Sinc function

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1: Diego Dominici

… ►Diego was very active in the SIAM activity group on Orthogonal Polynomials and Special Functions since 2010. …

2: 3.3 Interpolation

… ►For interpolation of a bounded function $f$ on $ℝ$ the cardinal function of $f$ is defined by ►

3.3.43 $$C(f,h)(x)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}f(kh)S(k,h)(x),$$

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Defines:

$C(f,h)$: cardinal function of $f$ (locally)

Symbols:

$f(z)$: function and $S(k,h)(x)$: Sinc function

Permalink:

http://dlmf.nist.gov/3.3.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(vi), §3.3 and Ch.3

►where ►

3.3.44 $$S(k,h)(x)=\frac{\sin \left(\pi (x-kh)/h\right)}{\pi (x-kh)/h},$$

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Defines:

$S(k,h)(x)$: Sinc function (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $\sin z$: sine function

Permalink:

http://dlmf.nist.gov/3.3.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(vi), §3.3 and Ch.3

►is called the Sinc function. …

3: Bibliography G

… ►

The incomplete gamma functions since Tricomi. In Tricomi’s Ideas and Contemporary Applied Mathematics (Rome/Turin, 1997), Atti Convegni Lincei, Vol. 147, pp. 203–237.

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External Links:

FullText, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

Ch.8.

Encodings:

BibTeX

…

4: 10.74 Methods of Computation

… ► ►Similar observations apply to the computation of modified Bessel functions, spherical Bessel functions, and Kelvin functions. …And since there are no error terms they could, in theory, be used for all values of $z$; however, there may be severe cancellation when $|z|$ is not large compared with $n^2$. … ► ►

§10.74(vi) Zeros and Associated Values

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5: 25.15 Dirichlet $L$-functions

… ►

25.15.3 $$L(s,\chi )=k^{-s}\sum _{r=1}^{k-1}\chi (r)\zeta (s,\frac{r}{k}),$$

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

finite sum

Source:

Apostol (1976, (8), p. 255)

Notes:

In the original source the upper-index of the finite sum is $k$. We use $k-1$ since $\chi (k)=0$.

Referenced by:

(25.11.36), (25.11.36), §25.11(x), §25.15(i), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

… ►

25.15.10

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Symbols:

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

Keywords:

zeros

Source:

Apostol (1976, Theorem 12.20, p. 268)

Referenced by:

Erratum (V1.1.4) for Equation (25.15.10)

Permalink:

http://dlmf.nist.gov/25.15.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

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

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.15(ii), §25.15 and Ch.25

6: 2.3 Integrals of a Real Variable

… ►converges for all sufficiently large $x$, and $q(t)$ is infinitely differentiable in a neighborhood of the origin. … ►For the function $\mathrm{\Gamma }$ see §5.2(i). ►This result is probably the most frequently used method for deriving asymptotic expansions of special functions. Since $q(t)$ need not be continuous (as long as the integral converges), the case of a finite integration range is included. … ►Another extension is to more general factors than the exponential function. …

7: Notices

… ►

In association with the DLMF we will provide an index of all software for the computation of special functions covered by the DLMF. It is our intention that this will become an exhaustive list of sources of software for special functions. In each case we will maintain a single link where readers can obtain more information about the listed software. We welcome requests from software authors (or distributors) for new items to list.

Note that here we will only include software with capabilities that go beyond the computation of elementary functions in standard precisions since such software is nearly universal in scientific computing environments.

…

8: 2.5 Mellin Transform Methods

… ►Similarly, since $ℳh_2\left(z\right)$ can be continued analytically to a meromorphic function (when $\kappa =0$) or to an entire function (when $\kappa \ne 0$), we can choose $\rho$ so that $ℳh_2\left(z\right)$ has no poles in . …

9: 2.6 Distributional Methods

… ►Since the functions $t^{-s-\alpha }$, $s=1,2,\mathrm{\dots }$, are not locally integrable on $[0,\mathrm{\infty })$, we cannot assign distributions to them in a similar manner. … ►Since the function $t^{\mu }\left(\ln t-\gamma -\psi \left(\mu +1\right)\right)$ and all its derivatives are locally absolutely continuous in $(0,\mathrm{\infty })$, the distributional derivatives in the first sum in (2.6.44) can be replaced by the corresponding ordinary derivatives. Furthermore, since $f_{n,n}^{(n)}(t)=f_n(t)$, it follows from (2.6.37) that the remainder terms $t^{\mu -1}\ast f_n$ in the last two equations can be associated with a locally integrable function in $(0,\mathrm{\infty })$. …

10: Bibliography S

… ►

Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

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Notes:

Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.

External Links:

ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt

Referenced by:

§3.3(vi), §3.4(i), §3.5(i).

Encodings:

BibTeX

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