auxiliary functions

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21: 6.5 Further Interrelations

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6.5.7 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=E_1\left(\mp \mathrm{i}z\right){\mathrm{e}}^{\mp \mathrm{i}z}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: 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, §6.5, §6.7(iii)

Permalink:

http://dlmf.nist.gov/6.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

22: 7.14 Integrals

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7.14.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(t\right)dt=\frac{1}{a}\mathrm{f}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.27 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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7.14.6 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}S\left(t\right)dt=\frac{1}{a}\mathrm{g}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.28 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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23: 9.18 Tables

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Miller (1946) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$ for $x=-20(.01)2;$ ${\log }_{10}\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)/\mathrm{Ai}\left(x\right)$ for $x=0(.1)25(1)75$; $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=-10(.1)2.5$; ${\log }_{10}\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)/\mathrm{Bi}\left(x\right)$ for $x=0(.1)10$; $M\left(x\right)$, $N\left(x\right)$, $\theta \left(x\right)$, $\varphi \left(x\right)$ (respectively $F(x)$, $G(x)$, $\chi (x)$, $\psi (x)$) for $x=-80(1)-30(.1)0$. Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

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Fox (1960, Table 3) tabulates $2{\pi }^{1/2}{x}^{1/4}\exp (\frac{2}{3}{x}^{3/2})\mathrm{Ai}\left(x\right)$, $2{\pi }^{1/2}{x}^{-1/4}\exp (\frac{2}{3}{x}^{3/2}){\mathrm{Ai}}^{\prime }\left(x\right)$, ${\pi }^{1/2}{x}^{1/4}\exp (-\frac{2}{3}{x}^{3/2})\mathrm{Bi}\left(x\right)$, and ${\pi }^{1/2}{x}^{-1/4}\exp (-\frac{2}{3}{x}^{3/2}){\mathrm{Bi}}^{\prime }\left(x\right)$ for $\frac{3}{2}{x}^{-3/2}=0(.001)0.05$, together with similar auxiliary functions for negative values of $x$. Precision is 10D.

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24: 10.75 Tables

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British Association for the Advancement of Science (1937) tabulates $J_0\left(x\right)$, $J_1\left(x\right)$, $x=0(.001)16(.01)25$, 10D; $Y_0\left(x\right)$, $Y_1\left(x\right)$, $x=0.01(.01)25$, 8–9S or 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $Y_0\left(x\right)$, $Y_1\left(x\right)$ for small values of $x$, as well as auxiliary functions to compute all four functions for large values of $x$.

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British Association for the Advancement of Science (1937) tabulates $I_0\left(x\right)$, $I_1\left(x\right)$, $x=0(.001)5$, 7–8D; $K_0\left(x\right)$, $K_1\left(x\right)$, $x=0.01(.01)5$, 7–10D; ${\mathrm{e}}^{-x}{I}_0\left(x\right)$, ${\mathrm{e}}^{-x}{I}_1\left(x\right)$, ${\mathrm{e}}^x{K}_0\left(x\right)$, ${\mathrm{e}}^x{K}_1\left(x\right)$, $x=5(.01)10(.1)20$, 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $K_0\left(x\right)$, $K_1\left(x\right)$ for small values of $x$.

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Young and Kirk (1964) tabulates ${\mathrm{ber}}_{n}x$, ${\mathrm{bei}}_{n}x$, ${\ker }_{n}x$, ${\mathrm{kei}}_{n}x$, $n=0,1$, $x=0(.1)10$, 15D; ${\mathrm{ber}}_{n}x$, ${\mathrm{bei}}_{n}x$, ${\ker }_{n}x$, ${\mathrm{kei}}_{n}x$, modulus and phase functions $M_n\left(x\right)$, ${\theta }_n\left(x\right)$, $N_n\left(x\right)$, ${\varphi }_n\left(x\right)$, $n=0,1,2$, $x=0(.01)2.5$, 8S, and $n=0(1)10$, $x=0(.1)10$, 7S. Also included are auxiliary functions to facilitate interpolation of the tables for $n=0(1)10$ for small values of $x$. (Concerning the phase functions see §10.68(iv).)

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25: 10.74 Methods of Computation

… ►Temme (1997) shows how to overcome this difficulty by use of the Maclaurin expansions for these coefficients or by use of auxiliary functions. …

26: 1.14 Integral Transforms

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1.14.21 $$ℒf\left(s-a\right)=ℒf_a\left(s\right),$$

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Symbols:

$ℒ\left(f\right)\left(s\right)$: Laplace transform and $\mathrm{e}$: base of natural logarithm

Keywords:

Laplace transform

Permalink:

http://dlmf.nist.gov/1.14.E21

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Laplace transform was changed to $ℒf\left(s\right)$ from $ℒ(f(t);s)$. In addition, the auxiliaryfunction $f_a(t)={\mathrm{e}}^{at}f(t)$ was introduced.

See also:

Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

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1.14.22 $$ℒf_a^{+}\left(s\right)={\mathrm{e}}^{-as}ℒf\left(s\right),$$

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Symbols:

$H\left(x\right)$: Heaviside function, $ℒ\left(f\right)\left(s\right)$: Laplace transform and $\mathrm{e}$: base of natural logarithm

Keywords:

Laplace transform

Permalink:

http://dlmf.nist.gov/1.14.E22

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Laplace transform was changed to $ℒ\left(f\right)\left(s\right)$ from $ℒ(f(t);s)$. In addition, the auxiliaryfunction $f_a^{+}(t)=H\left(t-a\right)f(t-a)$ was introduced.

See also:

Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

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1.14.23 $${(-1)}^n\frac{d^n}{{ds}^n}ℒf\left(s\right)=ℒf_n\left(s\right),$$ $n=1,2,3,\mathrm{\dots }$,

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Symbols:

$ℒ\left(f\right)\left(s\right)$: Laplace transform, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $n$: nonnegative integer

Keywords:

Laplace transform

Permalink:

http://dlmf.nist.gov/1.14.E23

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Laplace transform was changed to $ℒf\left(s\right)$ from $ℒ(f(t);s)$. In addition, the auxiliaryfunction $f_n(t)=t^{n}f(t)$ was introduced.

See also:

Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

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1.14.24 $${\int }_s^{\mathrm{\infty }}ℒf\left(u\right)du=ℒf_{-1}\left(s\right),$$

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Symbols:

$ℒ\left(f\right)\left(s\right)$: Laplace transform, $dx$: differential of $x$ and $\int$: integral

Keywords:

Laplace transform

Permalink:

http://dlmf.nist.gov/1.14.E24

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Laplace transform was changed to $ℒf\left(s\right)$ from $ℒ(f(t);s)$. In addition, the auxiliaryfunction $f_{-1}(t)=\frac{f(t)}{t}$ was introduced.

See also:

Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

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27: Bibliography F

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A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.

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Notes:

Table erratum: Math. Comp. v. 36 (1981), no. 153, p. 318. Part II of this report, with the same date but numbered ARL 69-0173, again puts $k=R\exp (i\theta )$ but with $R$ as parameter and $\theta$ as variable instead of the other way around. Part III, dated May 1970 and numbered ARL 70-0081, contains tables of auxiliary functions to help interpolation.

External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(ii), §19.37(ii).

Encodings:

BibTeX

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28: 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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29: 14.33 Tables

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Žurina and Karmazina (1964, 1965) tabulate the conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ for $\tau =0(.01)50$, $x=-0.9(.1)0.9$, 7S; $P_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ for $\tau =0(.01)50$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7D. Auxiliary tables are included to facilitate computation for larger values of $\tau$ when .

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Žurina and Karmazina (1963) tabulates the conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^1\left(x\right)$ for $\tau =0(.01)25$, $x=-0.9(.1)0.9$, 7S; $P_{-\frac{1}{2}+\mathrm{i}\tau }^1\left(x\right)$ for $\tau =0(.01)25$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7S. Auxiliary tables are included to assist computation for larger values of $\tau$ when .

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30: Bibliography K

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On the evaluation of the Gauss hypergeometric function. C. R. Acad. Bulgare Sci. 45 (6), pp. 35–36.

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External Links:

ISSN 0861-1459, MathReview, ZentralBlatt

Referenced by:

§15.15, §15.19(i).

Encodings:

BibTeX

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Generalized functions. Mathematics in Science and Engineering, Vol. 171, Academic Press, Inc., Orlando, FL.

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Notes:

Theory and technique

External Links:

ISBN 0-12-396560-8, MathReview (Qing Yi Chen), ZentralBlatt

Referenced by:

§1.16(viii).

Encodings:

BibTeX

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Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§19.12.

Encodings:

BibTeX

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Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.13(iii), §8.22(ii).

Encodings:

BibTeX

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Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument. Comput. Phys. Comm. 4, pp. 221–226.

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Notes:

Single-precision Fortran, accuracy 12-14S

External Links:

Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

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