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31: Bibliography F

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Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.

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Notes:

Single- and double-precision Fortran, maximum accuracy 32S.

External Links:

ISSN 0098-3500, Document, Code, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, 2nd item, 1st item.

Encodings:

BibTeX

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Tables of Values of the Function $w(z)=e^{-z^2}(1+2i{\pi }^{-1/2}{\int }_0^z{e}^{t^2}𝑑t)$ for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.

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External Links:

MathReview, ZentralBlatt

Referenced by:

V. N. Faddeeva and N. M. Terent’ev (1954).

Encodings:

BibTeX

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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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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On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.

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External Links:

Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§19.7(i).

Encodings:

BibTeX

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32: 19.28 Integrals of Elliptic Integrals

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19.28.4 $${\int }_0^1{t}^{\sigma -1}{(1-t)}^{c-1}{R}_{-a}(b_1,b_2;t,1)dt=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(\sigma \right)\mathrm{\Gamma }\left(\sigma +b_2-a\right)}{\mathrm{\Gamma }\left(\sigma +c-a\right)\mathrm{\Gamma }\left(\sigma +b_2\right)},$$ $c=b_1+b_2>0$, $\mathrm{\Re }\sigma >\max (0,a-b_2)$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\sigma$: parameter

Referenced by:

§19.28

Permalink:

http://dlmf.nist.gov/19.28.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

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19.28.9 $${\int }_0^{\pi /2}{R}_F({\sin }^2\theta {\cos }^2\left(x+y\right),{\sin }^2\theta {\cos }^2\left(x-y\right),1)d\theta =R_F(0,{\cos }^{2}x,1)R_F(0,{\cos }^{2}y,1),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $\sin z$: sine function

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

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33: 28.22 Connection Formulas

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28.22.9 $$f_{e,m}(h)=-\sqrt{\pi /2}{g}_{e,m}(h){\mathrm{Mc}}_m^{(2)}(0,h),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $f_{e,n}(h)$, $f_{o,n}(h)$: joining factors

A&S Ref:

20.8.12 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.22.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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28.22.10 $$f_{o,m}(h)=-\sqrt{\pi /2}{g}_{o,m}(h)\mathrm{Ms}_m^{(2)}{}_{}{}^{\prime }(0,h),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $f_{e,n}(h)$, $f_{o,n}(h)$: joining factors

A&S Ref:

20.8.12 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.22.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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$$\mathrm{Mc}_m^{(2)}{}_{}{}^{\prime }(0,h)=\sqrt{2/\pi }{g}_{e,m}(h),$$

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$${\mathrm{Ms}}_m^{(2)}(0,h)=-\sqrt{2/\pi }{g}_{o,m}(h),$$

… ►Here ${\mathrm{me}}_{\nu }(0,h^2)$ $(\ne 0)$ is given by (28.14.1) with $z=0$, and ${\mathrm{M}}_{\nu }^{(1)}(0,h)$ is given by (28.24.1) with $j=1$, $z=0$, and $n$ chosen so that $|c_{2n}^{\nu }(h^2)|=\max (|c_{2\mathrm{\ell }}^{\nu }(h^2)|)$, where the maximum is taken over all integers $\mathrm{\ell }$. …

34: 3.3 Interpolation

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3.3.12 $$c_n=\frac{1}{(n+1)!}\max \prod _{k=n_0}^{n_1}\left|t-k\right|,$$

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Defines:

$c_n$: bound coefficient (locally)

Symbols:

$!$: factorial (as in $n!$)

Referenced by:

§3.4(i)

Permalink:

http://dlmf.nist.gov/3.3.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(ii), §3.3 and Ch.3

►where the maximum is taken over $t$-intervals given in the formulas below. … ►

$$\left[z_0,z_1\right]f=(\left[z_1\right]f-\left[z_0\right]f)/(z_1-z_0),$$

… ►For example, for $k+1$ coincident points the limiting form is given by $\left[z_0,z_0,\mathrm{\dots },z_0\right]f=f^{(k)}(z_0)/k!$. … ►

3.3.44 $$S(k,h)(x)=\frac{\sin \left(\pi (x-kh)/h\right)}{\pi (x-kh)/h},$$

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Defines:

$S(k,h)(x)$: Sinc function (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $\sin z$: sine function

Permalink:

http://dlmf.nist.gov/3.3.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(vi), §3.3 and Ch.3

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35: Errata

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The asymptotic results were originally for $\nu$ real valued and $\nu \to +\mathrm{\infty }$. However, these results are also valid for complex values of $\nu$. The maximum sectors of validity are now specified.

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The second and the fourth lines containing $k^{\prime }/ik$ have both been replaced with $-i{k}^{\prime }/k$ to clarify the meaning.

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9.10.18 $$\mathrm{Ai}\left(z\right)=\frac{3z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{4\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt$$

The original equation taken from Schulten et al. (1979) was incorrect.

Reported 2015-03-20 by Walter Gautschi.

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9.10.19 $$\mathrm{Bi}\left(x\right)=\frac{3x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt$$

The original equation taken from Schulten et al. (1979) was incorrect.

Reported 2015-03-20 by Walter Gautschi.

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The entry for $\mathrm{sn}z$ at $z=\frac{3}{2}(K+\mathrm{i}{K}^{\prime })$ has been corrected. The correct entry is $(1+\mathrm{i})({(1+k^{\prime })}^{1/2}-\mathrm{i}{(1-k^{\prime })}^{1/2})/(2k^{1/2})$. Originally the terms ${(1+k^{\prime })}^{1/2}$ and ${(1-k^{\prime })}^{1/2}$ were given incorrectly as ${(1+k)}^{1/2}$ and ${(1-k)}^{1/2}$.

Similarly, the entry for $\mathrm{dn}z$ at $z=\frac{3}{2}(K+\mathrm{i}{K}^{\prime })$ has been corrected. The correct entry is $(-1+\mathrm{i})k_{}^{\prime }{}_{}{}^{1/2}({(1+k)}^{1/2}+i{(1-k)}^{1/2})/2$. Originally the terms ${(1+k)}^{1/2}$ and ${(1-k)}^{1/2}$ were given incorrectly as ${(1+k^{\prime })}^{1/2}$ and ${(1-k^{\prime })}^{1/2}$

Reported 2014-02-28 by Svante Janson.

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36: 29.5 Special Cases and Limiting Forms

… ►Let $\mu =\max (\nu -m,0)$. … ►

29.5.6 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{d{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)/dz|}_{z=0}}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{d{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)/dz|}_{z=0}}=\frac{\tanh z}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2},\frac{1}{2}\mu +\frac{1}{2}\nu +1}{\frac{3}{2}};{\tanh }^{2}z),$$ $m$ odd,

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

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37: 3.5 Quadrature

… ►where $h=(b-a)/n$. …To generate $G_k(h)$ the quantities $G_0(h),G_0(h/2),\mathrm{\dots },G_0(h/2^k)$ are needed. … ►Complex orthogonal polynomials $p_n(1/\zeta )$ of degree $n=0,1,2,\mathrm{\dots }$, in $1/\zeta$ that satisfy the orthogonality condition … ►A frequent problem with contour integrals is heavy cancellation, which occurs especially when the value of the integral is exponentially small compared with the maximum absolute value of the integrand. To avoid cancellation we try to deform the path to pass through a saddle point in such a way that the maximum contribution of the integrand is derived from the neighborhood of the saddle point. …

38: 3.11 Approximation Techniques

… ►A sufficient condition for $p_n(x)$ to be the minimax polynomial is that $\left|{ϵ}_n(x)\right|$ attains its maximum at $n+2$ distinct points in $[a,b]$ and ${ϵ}_n(x)$ changes sign at these consecutive maxima. … ►(Thus the $m_j$ are approximations to $m$, where $\pm m$ is the maximum value of $\left|{ϵ}_n(x)\right|$ on $[a,b]$.) … ►More precisely, it is known that for the interval $[a,b]$, the ratio of the maximum value of the remainder …to the maximum error of the minimax polynomial $p_n(x)$ is bounded by $1+L_n$, where $L_n$ is the $n$th Lebesgue constant for Fourier series; see §1.8(i). … ►and $\pm m$ is the maximum of $\left|{ϵ}_{k,\mathrm{\ell }}(x)\right|$ on $[a,b]$. …

39: Bibliography Z

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Computation of Special Functions. John Wiley & Sons Inc., New York.

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Notes:

Includes diskette containing a large collection of mathematical function software written in Fortran. Implementation in double precision. Maximum accuracy 16S.

External Links:

ISBN 0-471-11963-6, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

6th item, 3rd item, 1st item, 11st item, 7th item, 2nd item, 6th item, 3rd item, 2nd item, 5th item, 8th item, 4th item, 2nd item, §19.37(ii), §19.37(iii), §19.37(iii), §22.21, §24.19(i), §28.1, item (b), item (d), 8th item, 3rd item, 6th item, §5.22(ii), §5.22(iii), 2nd item, 2nd item, 4th item, 2nd item, 2nd item, §8.17(v), 4th item, 2nd item, 5th item, 3rd item, 3rd item, 3rd item, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Tablitsy funktsii Lezhandra $P_{-1/2+i\tau }^1(x)$ . Vyčisl. Centr Akad. Nauk SSSR, Moscow.

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Notes:

Translated title: Tables of the Legendre functions $P_{-1/2+i\tau }^1(x)$

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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Tables of the Legendre functions $P_{-1/2+i\tau }(x)$. Part I. Translated by D. E. Brown. Mathematical Tables Series, Vol. 22, Pergamon Press, Oxford.

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Notes:

Translation of Žurina and Karmazina, Tablitsy funktsii Lezhandra $P_{-1/2+i\tau }(x)$. Tom I, Izdat. Akad. Nauk SSSR, Moscow, 1960.

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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Tables of the Legendre functions $P_{-1/2+i\tau }(x)$. Part II. Translated by Prasenjit Basu. Mathematical Tables Series, Vol. 38. A Pergamon Press Book, The Macmillan Co., New York.

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Notes:

Translation of Žurina and Karmazina, Tablitsy funktsii Lezhandra $P_{-1/2+i\tau }(x)$. Tom II, Izdat. Akad. Nauk SSSR, Moscow, 1962.

External Links:

MathReview, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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Tables and formulae for the spherical functions $P_{-1/2+i\tau }^m(z)$ . Translated by E. L. Albasiny, Pergamon Press, Oxford.

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Notes:

Translation of Žurina and Karmazina, Tablitsy i formuly dlya sfericheskikh funktsii $P_{-1/2+i\tau }^m(z)$ , Vyčisl. Centr Akad Nauk SSSR, Moscow. 1962

External Links:

MathReview, ZentralBlatt

Referenced by:

§14.20(vii).

Encodings:

BibTeX

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40: 1.14 Integral Transforms

… ►If $\mathrm{\Re }s>\max (\mathrm{\Re }(a+\alpha ),\alpha )$, then … ►If also ${lim}_{t\to 0+}f(t)/t$ exists, then … ►where $A_p=\tan \left(\frac{1}{2}\pi /p\right)$ when , or $\cot \left(\frac{1}{2}\pi /p\right)$ when $p\ge 2$. … ►

Table 1.14.3: Fourier sine transforms.

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$f(t)$	$\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle {\int }_0^{\mathrm{\infty }}}f(t)\sin \left(xt\right)dt$,	$x>0$
…
$t^{-1/2}$	$x^{-1/2}$
$t^{-3/2}$	$2x^{1/2}$
…

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