cosine integrals

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31: 7.14 Integrals

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7.14.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(t\right)dt=\frac{1}{a}\mathrm{f}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: 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:

7.4.27 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E5

Encodings:

pMML, png, TeX

See also:

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

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7.14.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(\sqrt{\frac{2t}{\pi }}\right)dt=\frac{{(\sqrt{a^2+1}+a)}^{\frac{1}{2}}}{2a\sqrt{a^2+1}},$$ $\mathrm{\Re }a>0$,

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Symbols:

$C\left(z\right)$: 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:

7.4.29 in modified form

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E7

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. In the first quadrant of $ℂ$ $C\left(z\right)$ has an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. … ►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 7.13.3: Complex zeros $x_n+\mathrm{i}{y}_n$ of $C\left(z\right).$

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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 $C\left(z\right)$ satisfy …

33: Software Index

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	Open Source													With Book						Commercial
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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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34: 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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7.11.7 $$C\left(z\right)=z{}_1{}^{}F_2^{}(\frac{1}{4};\frac{5}{4},\frac{1}{2};-\frac{1}{16}{\pi }^2{z}^4),$$

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Symbols:

$C\left(z\right)$: 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.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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

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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

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36: 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.

37: 7.22 Methods of Computation

… ►The methods available for computing the main functions in this chapter are analogous to those described in §§6.18(i)–6.18(iv) for the exponential integral and sine and cosine integrals, and similar comments apply. …

38: Nico M. Temme

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6 Exponential, Logarithmic, Sine, and Cosine Integrals

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39: 11.10 Anger–Weber Functions

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11.10.1 $${𝐉}_{\nu }\left(z\right)=\frac{1}{\pi }{\int }_0^{\pi }\cos \left(\nu \theta -z\sin \theta \right)d\theta ,$$

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Defines:

${𝐉}_{\nu }\left(z\right)$: Anger function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order

A&S Ref:

12.3.1

Referenced by:

§11.10(v), §11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(i), §11.10 and Ch.11

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11.10.3 $$\frac{1}{\pi }{\int }_0^{2\pi }\cos \left(\nu \theta -z\sin \theta \right)d\theta =(1+\cos \left(2\pi \nu \right)){𝐉}_{\nu }\left(z\right)+\sin \left(2\pi \nu \right){𝐄}_{\nu }\left(z\right).$$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(i), §11.10 and Ch.11

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$A_{\pm }(\chi )=C\left(\chi \right)\pm S\left(\chi \right),$

… ►For the Fresnel integrals $C$ and $S$ see §7.2(iii). …

40: 7.12 Asymptotic Expansions

… ►The asymptotic expansions of $C\left(z\right)$ and $S\left(z\right)$ are given by (7.5.3), (7.5.4), and …