relations to exponential integrals

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21: 7.13 Zeros

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§7.13(iii) Zeros of the Fresnel Integrals

… ►As $n\to \mathrm{\infty }$ the $x_n$ and $y_n$ corresponding to the zeros of $C\left(z\right)$ satisfy … ►As $n\to \mathrm{\infty }$ the $x_n$ and $y_n$ corresponding to the zeros of $S\left(z\right)$ satisfy (7.13.5) with … ►In consequence of (7.5.5) and (7.5.10), zeros of $ℱ\left(z\right)$ are related to zeros of $\mathrm{erfc}z$. … ►For an asymptotic expansion of the zeros of ${\int }_0^z\exp \left(\frac{1}{2}\pi \mathrm{i}{t}^2\right)dt$ ($=ℱ\left(0\right)-ℱ\left(z\right)$ $=C\left(z\right)+\mathrm{i}S\left(z\right)$) see Tuẑilin (1971). …

22: 8.11 Asymptotic Approximations and Expansions

… ►in both cases uniformly with respect to bounded real values of $y$. For Dawson’s integral $F\left(y\right)$ see §7.2(ii). …For related expansions involving Hermite polynomials see Pagurova (1965). … ►This reference also contains explicit formulas for the coefficients in terms of Stirling numbers. … ►With $x=1$, an asymptotic expansion of $e_n(nx)/{\mathrm{e}}^{nx}$ follows from (8.11.14) and (8.11.16). …

23: 15.19 Methods of Computation

… ►For $z\in ℂ$ it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. This is because the linear transformations map the pair $\{{\mathrm{e}}^{\pi \mathrm{i}/3},{\mathrm{e}}^{-\pi \mathrm{i}/3}\}$ onto itself. … ►

§15.19(iii) Integral Representations

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§15.19(iv) Recurrence Relations

►The relations in §15.5(ii) can be used to compute $F(a,b;c;z)$, provided that care is taken to apply these relations in a stable manner; see §3.6(ii). …

24: 3.5 Quadrature

… ►For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. … ►The monic and orthonormal recursion relations of this section are both closely related to the Lanczos recursion relation in §3.2(vi). … ►are related to Bessel polynomials (§§10.49(ii) and 18.34). … … ►The standard Monte Carlo method samples points uniformly from the integration region to estimate the integral and its error. …

25: Bibliography G

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Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.

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Notes:

Includes Algol60 program, maximum accuracy 10S.

External Links:

ISSN 0001-0782, Document

Referenced by:

§6.21(ii), §8.28(vi).

Encodings:

BibTeX

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Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

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External Links:

MathReview (H. O. Pollak), ZentralBlatt

Referenced by:

§5.6(i), §6.8, §7.8, §8.10.

Encodings:

BibTeX

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Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.8(vi).

Encodings:

BibTeX

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Automatic reduction of elliptic integrals using Carlson’s relations. Math. Comp. 71 (237), pp. 311–318.

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External Links:

ISSN 0025-5718, Document, MathReview (David A. Richter), ZentralBlatt

Referenced by:

§19.29(ii).

Encodings:

BibTeX

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Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.

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External Links:

ISSN 1083-4362, Link, MathReview (Genkai Zhang), ZentralBlatt

Referenced by:

§18.28(xi), §18.38(iii).

Encodings:

BibTeX

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26: 7.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the argument. ►The main functions treated in this chapter are the error function $\mathrm{erf}z$; the complementary error functions $\mathrm{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the Goodwin–Staton integral $G\left(z\right)$; the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$; the Voigt functions $𝖴(x,t)$ and $𝖵(x,t)$. ►Alternative notations are $Q(z)=\frac{1}{2}\mathrm{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\mathrm{\Phi }(z)=\frac{1}{2}\mathrm{erfc}\left(-z/\sqrt{2}\right)$, $\mathrm{Erf}z=\frac{1}{2}\sqrt{\pi }\mathrm{erf}z$, $\mathrm{Erfi}z={\mathrm{e}}^{z^2}F\left(z\right)$, $C_1(z)=C\left(\sqrt{2/\pi }z\right)$, $S_1(z)=S\left(\sqrt{2/\pi }z\right)$, $C_2(z)=C\left(\sqrt{2z/\pi }\right)$, $S_2(z)=S\left(\sqrt{2z/\pi }\right)$. … ►

27: 19.5 Maclaurin and Related Expansions

§19.5 Maclaurin and Related Expansions

… ► … ►Coefficients of terms up to ${\lambda }^{49}$ are given in Lee (1990), along with tables of fractional errors in $K\left(k\right)$ and $E\left(k\right)$, $0.1\le k^2\le 0.9999$, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). … ►An infinite series for $\ln K\left(k\right)$ is equivalent to the infinite product … ►

28: 10.9 Integral Representations

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Poisson’s and Related Integrals

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Schläfli’s and Related Integrals

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Mehler–Sonine and Related Integrals

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§10.9(ii) Contour Integrals

… ►See Paris and Kaminski (2001, p. 116) for related results. …

29: 14.30 Spherical and Spheroidal Harmonics

… ► $P_n^m\left(x\right)$ and $Q_n^m\left(x\right)$ ($x>1$) are often referred to as the prolate spheroidal harmonics of the first and second kinds, respectively. …Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics $R_n^m(x)={\mathrm{e}}^{-\mathrm{i}\pi n/2}{P}_n^m\left(\mathrm{i}x\right)$ and $T_n^m(x)=\mathrm{i}{\mathrm{e}}^{\mathrm{i}\pi n/2}{Q}_n^m\left(\mathrm{i}x\right)$ which are real when $x>0$ and $n=0,1,2,\mathrm{\dots }$. … ►

14.30.3 $$Y_{l,m}(\theta ,\varphi )=\frac{{(-1)}^{l+m}}{2^{l}l!}{\left(\frac{(l-m)!(2l+1)}{4\pi (l+m)!}\right)}^{1/2}{\mathrm{e}}^{\mathrm{i}m\varphi }{\left(\sin \theta \right)}^m{\left(\frac{d}{d(\cos \theta )}\right)}^{l+m}{\left(\sin \theta \right)}^{2l}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(ii)

Permalink:

http://dlmf.nist.gov/14.30.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

… ►See also (34.3.22), and for further related integrals see Askey et al. (1986). … ►

14.30.8_5 $${\mathrm{e}}^{t𝐚\cdot 𝐱}=\sqrt{4\pi }\sum _{n=0}^{\mathrm{\infty }}\sum _{m=-n}^n\frac{t^n{r}^n{\lambda }^m{Y}_{n,m}(\theta ,\varphi )}{\sqrt{(2n+1)(n+m)!(n-m)!}},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $n$: nonnegative integer and $m$: integer

Source:

Courant and Hilbert (1953, §VII.7.3)

Permalink:

http://dlmf.nist.gov/14.30.E8_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

Suggested 2019-09-26 by Anupam Garg

See also:

Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

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30: Bibliography S

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A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.

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External Links:

ISSN 0036-1410, Document, MathReview (Ernest G. Kalnins), ZentralBlatt

Referenced by:

§28.32(i).

Encodings:

BibTeX

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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External Links:

ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

Encodings:

BibTeX

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Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§29.8.

Encodings:

BibTeX

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Algorithm 911: multiple-precision exponential integral and related functions. ACM Trans. Math. Software 37 (4), pp. Art. 46, 16.

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External Links:

Document, ZentralBlatt

Referenced by:

5th item, §4.48(iii), 3rd item, §6.21(ii).

Encodings:

BibTeX

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Automatic computing methods for special functions. III. The sine, cosine, exponential integrals, and related functions. J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 291–311.

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Notes:

Includes Fortran code, maximum accuracy 18S.

External Links:

MathReview (Martin E. Price), ZentralBlatt

Referenced by:

§6.21(ii).

Encodings:

BibTeX

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