auxiliary functions for sine and cosine integrals

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11: 6.18 Methods of Computation

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§6.18(ii) Auxiliary Functions

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12: 7.4 Symmetry

§7.4 Symmetry

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$C\left(-z\right)=-C\left(z\right),$

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$S\left(-z\right)=-S\left(z\right),$

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$C\left(\mathrm{i}z\right)=\mathrm{i}C\left(z\right),$

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$\mathrm{g}\left(-z\right)=\sqrt{2}\sin \left(\frac{1}{4}\pi +\frac{1}{2}\pi z^2\right)-\mathrm{g}\left(z\right).$

13: 7.5 Interrelations

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7.5.3 $$C\left(z\right)=\frac{1}{2}+\mathrm{f}\left(z\right)\sin \left(\frac{1}{2}\pi z^2\right)-\mathrm{g}\left(z\right)\cos \left(\frac{1}{2}\pi z^2\right),$$

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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, $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.9

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.4 $$S\left(z\right)=\frac{1}{2}-\mathrm{f}\left(z\right)\cos \left(\frac{1}{2}\pi z^2\right)-\mathrm{g}\left(z\right)\sin \left(\frac{1}{2}\pi z^2\right).$$

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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, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.10

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.6 $${\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}(\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right))=\frac{1}{2}(1\pm \mathrm{i})-(C\left(z\right)\pm \mathrm{i}S\left(z\right)).$$

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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, $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: 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

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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14: 7.2 Definitions

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7.2.10 $$\mathrm{f}\left(z\right)=\left(\frac{1}{2}-S\left(z\right)\right)\cos \left(\frac{1}{2}\pi z^2\right)-\left(\frac{1}{2}-C\left(z\right)\right)\sin \left(\frac{1}{2}\pi z^2\right),$$

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Defines:

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

Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.5

Referenced by:

§7.4, §7.5

Permalink:

http://dlmf.nist.gov/7.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(iv), §7.2 and Ch.7

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7.2.11 $$\mathrm{g}\left(z\right)=\left(\frac{1}{2}-C\left(z\right)\right)\cos \left(\frac{1}{2}\pi z^2\right)+\left(\frac{1}{2}-S\left(z\right)\right)\sin \left(\frac{1}{2}\pi z^2\right).$$

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Defines:

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

Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.6

Referenced by:

§7.4, §7.5

Permalink:

http://dlmf.nist.gov/7.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(iv), §7.2 and Ch.7

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15: 7.14 Integrals

§7.14 Integrals

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Fourier Transform

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Laplace Transforms

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Laplace Transforms

… ►In a series of ten papers Hadži (1968, 1969, 1970, 1972, 1973, 1975a, 1975b, 1976a, 1976b, 1978) gives many integrals containing error functions and Fresnel integrals, also in combination with the hypergeometric function, confluent hypergeometric functions, and generalized hypergeometric functions.

16: 7.24 Approximations

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§7.24(i) Approximations in Terms of Elementary Functions

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Hastings (1955) gives several minimax polynomial and rational approximations for $\mathrm{erf}x$, $\mathrm{erfc}x$ and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$.

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Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

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Bulirsch (1967) provides Chebyshev coefficients for the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ for $x\ge 3$ (15D).

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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.

17: 7.22 Methods of Computation

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§7.22(i) Main Functions

►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. … ►

§7.22(ii) Goodwin–Staton Integral

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§7.22(iii) Repeated Integrals of the Complementary Error Function

►The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$. …

18: 7.12 Asymptotic Expansions

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§7.12(i) Complementary Error Function

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§7.12(ii) Fresnel Integrals

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

§7.12(iii) Goodwin–Staton Integral

►See Olver (1997b, p. 115) for an expansion of $G\left(z\right)$ with bounds for the remainder for real and complex values of $z$.

19: Bibliography S

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Complements to asymptotic development of sine cosine integrals, and auxiliary functions. Univ. Beograd. Publ. Elecktrotehn. Fak., Ser. Mat. Fiz. 461–497, pp. 185–191.

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External Links:

MathReview (E. Riekstins), ZentralBlatt

Referenced by:

§6.15, §6.15.

Encodings:

BibTeX

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20: 1.14 Integral Transforms

§1.14 Integral Transforms

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§1.14(ii) Fourier Cosine and Sine Transforms

►The Fourier cosine transform and Fourier sine transform are defined respectively by … ►

Inversion

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§1.14(viii) Compendia

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