DLMF: Untitled Document

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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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§20.10 Integrals

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§20.10(i) Mellin Transforms with respect to the Lattice Parameter

… ►Here

\zeta\left(s\right)

again denotes the Riemann zeta function (§25.2). … ►

§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

… ►For corresponding results for argument derivatives of the theta functions see Erdélyi et al. (1954a, pp. 224–225) or Oberhettinger and Badii (1973, p. 193). …

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x=\tfrac{9}{20}z^{2}.

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x=-\tfrac{3}{20}z^{2},

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x=\tfrac{1}{3}z^{2}(-\cos\left(2\phi\right)-2\cos\phi),

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y=\tfrac{1}{3}z^{2}(\sin\left(2\phi\right)-2\sin\phi)

,

0\leq\phi\leq 2\pi

.

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36.4.11

x+iy=-z^{2}\exp\left(\tfrac{2}{3}i\pi m\right),

m=0,1,2

.

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $y$ : real parameter, $z$ : real parameter, $n$ : integer and $x$ : real parameter
Referenced by:: §36.7(iii)
Permalink:: http://dlmf.nist.gov/36.4.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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Abramowitz and Stegun (1964, Chapter 7) includes $\operatorname{erf}x$ , $(2/\sqrt{\pi})e^{-x^{2}}$ , $x\in[0,2]$ , 10D; $(2/\sqrt{\pi})e^{-x^{2}}$ , $x\in[2,10]$ , 8S; $xe^{x^{2}}\operatorname{erfc}x$ , $x^{-2}\in[0,0.25]$ , 7D; $2^{n}\Gamma\left(\frac{1}{2}n+1\right)\mathop{\mathrm{i}^{n}\mathrm{erfc}}% \left(x\right)$ , $n=1(1)6,10,11$ , $x\in[0,5]$ , 6S; $F\left(x\right)$ , $x\in[0,2]$ , 10D; $xF\left(x\right)$ , $x^{-2}\in[0,0.25]$ , 9D; $C\left(x\right)$ , $S\left(x\right)$ , $x\in[0,5]$ , 7D; $\mathrm{f}\left(x\right)$ , $\mathrm{g}\left(x\right)$ , $x\in[0,1]$ , $x^{-1}\in[0,1]$ , 15D.

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Zhang and Jin (1996, pp. 637, 639) includes $(2/\sqrt{\pi})e^{-x^{2}}$ , $\operatorname{erf}x$ , $x=0(.02)1(.04)3$ , 8D; $C\left(x\right)$ , $S\left(x\right)$ , $x=0(.2)10(2)100(100)500$ , 8D.

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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\operatorname{erf}z$ , $x\in[0,5]$ , $y=0.5(.5)3$ , 7D and 8D, respectively; the real and imaginary parts of $\int_{x}^{\infty}e^{\pm\mathrm{i}t^{2}}\,\mathrm{d}t$ , $(1/\sqrt{\pi})e^{\mp\mathrm{i}(x^{2}+(\pi/4))}\int_{x}^{\infty}e^{\pm\mathrm{i% }t^{2}}\,\mathrm{d}t$ , $x=0(.5)20(1)25$ , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

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Fettis et al. (1973) gives the first 100 zeros of $\operatorname{erf}z$ and $w\left(z\right)$ (the table on page 406 of this reference is for $w\left(z\right)$ , not for $\operatorname{erfc}z$ ), 11S.

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Zhang and Jin (1996, p. 642) includes the first 10 zeros of $\operatorname{erf}z$ , 9D; the first 25 distinct zeros of $C\left(z\right)$ and $S\left(z\right)$ , 8S.

… ►The notation

\operatorname{Li}_{2}\left(z\right)

was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828): … ►The special case

z=1

is the Riemann zeta function:

\zeta\left(s\right)=\operatorname{Li}_{s}\left(1\right)

. … ►Further properties include …and … ►In terms of polylogarithms …

§7.8 Inequalities

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7.8.5

\frac{x^{2}}{2x^{2}+1}\leq\frac{x^{2}(2x^{2}+5)}{4x^{4}+12x^{2}+3}\leq x% \mathsf{M}\left(x\right)<\frac{2x^{4}+9x^{2}+4}{4x^{4}+20x^{2}+15}<\frac{x^{2}% +1}{2x^{2}+3},

x\geq 0

.

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Symbols:: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio and $x$ : real variable
Referenced by:: §7.8
Permalink:: http://dlmf.nist.gov/7.8.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §7.8 and Ch.7

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7.8.7

\frac{\sinh x^{2}}{x}<{\mathrm{e}}^{x^{2}}F\left(x\right)=\int_{0}^{x}{\mathrm% {e}}^{t^{2}}\,\mathrm{d}t<\frac{{\mathrm{e}}^{x^{2}}-1}{x},

x>0

.

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Symbols:: $F\left(\NVar{z}\right)$ : Dawson’s integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral and $x$ : real variable
Referenced by:: Erratum (V1.1.5) for Additions
Permalink:: http://dlmf.nist.gov/7.8.E7
Encodings:: pMML, png, TeX
Addition (effective with 1.1.5):: A new inequality was added.
See also:: Annotations for §7.8 and Ch.7

►The function

F\left(x\right)/\sqrt{1-{\mathrm{e}}^{-2x^{2}}}

is strictly decreasing for

x>0

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7.8.8

\operatorname{erf}x<\sqrt{1-{\mathrm{e}}^{-4x^{2}/\pi}},

x>0

.

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm and $x$ : real variable
Source:: Pólya (1949, (1.5))
Referenced by:: §7.8, Erratum (V1.0.17) for Equation (7.8.8)
Permalink:: http://dlmf.nist.gov/7.8.E8
Encodings:: pMML, png, TeX
Addition (effective with 1.0.17):: This inequality was added. It is Pólya (1949, (1.5)). Note that it is a special case of (8.10.11).
Suggested by Roberto Iacono
See also:: Annotations for §7.8 and Ch.7

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§6.19(ii) Real Variables

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Abramowitz and Stegun (1964, Chapter 5) includes $x^{-1}\operatorname{Si}\left(x\right)$ , $-x^{-2}\operatorname{Cin}\left(x\right)$ , $x^{-1}\operatorname{Ein}\left(x\right)$ , $-x^{-1}\operatorname{Ein}\left(-x\right)$ , $x=0(.01)0.5$ ; $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=0.5(.01)2$ ; $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $xe^{-x}\operatorname{Ei}\left(x\right)$ , $xe^{x}E_{1}\left(x\right)$ , $x=2(.1)10$ ; $x\mathrm{f}\left(x\right)$ , $x^{2}\mathrm{g}\left(x\right)$ , $xe^{-x}\operatorname{Ei}\left(x\right)$ , $xe^{x}E_{1}\left(x\right)$ , $x^{-1}=0(.005)0.1$ ; $\operatorname{Si}\left(\pi x\right)$ , $\operatorname{Cin}\left(\pi x\right)$ , $x=0(.1)10$ . Accuracy varies but is within the range 8S–11S.

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Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $x=0(.5)20(2)30$ , 8D; $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=[0,100]$ , 8S.

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Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $ze^{z}E_{1}\left(z\right)$ , $x=-19(1)20$ , $y=0(1)20$ , 6D; $e^{z}E_{1}\left(z\right)$ , $x=-4(.5)-2$ , $y=0(.2)1$ , 6D; $E_{1}\left(z\right)+\ln z$ , $x=-2(.5)2.5$ , $y=0(.2)1$ , 6D.

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Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_{1}\left(z\right)$ , $\pm x=0.5,1,3,5,10,15,20,50,100$ , $y=0(.5)1(1)5(5)30,50,100$ , 8S.

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FDLIBM (free C library)

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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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S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §6.15.
Encodings:: BibTeX

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H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:: Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.
Referenced by:: §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).
Encodings:: BibTeX

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C. L. Frenzen (1990) Error bounds for a uniform asymptotic expansion of the Legendre function

{Q}^{-m}_{n}({\rm cosh}\,z)

. SIAM J. Math. Anal. 21 (2), pp. 523–535.

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External Links:: ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §14.26.
Encodings:: BibTeX

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G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:: ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)
Referenced by:: §18.32.
Encodings:: BibTeX

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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 $\operatorname{erf}x$ , $\operatorname{erfc}x$ and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ .

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Cody (1969) provides minimax rational approximations for $\operatorname{erf}x$ and $\operatorname{erfc}x$ . The maximum relative precision is about 20S.

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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, vol. 2, pp. 422–435) gives main diagonal Padé approximations for $F\left(z\right)$ , $\operatorname{erf}z$ , $\operatorname{erfc}z$ , $C\left(z\right)$ , and $S\left(z\right)$ ; approximate errors are given for a selection of $z$ -values.

§14.30 Spherical and Spheroidal Harmonics

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Herglotz generating function

►The following is the Herglotz generating function …where

\mathbf{a}=\left(\tfrac{1}{2\lambda}-\tfrac{\lambda}{2},-\tfrac{\mathrm{i}}{2% \lambda}-\tfrac{\mathrm{i}\lambda}{2},1\right)

and

\mathbf{x}=\left(r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta\right)

. … ►

trigonometric%20functions

1—10 of 27 matching pages

1: 6.20 Approximations

2: 20.10 Integrals

§20.10 Integrals

§20.10(i) Mellin Transforms with respect to the Lattice Parameter

§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

3: 36.4 Bifurcation Sets

4: 7.23 Tables

5: 25.12 Polylogarithms

6: 7.8 Inequalities

§7.8 Inequalities

7: 6.19 Tables

§6.19(ii) Real Variables

8: Bibliography F

9: 7.24 Approximations

§7.24(i) Approximations in Terms of Elementary Functions

10: 14.30 Spherical and Spheroidal Harmonics

§14.30 Spherical and Spheroidal Harmonics

Herglotz generating function