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1: 27.15 Chinese Remainder Theorem

… ►By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_1$), (mod $m_2$), (mod $m_3$), and (mod $m_4$), respectively. Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers $a_1(mod{m}_1)$, $a_2(mod{m}_2)$, $a_3(mod{m}_3)$, and $a_4(mod{m}_4)$, where each $a_j$ has no more than five digits. …

2: 1.10 Functions of a Complex Variable

… ►The coefficient $a_{-1}$ of ${(z-z_0)}^{-1}$ in the Laurent series for $f(z)$ is called the residue of $f(z)$ at $z_0$, and denoted by ${res}_{z=z_0}[f(z)]$, $\underset{z=z_0}{res}[f(z)]$, or (when there is no ambiguity) $res[f(z)]$. … ►

§1.10(iv) Residue Theorem

… ►Suppose that … ►is analytic in $ℂ$, except for simple poles at $z=z_n$ of residue $a_n$. …

3: 2.5 Mellin Transform Methods

… ►

2.5.6

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\mathrm{\Re }$: real part, $res$: residue, $f(x)$: locally integrable function, $c$: point, $I(x)$: convolution integral, $h(x)$: function, $d$: point and $E(x)$: function

Keywords:

Mellin transform

Referenced by:

§2.5(i), §2.5(i)

Permalink:

http://dlmf.nist.gov/2.5.E6

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(i), §2.5 and Ch.2

… ►

2.5.11 $${res}_{z=n}\left[x^{-z}ℳf\left(1-z\right)ℳh\left(z\right)\right]=(a_n\ln x+b_n)x^{-n},$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\ln z$: principal branch of logarithm function, $res$: residue, $h(t)=J_{\nu }^2\left(t\right)$: function, $a_n$: coefficients, $b_n$: coefficients and $f(t)=1/(1+t)$: function

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/2.5.E11

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

… ►

2.5.35

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Symbols:

$\mathrm{\Re }$: real part, $res$: residue, $G_{jk}(z)$: function, $p_{jk}$: real number, $q_{jk}>p_{jk}$: real number, $E_{jk}(x)$: function and $I(x)$: convolution integral

Permalink:

http://dlmf.nist.gov/2.5.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §2.5(ii), §2.5 and Ch.2

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2.5.46 $${res}_{z=k}\left[-{\zeta }^{z-1}\mathrm{\Gamma }\left(1-z\right)\pi \csc \left(\pi z\right)\right]=\left(-\ln \zeta +\psi \left(k\right)\right)\frac{{\zeta }^{k-1}}{(k-1)!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function and $res$: residue

Permalink:

http://dlmf.nist.gov/2.5.E46

Encodings:

pMML, png, TeX

See also:

Annotations for §2.5(iii), §2.5(iii), §2.5 and Ch.2

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2.5.47 $${res}_{z=1}\left[-{\zeta }^{z-1}\mathrm{\Gamma }\left(1-z\right)ℳh_2\left(z\right)\right]=\left(-\ln \zeta -\gamma \right)-ℳh_1\left(1\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\ln z$: principal branch of logarithm function, $res$: residue and $h(x)$: locally integrable function

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/2.5.E47

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(iii), §2.5(iii), §2.5 and Ch.2

…

4: 5.2 Definitions

… ►It is a meromorphic function with no zeros, and with simple poles of residue ${(-1)}^n/n!$ at $z=-n$. …$\psi \left(z\right)$ is meromorphic with simple poles of residue $-1$ at $z=-n$. …

5: 5.19 Mathematical Applications

… ►By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of $f(z)$ for large $|z|$, or small $|z|$, can be obtained complete with an integral representation of the error term. …

6: 8.15 Sums

… ►

8.15.2 $$a\sum _{k=1}^{\mathrm{\infty }}\left(\frac{{\mathrm{e}}^{2\pi \mathrm{i}k(z+h)}}{{\left(2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,2\pi \mathrm{i}kz)+\frac{{\mathrm{e}}^{-2\pi \mathrm{i}k(z+h)}}{{\left(-2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,-2\pi \mathrm{i}kz)\right)=\zeta (-a,z+h)+\frac{z^{a+1}}{a+1}+\left(h-\frac{1}{2}\right)z^a,$$ $h\in [0,1]$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Proof sketch:

Start with and $z>0$. For $\mathrm{\Gamma }(a,\pm 2\pi \mathrm{i}kz)$ take integral representation (8.2.2) and use the substitution $t=\pm 2\pi \mathrm{i}kz(1+\tau )$. The sum and the integral can be interchanged, and the sum can be evaluated via (4.6.1). Use integration by parts. This will result in $\left(h-\frac{1}{2}\right)z^a$ plus two integrals with infinitely many poles. The residue theorem (§1.10(iv)) will give us an infinite series which can be identified via (25.11.1). For other values of $a$ and $z$ use analytic continuation.

Referenced by:

§25.11(x), §25.11(x), §8.15, Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/8.15.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.3):

This equation was added.

See also:

Annotations for §8.15 and Ch.8

…

7: 27.2 Functions

… ►Such a set is a reduced residue system modulo $n$. …

8: Mathematical Introduction

… ► ►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
…
$res$	residue.
…

…

9: Bibliography V

… ►

Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.

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External Links:

ISSN 0377-9017, Document, MathReview (Laurent Habsieger), ZentralBlatt

Referenced by:

§19.35(i).

Encodings:

BibTeX

…

10: 2.3 Integrals of a Real Variable

… ►

2.3.18 $$b_s=\frac{1}{\mu }{res}_{t=a}\left[\frac{q(t)}{{(p(t)-p(a))}^{(\lambda +s)/\mu }}\right],$$ $s=0,1,2,\mathrm{\dots }$.

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Symbols:

$res$: residue, $a$: left endpoint, $b$: right endpoint, $p(t)$: real function, $q(t)$: function, $\lambda$: positive constant and $\mu$: positive constant

Referenced by:

§2.3(iii), §2.4(iv)

Permalink:

http://dlmf.nist.gov/2.3.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(iii), §2.3 and Ch.2

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