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

ⓘ

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

ⓘ

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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12: 6.18 Methods of Computation

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§6.18(ii) Auxiliary Functions

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13: 6.2 Definitions and Interrelations

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§6.2(iii) Auxiliary Functions

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6.2.17 $$\mathrm{f}\left(z\right)=\mathrm{Ci}\left(z\right)\sin z-\mathrm{si}\left(z\right)\cos z,$$

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Defines:

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

Symbols:

$\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{si}\left(z\right)$: sine integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

5.2.6

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.2.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(iii), §6.2 and Ch.6

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6.2.18 $$\mathrm{g}\left(z\right)=-\mathrm{Ci}\left(z\right)\cos z-\mathrm{si}\left(z\right)\sin z.$$

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Defines:

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

Symbols:

$\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{si}\left(z\right)$: sine integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

5.2.7

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.2.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(iii), §6.2 and Ch.6

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6.2.19 $$\mathrm{Si}\left(z\right)=\frac{1}{2}\pi -\mathrm{f}\left(z\right)\cos z-\mathrm{g}\left(z\right)\sin z,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{Si}\left(z\right)$: sine integral and $z$: complex variable

A&S Ref:

5.2.8

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(iii), §6.2 and Ch.6

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6.2.20 $$\mathrm{Ci}\left(z\right)=\mathrm{f}\left(z\right)\sin z-\mathrm{g}\left(z\right)\cos z.$$

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Symbols:

$\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\sin z$: sine function, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

A&S Ref:

5.2.9

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.2.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(iii), §6.2 and Ch.6

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14: 6.12 Asymptotic Expansions

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6.12.3 $$\mathrm{f}\left(z\right)\sim \frac{1}{z}\left(1-\frac{2!}{z^2}+\frac{4!}{z^4}-\frac{6!}{z^6}+\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

A&S Ref:

5.2.34

Referenced by:

§6.12(ii), §6.12(ii)

Permalink:

http://dlmf.nist.gov/6.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(ii), §6.12 and Ch.6

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6.12.4 $$\mathrm{g}\left(z\right)\sim \frac{1}{z^2}\left(1-\frac{3!}{z^2}+\frac{5!}{z^4}-\frac{7!}{z^6}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

A&S Ref:

5.2.35

Referenced by:

§6.12(ii), §6.12(ii)

Permalink:

http://dlmf.nist.gov/6.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(ii), §6.12 and Ch.6

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6.12.5 $$\mathrm{f}\left(z\right)=\frac{1}{z}\sum _{m=0}^{n-1}{(-1)}^m\frac{(2m)!}{z^{2m}}+R_n^{(\mathrm{f})}(z),$$

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Symbols:

$!$: factorial (as in $n!$), $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{f})}(z)$: remainder term

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(ii), §6.12 and Ch.6

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15: 8.21 Generalized Sine and Cosine Integrals

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§8.21(vii) Auxiliary Functions

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8.21.18 $$f(a,z)=\mathrm{si}(a,z)\cos z-\mathrm{ci}(a,z)\sin z,$$

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Defines:

$f(a,z)$: auxiliary function (locally)

Symbols:

$\cos z$: cosine function, $\mathrm{ci}(a,z)$: generalized cosine integral, $\mathrm{si}(a,z)$: generalized sine integral, $\sin z$: sine function, $z$: complex variable and $a$: parameter

A&S Ref:

5.2.6 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.20 $$\mathrm{si}(a,z)=f(a,z)\cos z+g(a,z)\sin z,$$

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Symbols:

$\cos z$: cosine function, $\mathrm{si}(a,z)$: generalized sine integral, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $f(a,z)$: auxiliary function and $g(a,z)$: auxiliary function

Referenced by:

§8.21(viii)

Permalink:

http://dlmf.nist.gov/8.21.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.21 $$\mathrm{ci}(a,z)=-f(a,z)\sin z+g(a,z)\cos z.$$

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Symbols:

$\cos z$: cosine function, $\mathrm{ci}(a,z)$: generalized cosine integral, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $f(a,z)$: auxiliary function and $g(a,z)$: auxiliary function

Referenced by:

§8.21(viii)

Permalink:

http://dlmf.nist.gov/8.21.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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§8.21(viii) Asymptotic Expansions

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16: 6.7 Integral Representations

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§6.7(iii) Auxiliary Functions

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6.7.12 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)={\mathrm{e}}^{-\mathrm{i}z}{\int }_z^{\mathrm{\infty }}\frac{{\mathrm{e}}^{\mathrm{i}t}}{t}dt,$$ $|\mathrm{ph}z|\le \pi .$

ⓘ

Symbols:

$\pi$: 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, $\mathrm{ph}$: phase, $\mathrm{f}\left(z\right)$: 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:

§6.7(iii)

Permalink:

http://dlmf.nist.gov/6.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(iii), §6.7 and Ch.6

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6.7.13 $$\mathrm{f}\left(z\right)={\int }_0^{\mathrm{\infty }}\frac{\sin t}{t+z}dt={\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}}{t^2+1}dt,$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\sin z$: sine function, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.12(ii), §6.7(iii)

Permalink:

http://dlmf.nist.gov/6.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(iii), §6.7 and Ch.6

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6.7.15 $$\mathrm{f}\left(z\right)=2{\int }_0^{\mathrm{\infty }}{K}_0\left(2\sqrt{zt}\right)\cos tdt,$$

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.7(iii)

Permalink:

http://dlmf.nist.gov/6.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(iii), §6.7 and Ch.6

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6.7.16 $$\mathrm{g}\left(z\right)=2{\int }_0^{\mathrm{\infty }}{K}_0\left(2\sqrt{zt}\right)\sin tdt.$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.7(iii)

Permalink:

http://dlmf.nist.gov/6.7.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(iii), §6.7 and Ch.6

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17: 8.24 Physical Applications

… ►With more general values of $p$, $E_p\left(x\right)$ supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).

18: 12.19 Tables

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Miller (1955) includes $W(a,x)$, $W(a,-x)$, and reduced derivatives for $a=-10(1)10$, $x=0(.1)10$, 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.

►

Fox (1960) includes modulus and phase functions for $W(a,x)$ and $W(a,-x)$, and several auxiliary functions for $x^{-1}=0(.005)0.1$, $a=-10(1)10$, 8S.

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19: 6.19 Tables

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§6.19(ii) Real Variables

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20: 7.12 Asymptotic Expansions

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§7.12(ii) Fresnel Integrals

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7.12.2 $$\mathrm{f}\left(z\right)\sim \frac{1}{\pi z}\sum _{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m}}{{(\pi z^2/2)}^{2m}},$$

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

A&S Ref:

7.3.27

Referenced by:

§7.12(ii), §7.12(ii), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E2

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{\pi z}{\sum }_{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{1\cdot 3\cdot 5\mathrm{\cdots }(4m-1)}{{(\pi z^2)}^{2m}}$. We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

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7.12.3 $$\mathrm{g}\left(z\right)\sim \frac{1}{\pi z}\sum _{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m+1}}{{(\pi z^2/2)}^{2m+1}},$$

ⓘ

Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

A&S Ref:

7.3.28

Referenced by:

§7.12(ii), §7.12(ii), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E3

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{{\pi }^2{z}^3}{\sum }_{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{1\cdot 3\cdot 5\mathrm{\cdots }(4m+1)}{{(\pi z^2)}^{2m}}$ We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

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7.12.4 $$\mathrm{f}\left(z\right)=\frac{1}{\pi z}\sum _{m=0}^{n-1}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m}}{{(\pi z^2/2)}^{2m}}+R_n^{(\mathrm{f})}(z),$$

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{f})}(z)$: remainder term

A&S Ref:

7.3.27 (in different form)

Referenced by:

§7.12(ii), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E4

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{\pi z}{\sum }_{m=0}^{n-1}{(-1)}^m\frac{1\cdot 3\mathrm{\cdots }(4m-1)}{{(\pi z^2)}^{2m}}+R_n^{(\mathrm{f})}(z)$. We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

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7.12.5 $$\mathrm{g}\left(z\right)=\frac{1}{\pi z}\sum _{m=0}^{n-1}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m+1}}{{(\pi z^2/2)}^{2m+1}},+R_n^{(\mathrm{g})}(z),$$

ⓘ

Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{g})}(z)$: remainder term

A&S Ref:

7.3.28 (in different form)

Referenced by:

Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E5

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{{\pi }^2{z}^3}{\sum }_{m=0}^{n-1}{(-1)}^m\frac{1\cdot 3\mathrm{\cdots }(4m+1)}{{(\pi z^2)}^{2m}}+R_n^{(\mathrm{g})}(z)$. We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

…