subtraction

SearchAdvancedHelp

(0.001 seconds)

21—30 of 30 matching pages

21: 2.4 Contour Integrals

… ►By subtraction from (2.4.3) …

22: 19.16 Definitions

… ►

19.16.9 $$R_{-a}(𝐛;𝐳)=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\mathrm{\infty }}{t}^{a^{\prime }-1}\prod _{j=1}^n{(t+z_j)}^{-b_j}dt=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\mathrm{\infty }}{t}^{a-1}\prod _{j=1}^n{(1+t{z}_j)}^{-b_j}dt,$$ $b_1+\mathrm{\cdots }+b_n>a>0$, $b_j\in ℝ$, $z_j\in ℂ\setminus (-\mathrm{\infty },0]$,

ⓘ

Defines:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function

Symbols:

$\mathrm{B}(a,b)$: beta function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $(a,b]$: half-closed interval, $ℝ$: real line, $\setminus$: set subtraction and $n$: nonnegative integer

Referenced by:

§19.16(ii), §19.16(ii), §19.19, §19.20(iv), Erratum (V1.0.18) for Equation (19.16.9)

Permalink:

http://dlmf.nist.gov/19.16.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.18):

The constraint $a,a^{\prime }>0$, was replaced with $b_1+\mathrm{\cdots }+b_n>a>0$, $b_j\in ℝ$. It therefore follows from (19.16.10) that $a^{\prime }>0$.

Suggested 2017-07-25 by Bastien Roucariès

See also:

Annotations for §19.16(ii), §19.16 and Ch.19

…

23: 25.5 Integral Representations

… ►

25.5.7 $$\zeta \left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum _{m=1}^n\frac{B_{2m}}{(2m)!}{\left(s\right)}_{2m-1}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}-\sum _{m=1}^n\frac{B_{2m}}{(2m)!}{x}^{2m-1}\right)\frac{x^{s-1}}{{\mathrm{e}}^x}dx,$$ $\mathrm{\Re }s>-(2n+1)$, $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Proof sketch:

Derivable from (25.5.6) by adding and subtracting ${\sum }_{m=1}^n{B}_{2m}{x}^{2m-1}/(2m)!$ in the integrand, using (5.2.1), (5.2.5), and recognizing that ${({\mathrm{e}}^x-1)}^{-1}-x^{-1}+1/2-{\sum }_{m=1}^n{B}_{2m}{x}^{2m-1}/(2m)!=O\left(x^{2n+1}\right)$ as $x\to 0$, demonstrating the region of convergence.

Referenced by:

Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/25.5.E7

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

We have rewritten the original summation ${\sum }_{m=1}^n\frac{B_{2m}}{(2m)!}\frac{\mathrm{\Gamma }\left(s+2m-1\right)}{\mathrm{\Gamma }\left(s\right)}$ more concisely as ${\sum }_{m=1}^n\frac{B_{2m}}{(2m)!}{\left(s\right)}_{2m-1}$ using the Pochhammer symbol.

See also:

Annotations for §25.5(i), §25.5 and Ch.25

…

24: 1.9 Calculus of a Complex Variable

… ►

Operations

…

25: 1.2 Elementary Algebra

… ►

1.2.67 $$\Vert 𝐀\Vert =\underset{𝐱\in {𝐄}_n\setminus \left\{𝟎\right\}}{\max }\frac{\Vert 𝐀𝐱\Vert }{\Vert 𝐱\Vert }=\underset{\Vert 𝐱\Vert =1}{\max }\Vert 𝐀𝐱\Vert .$$

ⓘ

Defines:

$\Vert 𝐀\Vert$: matrix norm

Symbols:

$\in$: element of, $\setminus$: set subtraction, $\Vert 𝐯\Vert$: vector norm ($l$²), ${𝐄}_n$: space of $n$-dimensional vectors, real or complex, $𝐀$: non-defective matrix and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/1.2.E67

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

…

26: 1.5 Calculus of Two or More Variables

… ►

1.5.28

ⓘ

Symbols:

$\in$: element of, $(a,b)$: open interval, $\setminus$: set subtraction, $R$: closed rectangle and $D$: region contained in $R$

Permalink:

http://dlmf.nist.gov/1.5.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(v), §1.5(v), §1.5 and Ch.1

…

27: 2.11 Remainder Terms; Stokes Phenomenon

… ►Subtraction of this result from the sum of the first 5 terms in (2.11.25) yields 0. …

28: 19.29 Reduction of General Elliptic Integrals

… ►

19.29.8 $${\int }_y^x\frac{a_{\alpha }+b_{\alpha }t}{a_5+b_{5}t}\frac{dt}{s(t)}=\frac{2}{3}\frac{d_{\alpha \beta }{d}_{\alpha \gamma }{d}_{\alpha \delta }}{d_{\alpha 5}}{R}_J(U_{12}^2,U_{13}^2,U_{23}^2,U_{\alpha 5}^2)+2R_C(S_{\alpha 5}^2,Q_{\alpha 5}^2),$$ $S_{\alpha 5}^2\in ℂ\setminus (-\mathrm{\infty },0)$,

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $(a,b)$: open interval, $\setminus$: set subtraction, $U_n$ , $S_n$ , $Q_{\alpha \beta }$ , $d_{\alpha \beta }$ , $s(t)$: product, $\alpha$: parameter, $\beta$: parameter, $\gamma$: parameter and $\delta$: parameter

Referenced by:

§19.29(i), §19.29(ii), §19.29(ii), §19.29(ii)

Permalink:

http://dlmf.nist.gov/19.29.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.29(i), §19.29 and Ch.19

…

29: 19.25 Relations to Other Functions

… ►

19.25.36 $$\mathrm{\wp }\left(z\right)-e_j\in ℂ\setminus (-\mathrm{\infty },0],$$ $j=1,2,3.$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $ℂ$: complex plane, $\in$: element of, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $j$: $(=1,2,3)$ and $z$: complex number

Referenced by:

(19.25.39)

Permalink:

http://dlmf.nist.gov/19.25.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

…

30: 25.11 Hurwitz Zeta Function

… ►

25.11.28 $$\zeta (s,a)=\frac{1}{2}{a}^{-s}+\frac{a^{1-s}}{s-1}+\sum _{k=1}^n\frac{B_{2k}}{(2k)!}{\left(s\right)}_{2k-1}{a}^{1-s-2k}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}-\sum _{k=1}^n\frac{B_{2k}}{(2k)!}{x}^{2k-1}\right)x^{s-1}{\mathrm{e}}^{-ax}dx,$$ $\mathrm{\Re }s>-(2n+1)$, $s\ne 1$, $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

improper integral, integral representation

Proof sketch:

Derivable from (25.11.27) by adding and subtracting ${\sum }_{k=1}^n{B}_{2k}{x}^{2k-1}/(2k)!$ in the integrand, using (5.2.1), (5.2.5), and recognizing that ${({\mathrm{e}}^x-1)}^{-1}-x^{-1}+1/2-{\sum }_{k=1}^n{B}_{2k}{x}^{2k-1}/(2k)!=O\left(x^{2n+1}\right)$ as $x\to 0$, demonstrating the region of convergence.

Referenced by:

Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/25.11.E28

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

We have rewritten the original summation ${\sum }_{k=1}^n\frac{\mathrm{\Gamma }\left(s+2k-1\right)}{\mathrm{\Gamma }\left(s\right)}\frac{B_{2k}}{(2k)!}{a}^{-2k-s+1}$ more concisely as ${\sum }_{k=1}^n\frac{B_{2k}}{(2k)!}{\left(s\right)}_{2k-1}{a}^{1-s-2k}$ using the Pochhammer symbol.

See also:

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

…