sine integral

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31: 7.14 Integrals

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7.14.6 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}S\left(t\right)dt=\frac{1}{a}\mathrm{g}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.28 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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7.14.8 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}S\left(\sqrt{\frac{2t}{\pi }}\right)dt=\frac{{(\sqrt{a^2+1}-a)}^{\frac{1}{2}}}{2a\sqrt{a^2+1}},$$ $\mathrm{\Re }a>0$.

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Symbols:

$S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.30 in modified form

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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32: 7.13 Zeros

… ►At $z=0$, $C\left(z\right)$ has a simple zero and $S\left(z\right)$ has a triple zero. …Similarly for $S\left(z\right)$. … ►Tables 7.13.3 and 7.13.4 give 10D values of the first five $x_n$ and $y_n$ of $C\left(z\right)$ and $S\left(z\right)$, respectively. … ►

Table 7.13.4: Complex zeros $x_n+\mathrm{i}{y}_n$ of $S\left(z\right)$.

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$n$	$x_n$	$y_n$
…

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►As $n\to \mathrm{\infty }$ the $x_n$ and $y_n$ corresponding to the zeros of $S\left(z\right)$ satisfy (7.13.5) with …

33: 7.11 Relations to Other Functions

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7.11.6 $$C\left(z\right)+\mathrm{i}S\left(z\right)=zM(\frac{1}{2},\frac{3}{2},\frac{1}{2}\pi \mathrm{i}{z}^2)=z{\mathrm{e}}^{\pi \mathrm{i}{z}^2/2}M(1,\frac{3}{2},-\frac{1}{2}\pi \mathrm{i}{z}^2).$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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7.11.8 $$S\left(z\right)=\frac{1}{6}\pi z^3{}_1{}^{}F_2^{}(\frac{3}{4};\frac{7}{4},\frac{3}{2};-\frac{1}{16}{\pi }^2{z}^4).$$

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Symbols:

$S\left(z\right)$: Fresnel integral, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

34: Software Index

… ► ►►►►►►►

	Open Source													With Book						Commercial
…
6 Exponential, Logarithmic, Sine, and Cosine Integrals
6.21(ii) $E_1\left(x\right)$, $\mathrm{Ei}\left(x\right)$, $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $\mathrm{Shi}\left(x\right)$, $\mathrm{Chi}\left(x\right)$, $x\in ℝ$	✓	✓	✓	✓	✓	✓	✓		✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
6.21(iii) $E_1\left(z\right)$, $\mathrm{Si}\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Shi}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $z\in ℂ$	✓	✓				✓			✓			✓	✓	✓							✓	✓	✓
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7.25(iv) $C\left(x\right)$, $S\left(x\right)$, $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in ℝ$	✓		✓						✓			a	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
7.25(v) $C\left(z\right)$, $S\left(z\right)$, $z\in ℂ$	✓											a	✓								✓	✓	✓
…

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35: 6.10 Other Series Expansions

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§6.10(ii) Expansions in Series of Spherical Bessel Functions

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

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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

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36: 10.64 Integral Representations

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10.64.1 $${\mathrm{ber}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\cos \left(x\sin t-nt\right)\cosh \left(x\sin t\right)dt,$$

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Symbols:

${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

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10.64.2 $${\mathrm{bei}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\sin \left(x\sin t-nt\right)\sinh \left(x\sin t\right)dt.$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

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37: 7.24 Approximations

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Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for $F\left(z\right)$, $\mathrm{erf}z$, $\mathrm{erfc}z$, $C\left(z\right)$, and $S\left(z\right)$; approximate errors are given for a selection of $z$-values.

38: 10.60 Sums

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10.60.11 $$\sum _{n=0}^{\mathrm{\infty }}{𝗃}_n^2\left(z\right)=\frac{\mathrm{Si}\left(2z\right)}{2z}.$$

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Symbols:

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

A&S Ref:

10.1.52

Referenced by:

§10.60(iii), Ch.10

Permalink:

http://dlmf.nist.gov/10.60.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(iii), §10.60 and Ch.10

►For $\mathrm{Si}$ see §6.2(ii). …

39: 4.26 Integrals

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4.26.1 $$\int \sin xdx=-\cos x,$$

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.113

Permalink:

http://dlmf.nist.gov/4.26.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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4.26.2 $$\int \cos xdx=\sin x.$$

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.114

Permalink:

http://dlmf.nist.gov/4.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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4.26.9 $${\int }_0^{\pi }\sin \left(mt\right)\sin \left(nt\right)dt=0,$$ $m\ne n$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $m$: integer and $n$: integer

A&S Ref:

4.3.140

Permalink:

http://dlmf.nist.gov/4.26.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(iii), §4.26(iii), §4.26 and Ch.4

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4.26.14 $$\int \arcsin xdx=x\arcsin x+{(1-x^2)}^{1/2},$$ ,

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\arcsin z$: arcsine function and $x$: real variable

A&S Ref:

4.4.58 (modified)

Permalink:

http://dlmf.nist.gov/4.26.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(iv), §4.26 and Ch.4

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4.26.20 $$\int x\arcsin xdx=\left(\frac{x^2}{2}-\frac{1}{4}\right)\arcsin x+\frac{x}{4}{(1-x^2)}^{1/2},$$ ,

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\arcsin z$: arcsine function and $x$: real variable

A&S Ref:

4.4.64 (modified)

Permalink:

http://dlmf.nist.gov/4.26.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(iv), §4.26 and Ch.4

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40: 4.40 Integrals

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4.40.1 $$\int \sinh xdx=\cosh x,$$

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Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.77

Permalink:

http://dlmf.nist.gov/4.40.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

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4.40.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-x}\frac{\sin \left(ax\right)}{\sinh x}dx=\frac{1}{2}\pi \coth \left(\frac{1}{2}\pi a\right)-\frac{1}{a},$$ $a\ne 0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iii), §4.40 and Ch.4

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4.40.9 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{ax}}{{\left(\cosh \left(\frac{1}{2}x\right)\right)}^2}dx=\frac{4\pi a}{\sin \left(\pi a\right)},$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iii), §4.40 and Ch.4

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4.40.11 $$\int \mathrm{arcsinh}xdx=x\mathrm{arcsinh}x-{(1+x^2)}^{1/2}.$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.6.43 (modified)

Permalink:

http://dlmf.nist.gov/4.40.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iv), §4.40 and Ch.4

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4.40.15 $$\int \mathrm{arcsech}xdx=x\mathrm{arcsech}x+\arcsin x,$$ ,

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Symbols:

$dx$: differential of $x$ , $\mathrm{arcsech}z$: inverse hyperbolic secant function, $\int$: integral, $\arcsin z$: arcsine function and $x$: real variable

A&S Ref:

4.6.47 (modified. The first printing had an error.)

Permalink:

http://dlmf.nist.gov/4.40.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iv), §4.40 and Ch.4

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