line integral

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1: 21.1 Special Notation

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$g,h$	positive integers.
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${\oint }_a\omega$	line integral of the differential $\omega$ over the cycle $a$.

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2: 1.6 Vectors and Vector-Valued Functions

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§1.6(iv) Path and Line Integrals

… ►The line integral of a vector-valued function $𝐅=F_1𝐢+F_2𝐣+F_3𝐤$ along $𝐜$ is given by …

3: 19.31 Probability Distributions

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

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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

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4: 6.7 Integral Representations

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6.7.5 $${\int }_x^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{a^2+t^2}dt=-\frac{1}{2a\mathrm{i}}\left({\mathrm{e}}^{\mathrm{i}a}{E}_1\left(x+\mathrm{i}a\right)-{\mathrm{e}}^{-\mathrm{i}a}{E}_1\left(x-\mathrm{i}a\right)\right),$$ $a>0$, $x\in ℝ$,

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line and $x$: real variable

A&S Ref:

5.1.43 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.6 $${\int }_x^{\mathrm{\infty }}\frac{t{\mathrm{e}}^{-t}}{a^2+t^2}dt=\frac{1}{2}\left({\mathrm{e}}^{\mathrm{i}a}{E}_1\left(x+\mathrm{i}a\right)+{\mathrm{e}}^{-\mathrm{i}a}{E}_1\left(x-\mathrm{i}a\right)\right),$$ $a>0$, $x\in ℝ$.

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line and $x$: real variable

A&S Ref:

5.1.44 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.7 $${\int }_0^1\frac{{\mathrm{e}}^{-at}\sin \left(bt\right)}{t}dt=\mathrm{\Im }\mathrm{Ein}\left(a+\mathrm{i}b\right),$$ $a,b\in ℝ$,

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line and $\sin z$: sine function

A&S Ref:

5.1.36 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.8 $${\int }_0^1\frac{{\mathrm{e}}^{-at}(1-\cos \left(bt\right))}{t}dt=\mathrm{\Re }\mathrm{Ein}\left(a+\mathrm{i}b\right)-\mathrm{Ein}\left(a\right),$$ $a,b\in ℝ$.

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\cos z$: cosine function, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part and $ℝ$: real line

A&S Ref:

5.1.38 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.11 $${\int }_0^1\frac{(1-{\mathrm{e}}^{-at})\cos \left(bt\right)}{t}dt=\mathrm{\Re }\mathrm{Ein}\left(a+\mathrm{i}b\right)-\mathrm{Cin}\left(b\right),$$ $a,b\in ℝ$.

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\cos z$: cosine function, $\mathrm{Cin}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part and $ℝ$: real line

A&S Ref:

5.2.33 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(ii), §6.7 and Ch.6

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5: 13.4 Integral Representations

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§13.4(i) Integrals Along the Real Line

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6: 8.6 Integral Representations

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§8.6(i) Integrals Along the Real Line

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7: 5.20 Physical Applications

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5.20.3 $${\psi }_n(\beta )={\int }_{{ℝ}^n}{\mathrm{e}}^{-\beta W}dx={(2\pi )}^{n/2}{\beta }^{-(n/2)-(\beta n(n-1)/4)}{\left(\mathrm{\Gamma }\left(1+\frac{1}{2}\beta \right)\right)}^{-n}\prod _{j=1}^n\mathrm{\Gamma }\left(1+\frac{1}{2}j\beta \right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $ℝ$: real line, $j$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $W$: potential energy, ${\psi }_n(\beta )$: partition function and $\beta =1/(kT)$

Permalink:

http://dlmf.nist.gov/5.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.20, §5.20 and Ch.5

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8: 13.16 Integral Representations

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§13.16(i) Integrals Along the Real Line

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9: 11.5 Integral Representations

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§11.5(i) Integrals Along the Real Line

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10: 29.17 Other Solutions

… ►Lamé–Wangerin functions are solutions of (29.2.1) with the property that ${(\mathrm{sn}(z,k))}^{1/2}w(z)$ is bounded on the line segment from $\mathrm{i}{K}^{\prime }$ to $2K+\mathrm{i}{K}^{\prime }$. …