connection with Fresnel integrals

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21: 7.24 Approximations

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§7.24(i) Approximations in Terms of Elementary Functions

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Cody (1969) provides minimax rational approximations for $\mathrm{erf}x$ and $\mathrm{erfc}x$. The maximum relative precision is about 20S.

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Cody (1968) gives minimax rational approximations for the Fresnel integrals (maximum relative precision 19S); for a Fortran algorithm and comments see Snyder (1993).

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Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for $(1+2x){\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $(0,\mathrm{\infty })$ (22D).

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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.

22: 7.10 Derivatives

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${\displaystyle \frac{d\mathrm{f}\left(z\right)}{dz}}=-\pi z\mathrm{g}\left(z\right),$

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${\displaystyle \frac{d\mathrm{g}\left(z\right)}{dz}}=\pi z\mathrm{f}\left(z\right)-1.$

23: 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 ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$. …

24: 7.14 Integrals

§7.14 Integrals

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Fourier Transform

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§7.14(ii) Fresnel Integrals

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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.

25: 7.21 Physical Applications

§7.21 Physical Applications

►The error functions, Fresnel integrals, and related functions occur in a variety of physical applications. Fresnel integrals and Cornu’s spiral occurred originally in the analysis of the diffraction of light; see Born and Wolf (1999, §8.7). … ►Carslaw and Jaeger (1959) gives many applications and points out the importance of the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$. … ►Dawson’s integral appears in de-convolving even more complex motional effects; see Pratt (2007). …

26: 7.13 Zeros

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§7.13(iii) Zeros of the Fresnel Integrals

… ►Similarly for $S\left(z\right)$. Let $z_n$ be a zero of one of the Fresnel integrals. … ►

§7.13(iv) Zeros of $ℱ\left(z\right)$

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27: 7.23 Tables

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Abramowitz and Stegun (1964, Chapter 7) includes $\mathrm{erf}x$, $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [0,2]$, 10D; $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [2,10]$, 8S; $x{\mathrm{e}}^{x^2}\mathrm{erfc}x$, $x^{-2}\in [0,0.25]$, 7D; $2^n\mathrm{\Gamma }\left(\frac{1}{2}n+1\right){\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $n=1(1)6,10,11$, $x\in [0,5]$, 6S; $F\left(x\right)$, $x\in [0,2]$, 10D; $xF\left(x\right)$, $x^{-2}\in [0,0.25]$, 9D; $C\left(x\right)$, $S\left(x\right)$, $x\in [0,5]$, 7D; $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in [0,1]$, $x^{-1}\in [0,1]$, 15D.

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Abramowitz and Stegun (1964, Table 27.6) includes the Goodwin–Staton integral $G\left(x\right)$, $x=1(.1)3(.5)8$, 4D; also $G\left(x\right)+\ln x$, $x=0(.05)1$, 4D.

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Zhang and Jin (1996, pp. 637, 639) includes $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $\mathrm{erf}x$, $x=0(.02)1(.04)3$, 8D; $C\left(x\right)$, $S\left(x\right)$, $x=0(.2)10(2)100(100)500$, 8D.

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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\mathrm{erf}z$, $x\in [0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of ${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $(1/\sqrt{\pi }){\mathrm{e}}^{\mp \mathrm{i}(x^2+(\pi /4))}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

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Zhang and Jin (1996, p. 642) includes the first 10 zeros of $\mathrm{erf}z$, 9D; the first 25 distinct zeros of $C\left(z\right)$ and $S\left(z\right)$, 8S.

28: 7.7 Integral Representations

§7.7 Integral Representations

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§7.7(i) Error Functions and Dawson’s Integral

►Integrals of the type $\int {\mathrm{e}}^{-z^2}R(z)dz$, where $R(z)$ is an arbitrary rational function, can be written in closed form in terms of the error functions and elementary functions. … ►

§7.7(ii) Auxiliary Functions

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Mellin–Barnes Integrals

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29: 7.6 Series Expansions

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§7.6(i) Power Series

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7.6.4 $$C\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{(\frac{1}{2}\pi )}^{2n}}{(2n)!(4n+1)}{z}^{4n+1},$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.11

Referenced by:

§7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(i), §7.6 and Ch.7

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7.6.6 $$S\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{(\frac{1}{2}\pi )}^{2n+1}}{(2n+1)!(4n+3)}{z}^{4n+3},$$

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Symbols:

$S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.13

Referenced by:

§7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(i), §7.6 and Ch.7

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

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7.6.10 $$C\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{𝗃}_{2n}\left(\frac{1}{2}\pi z^2\right),$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(ii), §7.6 and Ch.7

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30: 13.27 Mathematical Applications

… ►Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. …The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. …