Fresnel integrals

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21: 11.10 Anger–Weber Functions

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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). …

22: 7.7 Integral Representations

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§7.7(ii) Auxiliary Functions

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7.7.10 $$\mathrm{f}\left(z\right)=\frac{1}{\pi \sqrt{2}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-\pi z^{2}t/2}}{\sqrt{t}(t^2+1)}dt,$$ $|\mathrm{ph}z|\le \frac{1}{4}\pi$,

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

7.4.26 (in different form)

Referenced by:

§7.12(ii), §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7 and Ch.7

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7.7.11 $$\mathrm{g}\left(z\right)=\frac{1}{\pi \sqrt{2}}{\int }_0^{\mathrm{\infty }}\frac{\sqrt{t}{\mathrm{e}}^{-\pi z^{2}t/2}}{t^2+1}dt,$$ $|\mathrm{ph}z|\le \frac{1}{4}\pi$,

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Symbols:

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

A&S Ref:

7.4.25 (in different form)

Referenced by:

§7.12(ii), §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7 and Ch.7

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7.7.12 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)={\mathrm{e}}^{-\pi \mathrm{i}{z}^2/2}{\int }_z^{\mathrm{\infty }}{\mathrm{e}}^{\pi \mathrm{i}{t}^2/2}dt.$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

Referenced by:

§7.7(ii)

Permalink:

http://dlmf.nist.gov/7.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7 and Ch.7

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Mellin–Barnes Integrals

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23: Nico M. Temme

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7 Error Functions, Dawson’s and Fresnel Integrals

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24: Bibliography T

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Theory of the Fresnel integral. USSR Comput. Math. and Math. Phys. 9 (4), pp. 271–279.

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Notes:

Translation of the paper in Zh. Vychisl. Mat. Mat. Fiz., Vol. 9, No. 4, 938-944, 1969

External Links:

Document, ZentralBlatt

Referenced by:

§7.13(iv).

Encodings:

BibTeX

25: Bibliography F

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Table errata; Complex zeros of Fresnel integrals. Math. Comp. 27 (121), pp. 219.

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Notes:

Errata in Abramowitz and Stegun (1964)

External Links:

FullText, MathReview

Referenced by:

(7.13.6), §7.13(iii).

Encodings:

BibTeX

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26: Bibliography S

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Algorithm 723: Fresnel integrals. ACM Trans. Math. Software 19 (4), pp. 452–456.

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Notes:

Single- and double-precision Fortran, maximum accuracy 16D. For remarks see same journal v. 22 (1996), p. 498-500 and v. 26 (2000), p. 617

External Links:

ISSN 0098-3500, Document, GAMS (module 723; package TOMS), ZentralBlatt

Referenced by:

3rd item, 1st item.

Encodings:

BibTeX

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Automatic computing methods for special functions. IV. Complex error function, Fresnel integrals, and other related functions. J. Res. Nat. Bur. Standards 86 (6), pp. 661–686.

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Notes:

Includes Fortran code, maximum accuracy 18S.

External Links:

ISSN 0022-4340, MathReview, ZentralBlatt

Referenced by:

§7.25(iii).

Encodings:

BibTeX

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27: Bibliography

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Recurrence relations for the Fresnel integral ${\int }_0^{\mathrm{\infty }}\frac{\exp (-ct)dt}{\sqrt{t}(1+t^2)}$ and similar integrals. Comm. ACM 17 (8), pp. 480–481.

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External Links:

ISSN 0001-0782, Document, MathReview, ZentralBlatt

Referenced by:

§6.18(ii).

Encodings:

BibTeX

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28: Bibliography K

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On the zeros of the Fresnel integrals. Canad. J. Math. 9, pp. 118–131.

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External Links:

MathReview (W. C. Rheinboldt), ZentralBlatt

Referenced by:

§7.13(iii).

Encodings:

BibTeX

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29: Bibliography L

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Algorithm 244: Fresnel integrals. Comm. ACM 7 (11), pp. 660–661.

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Notes:

Includes Algol program, maximum accuracy 12D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§7.25(iv).

Encodings:

BibTeX

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30: Bibliography B

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Numerical calculation of the sine, cosine and Fresnel integrals. Numer. Math. 9 (5), pp. 380–385.

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Notes:

Algol 60 procedures for sine, cosine, and 2 Fresnel integrals are included.

External Links:

ISSN 0029-599X, ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt

Referenced by:

2nd item, §7.25(iv).

Encodings:

BibTeX

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