auxiliary functions

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1: 6.1 Special Notation

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2: 7.10 Derivatives

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

3: 7.5 Interrelations

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

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

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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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4: 6.4 Analytic Continuation

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6.4.6 $$\mathrm{f}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pi {\mathrm{e}}^{\mp \mathrm{i}z}-\mathrm{f}\left(z\right),$$

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Symbols:

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

Referenced by:

§6.12(ii), §6.4

Permalink:

http://dlmf.nist.gov/6.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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6.4.7 $$\mathrm{g}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\mp \pi \mathrm{i}{\mathrm{e}}^{\mp \mathrm{i}z}+\mathrm{g}\left(z\right).$$

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Symbols:

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

Referenced by:

§6.12(ii), §6.4

Permalink:

http://dlmf.nist.gov/6.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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5: 7.4 Symmetry

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

6: 7.2 Definitions

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§7.2(iv) Auxiliary Functions

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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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7: 6.11 Relations to Other Functions

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6.11.3 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)=U(1,1,-\mathrm{i}z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.11, 5th item

Permalink:

http://dlmf.nist.gov/6.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

8: 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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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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9: 7.22 Methods of Computation

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§7.22(i) Main Functions

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10: 6.20 Approximations

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Hastings (1955) gives several minimax polynomial and rational approximations for $E_1\left(x\right)+\ln x$, $x{\mathrm{e}}^x{E}_1\left(x\right)$, and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$. These are included in Abramowitz and Stegun (1964, Ch. 5).

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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$, with accuracies up to 20S.

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Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric $U$-function (§13.2(i)) from which Chebyshev expansions near infinity for $E_1\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the $U$ functions. If the scheme can be used in backward direction.

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