sums and integrals

SearchAdvancedHelp

(0.001 seconds)

11—20 of 173 matching pages

11: 8.19 Generalized Exponential Integral

… ►

8.19.7 $$E_n\left(z\right)=\frac{{(-z)}^{n-1}}{(n-1)!}{E}_1\left(z\right)+\frac{{\mathrm{e}}^{-z}}{(n-1)!}\sum _{k=0}^{n-2}(n-k-2)!{(-z)}^k,$$ $n=2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $E_p\left(z\right)$: generalized exponential integral, $!$: factorial (as in $n!$), $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(iii), §8.19 and Ch.8

… ►

8.19.8 $$E_n\left(z\right)=\frac{{(-z)}^{n-1}}{(n-1)!}(\psi \left(n\right)-\ln z)-\sum _{\begin{array}{c}k=0\\ k\ne n-1\end{array}}^{\mathrm{\infty }}\frac{{(-z)}^k}{k!(1-n+k)},$$

ⓘ

Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $E_p\left(z\right)$: generalized exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

5.1.12

Referenced by:

§8.19(iv)

Permalink:

http://dlmf.nist.gov/8.19.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(iv), §8.19 and Ch.8

… ►

8.19.10 $$E_p\left(z\right)=z^{p-1}\mathrm{\Gamma }\left(1-p\right)-\sum _{k=0}^{\mathrm{\infty }}\frac{{(-z)}^k}{k!(1-p+k)},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $E_p\left(z\right)$: generalized exponential integral, $!$: factorial (as in $n!$), $z$: complex variable, $p$: parameter and $k$: nonnegative integer

Referenced by:

§8.19(iv)

Permalink:

http://dlmf.nist.gov/8.19.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(iv), §8.19 and Ch.8

►

8.19.11 $$E_p\left(z\right)=\mathrm{\Gamma }\left(1-p\right)\left(z^{p-1}-{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{\mathrm{\Gamma }\left(2-p+k\right)}\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable, $p$: parameter and $k$: nonnegative integer

Referenced by:

§8.19(iv)

Permalink:

http://dlmf.nist.gov/8.19.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(iv), §8.19 and Ch.8

… ►

8.19.24 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{E}_n\left(t\right)dt=\frac{{(-1)}^{n-1}}{a^n}\left(\ln \left(1+a\right)+\sum _{k=1}^{n-1}\frac{{(-1)}^k{a}^k}{k}\right),$$ $n=1,2,\mathrm{\dots }$, $\mathrm{\Re }a>-1$,

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $a$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

5.1.34

Referenced by:

§8.19(x)

Permalink:

http://dlmf.nist.gov/8.19.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(x), §8.19 and Ch.8

…

12: 4.10 Integrals

§4.10 Integrals

►

§4.10(i) Logarithms

… ►For $\mathrm{li}\left(x\right)$ see §6.2(i). ►

§4.10(ii) Exponentials

… ►Extensive compendia of indefinite and definite integrals of logarithms and exponentials include Apelblat (1983, pp. 16–47), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 107–116), Gröbner and Hofreiter (1950, pp. 52–90), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.3, 1.6, 2.3, 2.6).

13: 6.16 Mathematical Applications

… ►

6.16.2 $$S_n(x)=\sum _{k=0}^{n-1}\frac{\sin \left((2k+1)x\right)}{2k+1}=\frac{1}{2}{\int }_0^x\frac{\sin \left(2nt\right)}{\sin t}dt=\frac{1}{2}\mathrm{Si}\left(2nx\right)+R_n(x),$$

ⓘ

Defines:

$S_n(x)$: partial sum (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable, $n$: nonnegative integer and $R_n(x)$: remainder term

Permalink:

http://dlmf.nist.gov/6.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

…

14: 8.21 Generalized Sine and Cosine Integrals

… ►

8.21.14 $$\mathrm{Si}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{z}^{2k+1}}{(2k+a+1)(2k+1)!},$$ $a\ne -1,-3,-5,\mathrm{\dots }$,

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\mathrm{Si}(a,z)$: generalized sine integral, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

►

8.21.15 $$\mathrm{Ci}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{z}^{2k}}{(2k+a)(2k)!},$$ $a\ne 0,-2,-4,\mathrm{\dots }$.

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\mathrm{Ci}(a,z)$: generalized cosine integral, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

… ►

8.21.16 $$\mathrm{Si}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{\left(2k+\frac{3}{2}\right){\left(1-\frac{1}{2}a\right)}_k}{{\left(\frac{1}{2}+\frac{1}{2}a\right)}_{k+1}}{𝗃}_{2k+1}\left(z\right),$$ $a\ne -1,-3,-5,\mathrm{\dots }$,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{Si}(a,z)$: generalized sine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

►

8.21.17 $$\mathrm{Ci}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{\left(2k+\frac{1}{2}\right){\left(\frac{1}{2}-\frac{1}{2}a\right)}_k}{{\left(\frac{1}{2}a\right)}_{k+1}}{𝗃}_{2k}\left(z\right),$$ $a\ne 0,-2,-4,\mathrm{\dots }$.

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{Ci}(a,z)$: generalized cosine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

…

15: 19.19 Taylor and Related Series

… ►

19.19.2 $$R_{-a}(𝐛;𝐳)=\sum _{N=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_N}{{\left(c\right)}_N}{T}_N(𝐛,\mathrm{𝟏}-𝐳),$$ $c={\sum }_{j=1}^n{b}_j$, ,

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: nonnegative integer and $T_N(𝐛,𝐳)$: polynomial

Referenced by:

§19.19

Permalink:

http://dlmf.nist.gov/19.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

►

19.19.3 $$R_{-a}(𝐛;𝐳)=z_n^{-a}\sum _{N=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_N}{{\left(c\right)}_N}{T}_N(b_1,\mathrm{\dots },b_{n-1};1-(z_1/z_n),\mathrm{\dots },1-(z_{n-1}/z_n)),$$ $c={\sum }_{j=1}^n{b}_j$, .

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: nonnegative integer and $T_N(𝐛,𝐳)$: polynomial

Referenced by:

§19.19, §19.19, §19.28

Permalink:

http://dlmf.nist.gov/19.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

… ►

19.19.7 $$R_{-a}(\frac{\mathrm{𝟏}}{\mathrm{𝟐}};𝐳)=A^{-a}\sum _{N=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_N}{{\left(\frac{1}{2}n\right)}_N}{T}_N(\frac{\mathrm{𝟏}}{\mathrm{𝟐}},𝐙),$$

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: nonnegative integer, $T_N(𝐛,𝐳)$: polynomial, $𝐙$: variable and $A$

Referenced by:

§19.15, §19.36(i), §19.36(i), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

…

16: 19.16 Definitions

… ►

§19.16(i) Symmetric Integrals

… ► … ►All other elliptic cases are integrals of the second kind. …(Note that $R_C(x,y)$ is not an elliptic integral.) … ►Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral: …

17: 6.10 Other Series Expansions

… ►

6.10.4 $$\mathrm{Si}\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{\left({𝗃}_n\left(\frac{1}{2}z\right)\right)}^2,$$

ⓘ

Symbols:

$\mathrm{Si}\left(z\right)$: sine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.10(ii)

Permalink:

http://dlmf.nist.gov/6.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

►

6.10.5 $$\mathrm{Cin}\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}{a}_n{\left({𝗃}_n\left(\frac{1}{2}z\right)\right)}^2,$$

ⓘ

Symbols:

$\mathrm{Cin}\left(z\right)$: cosine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $n$: nonnegative integer and $a_n$: coefficients

Permalink:

http://dlmf.nist.gov/6.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

►

6.10.6 $$\mathrm{Ei}\left(x\right)=\gamma +\ln \left|x\right|+\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n(x-a_n){\left({𝗂}_n^{(1)}\left(\frac{1}{2}x\right)\right)}^2,$$ $x\ne 0$,

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $x$: real variable, $n$: nonnegative integer and $a_n$: coefficients

Referenced by:

§6.18(i)

Permalink:

http://dlmf.nist.gov/6.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

… ►

6.10.8 $$\mathrm{Ein}\left(z\right)=z{\mathrm{e}}^{-z/2}\left({𝗂}_0^{(1)}\left(\frac{1}{2}z\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{2n+1}{n(n+1)}{𝗂}_n^{(1)}\left(\frac{1}{2}z\right)\right).$$

ⓘ

Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.10(ii)

Permalink:

http://dlmf.nist.gov/6.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

…

18: 36.8 Convergent Series Expansions

… ►

${\mathrm{\Psi }}_K\left(𝐱\right)={\displaystyle \frac{2}{K+2}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}\exp \left(\mathrm{i}{\displaystyle \frac{\pi (2n+1)}{2(K+2)}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{2n+1}{K+2}}\right)a_{2n}(𝐱),$ $K$ even,

►

${\mathrm{\Psi }}_K\left(𝐱\right)={\displaystyle \frac{2}{K+2}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\mathrm{i}}^n\cos \left({\displaystyle \frac{\pi (n(K+1)-1)}{2(K+2)}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{n+1}{K+2}}\right)a_n(𝐱),$ $K$ odd,

… ►

36.8.3 $$\frac{3^{2/3}}{4{\pi }^2}{\mathrm{\Psi }}^{(\mathrm{H})}\left(3^{1/3}𝐱\right)=\mathrm{Ai}\left(x\right)\mathrm{Ai}\left(y\right)\sum _{n=0}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{c_n(x)c_n(y)}{n!}+\mathrm{Ai}\left(x\right){\mathrm{Ai}}^{\prime }\left(y\right)\sum _{n=2}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{c_n(x)d_n(y)}{n!}+{\mathrm{Ai}}^{\prime }\left(x\right)\mathrm{Ai}\left(y\right)\sum _{n=2}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{d_n(x)c_n(y)}{n!}+{\mathrm{Ai}}^{\prime }\left(x\right){\mathrm{Ai}}^{\prime }\left(y\right)\sum _{n=1}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{d_n(x)d_n(y)}{n!},$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $z$: real parameter, $m$: integer, $x$: real parameter, $c_n(t)$: coefficients and $d_n(t)$: coefficients

Referenced by:

§36.8

Permalink:

http://dlmf.nist.gov/36.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.8 and Ch.36

… ►

36.8.4 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)=2{\pi }^2{\left(\frac{2}{3}\right)}^{2/3}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-\mathrm{i}{(2/3)}^{2/3}z\right)}^n}{n!}\mathrm{\Re }\left(f_n(\frac{x+\mathrm{i}y}{{12}^{1/3}},\frac{x-\mathrm{i}y}{{12}^{1/3}})\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter, $m$: integer, $x$: real parameter and $f_n$: function

Referenced by:

§36.8

Permalink:

http://dlmf.nist.gov/36.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.8 and Ch.36

…

19: 19.31 Probability Distributions

§19.31 Probability Distributions

► $R_G(x,y,z)$ and $R_F(x,y,z)$ occur as the expectation values, relative to a normal probability distribution in ${ℝ}^2$ or ${ℝ}^3$, of the square root or reciprocal square root of a quadratic form. … ►

19.31.1 $${𝐱}^{\mathrm{T}}𝐀𝐱=\sum _{r=1}^n\sum _{s=1}^n{a}_{r,s}{x}_r{x}_s,$$

ⓘ

Symbols:

${𝐀}^{\mathrm{T}}$: transpose of matrix and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/19.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.31 and Ch.19

… ►

19.31.2 $${\int }_{{ℝ}^n}{({𝐱}^{\mathrm{T}}𝐀𝐱)}^{\mu }\exp \left(-{𝐱}^{\mathrm{T}}𝐁𝐱\right)d{x}_1\mathrm{\cdots }d{x}_n=\frac{{\pi }^{n/2}\mathrm{\Gamma }\left(\mu +\frac{1}{2}n\right)}{\sqrt{det𝐁}\mathrm{\Gamma }\left(\frac{1}{2}n\right)}{R}_{\mu }(\frac{1}{2},\mathrm{\dots },\frac{1}{2};{\lambda }_1,\mathrm{\dots },{\lambda }_n),$$ $\mu >-\frac{1}{2}n$.

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $ℝ$: real line, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $n$: nonnegative integer and ${\lambda }_n$: eigenvalues

Referenced by:

§19.31

Permalink:

http://dlmf.nist.gov/19.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.31 and Ch.19

►§19.16(iii) shows that for $n=3$ the incomplete cases of $R_F$ and $R_G$ occur when $\mu =-1/2$ and $\mu =1/2$, respectively, while their complete cases occur when $n=2$. …

20: 36.6 Scaling Relations

§36.6 Scaling Relations

… ►

${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)=k^{{\beta }^{(\mathrm{U})}}{\mathrm{\Psi }}^{(\mathrm{U})}\left({𝐲}^{(\mathrm{U})}(k)\right),$

… ►

Indices for $k$-Scaling of Magnitude of ${\mathrm{\Psi }}_K$ or ${\mathrm{\Psi }}^{(\mathrm{U})}$ (Singularity Index)

… ►

$\text{cuspoids: }{\gamma }_K={\displaystyle \sum _{m=1}^K}{\gamma }_{mK}={\displaystyle \frac{K(K+3)}{2(K+2)}},$

►

$\text{umbilics: }{\gamma }^{(\mathrm{U})}={\displaystyle \sum _{m=1}^3}{\gamma }_m^{(\mathrm{U})}=\frac{5}{3}.$

…