functions f(ϵ,ℓ;r),h(ϵ,ℓ;r)

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1: 7.4 Symmetry

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$\mathrm{f}\left(\mathrm{i}z\right)=(1/\sqrt{2}){\mathrm{e}}^{\frac{1}{4}\pi \mathrm{i}-\frac{1}{2}\pi \mathrm{i}{z}^2}-\mathrm{i}\mathrm{f}\left(z\right),$

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$\mathrm{f}\left(-z\right)=\sqrt{2}\cos \left(\frac{1}{4}\pi +\frac{1}{2}\pi z^2\right)-\mathrm{f}\left(z\right),$

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2: 7.5 Interrelations

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7.5.5 $${\mathrm{e}}^{-\frac{1}{2}\pi \mathrm{i}{z}^2}ℱ\left(z\right)=\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right).$$

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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, $ℱ\left(z\right)$: 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:

§7.13(iv), §7.5, §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.5.E5

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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7.5.10 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=\frac{1}{2}(1\pm \mathrm{i}){\mathrm{e}}^{{\zeta }^2}\mathrm{erfc}\zeta .$$

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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, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

A&S Ref:

7.3.22 (in different form)

Referenced by:

§7.12(ii), §7.13(iv), §7.5, §7.5

Permalink:

http://dlmf.nist.gov/7.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.11 $${|ℱ\left(x\right)|}^2={\mathrm{f}}^2\left(x\right)+{\mathrm{g}}^2\left(x\right),$$ $x\ge 0$,

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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, $ℱ\left(z\right)$: Fresnel integral and $x$: real variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.12 $${|ℱ\left(x\right)|}^2=2+{\mathrm{f}}^2\left(-x\right)+{\mathrm{g}}^2\left(-x\right)-2\sqrt{2}\cos \left(\frac{1}{4}\pi +\frac{1}{2}\pi x^2\right)\mathrm{f}\left(-x\right)-2\sqrt{2}\cos \left(\frac{1}{4}\pi -\frac{1}{2}\pi x^2\right)\mathrm{g}\left(-x\right),$$ $x\le 0$.

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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, $ℱ\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $x$: real variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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3: 7.10 Derivatives

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7.10.1 $$\frac{d^{n+1}\mathrm{erf}z}{{dz}^{n+1}}={(-1)}^n\frac{2}{\sqrt{\pi }}{H}_n\left(z\right){\mathrm{e}}^{-z^2},$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.19

Permalink:

http://dlmf.nist.gov/7.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

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

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

4: 8.22 Mathematical Applications

… ►The so-called terminant function $F_p\left(z\right)$, defined by ►

8.22.1 $$F_p\left(z\right)=\frac{\mathrm{\Gamma }\left(p\right)}{2\pi }{z}^{1-p}{E}_p\left(z\right)=\frac{\mathrm{\Gamma }\left(p\right)}{2\pi }\mathrm{\Gamma }(1-p,z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $F_p\left(z\right)$: terminant function, $z$: complex variable and $p$: parameter

Permalink:

http://dlmf.nist.gov/8.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.22(i), §8.22 and Ch.8

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5: 7.24 Approximations

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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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Luke (1969b, pp. 323–324) covers $\frac{1}{2}\sqrt{\pi }\mathrm{erf}x$ and ${\mathrm{e}}^{x^2}F\left(x\right)$ for $-3\le x\le 3$ (the Chebyshev coefficients are given to 20D); $\sqrt{\pi }x{\mathrm{e}}^{x^2}\mathrm{erfc}x$ and $2xF\left(x\right)$ for $x\ge 3$ (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

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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.

6: 7.25 Software

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§7.25(iv) $C\left(x\right)$, $S\left(x\right)$, $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in ℝ$

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7: 7.7 Integral Representations

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

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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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7.7.13 $$\mathrm{f}\left(z\right)=\frac{{(2\pi )}^{-3/2}}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\zeta }^{-s}\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(s+\frac{1}{2}\right)\mathrm{\Gamma }\left(s+\frac{3}{4}\right)\mathrm{\Gamma }\left(\frac{1}{4}-s\right)ds,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\zeta$: change of variable and $c$: constant

Keywords:

Mellin transform

Referenced by:

§7.7(ii), §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.7.E13

Encodings:

pMML, png, TeX

See also:

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

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7.7.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\cos \left(t^2\right)dt=\sqrt{\frac{\pi }{2}}\mathrm{f}\left(\frac{a}{\sqrt{2\pi }}\right),$$ $\mathrm{\Re }a>0$,

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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, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.22 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.7.E15

Encodings:

pMML, png, TeX

See also:

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

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8: 2.11 Remainder Terms; Stokes Phenomenon

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2.11.11 $$F_{n+p}\left(z\right)=\frac{{\mathrm{e}}^{-z}}{2\pi }{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}{t}^{n+p-1}}{1+t}dt=\frac{\mathrm{\Gamma }\left(n+p\right)}{2\pi }\frac{E_{n+p}\left(z\right)}{z^{n+p-1}}.$$

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Defines:

$F_p\left(z\right)$: terminant function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $\int$: integral

Permalink:

http://dlmf.nist.gov/2.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

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2.11.12 $$F_{n+p}\left(z\right)=\frac{{\mathrm{e}}^{-z}}{2\pi }{\int }_0^{\mathrm{\infty }}\exp \left(-\rho \left(t{\mathrm{e}}^{\mathrm{i}\theta }-\ln t\right)\right)\frac{t^{\alpha -1}}{1+t}dt.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\ln z$: principal branch of logarithm function, $F_p\left(z\right)$: terminant function, $\rho$: radius and $\theta$: radius

Permalink:

http://dlmf.nist.gov/2.11.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

… ►Owing to the factor ${\mathrm{e}}^{-\rho }$, that is, ${\mathrm{e}}^{-|z|}$ in (2.11.13), $F_{n+p}\left(z\right)$ is uniformly exponentially small compared with $E_p\left(z\right)$. … ►However, to enjoy the resurgence property (§2.7(ii)) we often seek instead expansions in terms of the $F$-functions introduced in §2.11(iii), leaving the connection of the error-function type behavior as an implicit consequence of this property of the $F$-functions. In this context the $F$-functions are called terminants, a name introduced by Dingle (1973). …

9: 7.2 Definitions

… ► $\mathrm{erf}z$, $\mathrm{erfc}z$, and $w\left(z\right)$ are entire functions of $z$, as is $F\left(z\right)$ in the next subsection. … ► $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$ are entire functions of $z$, as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection. … ►

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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10: 9 Airy and Related Functions

Chapter 9 Airy and Related Functions