repeated%20integrals%20of%20the%20complementary%20error%20function

(0.005 seconds)

21—30 of 979 matching pages

21: 7.21 Physical Applications

§7.21 Physical Applications

►The error functions, Fresnel integrals, and related functions occur in a variety of physical applications. … ►Carslaw and Jaeger (1959) gives many applications and points out the importance of the repeated integrals of the complementary error function

\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)

. …Efficient algorithms for computing the Faddeeva (or Faddeyeva) function are discussed in Wells (1999), a paper frequently cited in the astrophysics literature. … ►Dawson’s integral appears in de-convolving even more complex motional effects; see Pratt (2007). …

22: 2.11 Remainder Terms; Stokes Phenomenon

… ►As an example consider … ►From §8.19(i) the generalized exponential integral is given by … ►Two different asymptotic expansions in terms of elementary functions, (2.11.6) and (2.11.7), are available for the generalized exponential integral in the sector

\frac{1}{2}\pi<\operatorname{ph}z<\frac{3}{2}\pi

. … ►Often the process of re-expansion can be repeated any number of times. … ►For example, using double precision

d_{20}

is found to agree with (2.11.31) to 13D. …

23: Peter L. Walker

… ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. … ►

20 Theta Functions

►

22 Jacobian Elliptic Functions

►

23 Weierstrass Elliptic and Modular Functions

…

24: 7.8 Inequalities

§7.8 Inequalities

… ►

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

ⓘ

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

… ►

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

ⓘ

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

. For these and similar results for Dawson’s integral

F\left(x\right)

see Janssen (2021). …

25: Bibliography

… ►

M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

ⓘ

External Links:: Document, MathReview (Mark Adler)
Referenced by:: §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
Encodings:: BibTeX

… ►

A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

ⓘ

External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

… ►

S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

ⓘ

Notes:: Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.
External Links:: Code, Document
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

Z. Altaç (1996) Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.

ⓘ

External Links:: Document
Referenced by:: §10.73(iv), §8.24(iii).
Encodings:: BibTeX

… ►

D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

…

26: 10.75 Tables

… ►

•

Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

… ►

•

Zhang and Jin (1996, p. 270) tabulates $\int_{0}^{x}J_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{0}^{x}t^{-1}(1-J_{0}\left(t\right))\,\mathrm{d}t$ , $\int_{0}^{x}Y_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{x}^{\infty}t^{-1}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)1(.5)20$ , 8D.

… ►

•

Bickley et al. (1952) tabulates $x^{-n}I_{n}\left(x\right)$ or $e^{-x}I_{n}\left(x\right)$ , $x^{n}K_{n}\left(x\right)$ or $e^{x}K_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ (.01 or .1) 10(.1) 20, 8S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)20$ , 10S.

… ►

•

Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_{n}\left(z\right)$ and $K_{n}'\left(z\right)$ , for $n=2(1)20$ , 9S.

… ►

•

Zhang and Jin (1996, p. 271) tabulates $e^{-x}\int_{0}^{x}I_{0}\left(t\right)\,\mathrm{d}t$ , $e^{-x}\int_{0}^{x}t^{-1}(I_{0}\left(t\right)-1)\,\mathrm{d}t$ , $e^{x}\int_{x}^{\infty}K_{0}\left(t\right)\,\mathrm{d}t$ , $xe^{x}\int_{x}^{\infty}t^{-1}K_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)1(.5)20$ , 8D.

…

27: 22.3 Graphics

… ►

§22.3(i) Real Variables: Line Graphs

… ►

§22.3(iii) Complex $z$ ; Real $k$

►In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. … ►

§22.3(iv) Complex $k$

… ►In Figures 22.3.24 and 22.3.25, height corresponds to the absolute value of the function and color to the phase. …

28: 7.25 Software

… ►

§7.25(ii) $\operatorname{erf}x$ , $\operatorname{erfc}x$ , $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(x\right)$ , $x\in\mathbb{R}$

… ►

§7.25(iv) $C\left(x\right)$ , $S\left(x\right)$ , $\mathrm{f}\left(x\right)$ , $\mathrm{g}\left(x\right)$ , $x\in\mathbb{R}$

… ►

§7.25(v) $C\left(z\right)$ , $S\left(z\right)$ , $z\in\mathbb{C}$

… ►

§7.25(vi) $\mathcal{F}\left(x\right)$ , $G\left(x\right)$ , $\mathsf{U}\left(x,t\right)$ , $\mathsf{V}\left(x,t\right)$ , $x\in\mathbb{R}$

… ►

§7.25(vii) $\mathcal{F}\left(z\right)$ , $G\left(z\right)$ , $z\in\mathbb{C}$

…

29: 19.36 Methods of Computation

… ►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). … ►Lee (1990) compares the use of theta functions for computation of

K\left(k\right)

E\left(k\right)

, and

K\left(k\right)-E\left(k\right)

0\leq k^{2}\leq 1

, with four other methods. …For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … ►Similarly, §19.26(ii) eases the computation of functions such as

R_{F}\left(x,y,z\right)

when

x

(

>0

) is small compared with

\min(y,z)

. … ►

30: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

…