sine and cosine integrals

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31: 7.14 Integrals

§7.14 Integrals

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Fourier Transform

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Laplace Transforms

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§7.14(ii) Fresnel Integrals

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Laplace Transforms

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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. … ►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. … ►For an asymptotic expansion of the zeros of ${\int }_0^z\exp \left(\frac{1}{2}\pi \mathrm{i}{t}^2\right)dt$ ($=ℱ\left(0\right)-ℱ\left(z\right)$ $=C\left(z\right)+\mathrm{i}S\left(z\right)$) see Tuẑilin (1971). …

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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35: 6.10 Other Series Expansions

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

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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: 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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4.26.8 $$\int {\mathrm{e}}^{ax}\cos \left(bx\right)dx=\frac{{\mathrm{e}}^{ax}}{a^2+b^2}(a\cos \left(bx\right)+b\sin \left(bx\right)).$$

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\sin z$: sine function, $a$: real or complex constant and $x$: real variable

A&S Ref:

4.3.137

Permalink:

http://dlmf.nist.gov/4.26.E8

Encodings:

pMML, png, TeX

See also:

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

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4.26.11 $${\int }_0^{\pi }{\sin }^2\left(nt\right)dt={\int }_0^{\pi }{\cos }^2\left(nt\right)dt=\frac{1}{2}\pi ,$$ $n\ne 0$.

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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 and $n$: integer

A&S Ref:

4.3.141

Permalink:

http://dlmf.nist.gov/4.26.E11

Encodings:

pMML, png, TeX

See also:

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

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4.26.13 $${\int }_0^{\mathrm{\infty }}\sin \left(t^2\right)dt={\int }_0^{\mathrm{\infty }}\cos \left(t^2\right)dt=\frac{1}{2}\sqrt{\frac{\pi }{2}}.$$

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Symbols:

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

A&S Ref:

4.3.144

Referenced by:

§4.26(iii)

Permalink:

http://dlmf.nist.gov/4.26.E13

Encodings:

pMML, png, TeX

See also:

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

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