{1}{2}}{\displaystyle \sum _{\begin{array}{c}k=1\\ k\ne j\end{array}}^N}{\displaystyle \frac{{\gamma }_j{\gamma }_k}{a_j-a_k}}-q_j,$

… ►An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). …

20: 1.8 Fourier Series

… ►

1.8.3 $$f(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n{\mathrm{e}}^{\mathrm{i}nx},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $n$: nonnegative integer

Referenced by:

§1.17(iii), §1.18(v)

Permalink:

http://dlmf.nist.gov/1.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

… ►(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda$. … ►Let $f(x)$ be an absolutely integrable function of period $2\pi$, and continuous except at a finite number of points in any bounded interval. Then the series (1.8.1) converges to the sum … ►

1.8.15 $$\frac{1}{2}f(0)+\sum _{n=1}^{\mathrm{\infty }}f(n)={\int }_0^{\mathrm{\infty }}f(x)dx+2\sum _{n=1}^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}f(x)\cos \left(2\pi nx\right)dx.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(iv), §1.8 and Ch.1

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