Neumann%20integral

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11: 10.23 Sums

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Neumann’s Addition Theorem

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Neumann’s Expansion

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10.23.11 $$a_k=\frac{1}{2\pi \mathrm{i}}{\int }_{|t|=c^{\prime }}f(t)O_k\left(t\right)dt,$$ ,

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Defines:

$a_k$ (locally)

Symbols:

$O_n\left(x\right)$: Neumann’s polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $k$: nonnegative integer, $z$: complex variable and $f(t)$

A&S Ref:

9.1.83

Referenced by:

Erratum (V1.1.2) for Equation (10.23.11)

Permalink:

http://dlmf.nist.gov/10.23.E11

Encodings:

pMML, png, TeX

Errata (effective with 1.1.2):

Originally the contour of integration written incorrectly as $|z|=c^{\prime }$, has been corrected to be $|t|=c^{\prime }$.

Reported 2021-03-22 by Mark Dunster

See also:

Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

►and $O_k\left(t\right)$ is Neumann’s polynomial, defined by the generating function: … ► $O_n\left(t\right)$ is a polynomial of degree $n+1$ in $1/t:O_0\left(t\right)=1/t$ and …

12: 14.12 Integral Representations

§14.12 Integral Representations

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§14.12(ii)

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Neumann’s Integral

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Heine’s Integral

… ►For further integral representations see Erdélyi et al. (1953a, pp. 158–159) and Magnus et al. (1966, pp. 184–190), and for contour integrals and other representations see §14.25.

13: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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§8.26(iv) Generalized Exponential Integral

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

14: 6.19 Tables

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§6.19(ii) Real Variables

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Abramowitz and Stegun (1964, Chapter 5) includes $x^{-1}\mathrm{Si}\left(x\right)$, $-x^{-2}\mathrm{Cin}\left(x\right)$, $x^{-1}\mathrm{Ein}\left(x\right)$, $-x^{-1}\mathrm{Ein}\left(-x\right)$, $x=0(.01)0.5$; $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $\mathrm{Ei}\left(x\right)$, $E_1\left(x\right)$, $x=0.5(.01)2$; $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $x{\mathrm{e}}^{-x}\mathrm{Ei}\left(x\right)$, $x{\mathrm{e}}^x{E}_1\left(x\right)$, $x=2(.1)10$; $x\mathrm{f}\left(x\right)$, $x^2\mathrm{g}\left(x\right)$, $x{\mathrm{e}}^{-x}\mathrm{Ei}\left(x\right)$, $x{\mathrm{e}}^x{E}_1\left(x\right)$, $x^{-1}=0(.005)0.1$; $\mathrm{Si}\left(\pi x\right)$, $\mathrm{Cin}\left(\pi x\right)$, $x=0(.1)10$. Accuracy varies but is within the range 8S–11S.

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Zhang and Jin (1996, pp. 652, 689) includes $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $x=0(.5)20(2)30$, 8D; $\mathrm{Ei}\left(x\right)$, $E_1\left(x\right)$, $x=[0,100]$, 8S.

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Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $z{\mathrm{e}}^z{E}_1\left(z\right)$, $x=-19(1)20$, $y=0(1)20$, 6D; ${\mathrm{e}}^z{E}_1\left(z\right)$, $x=-4(.5)-2$, $y=0(.2)1$, 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.

15: 36 Integrals with Coalescing Saddles

Chapter 36 Integrals with Coalescing Saddles

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16: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

Encodings:

BibTeX

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.

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External Links:

ISSN 0285-4821, MathReview (Peter M. Neumann), ZentralBlatt

Referenced by:

§24.10(ii).

Encodings:

BibTeX

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

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Notes:

Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.

External Links:

ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt

Referenced by:

§3.3(vi), §3.4(i), §3.5(i).

Encodings:

BibTeX

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17: 6.20 Approximations

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Hastings (1955) gives several minimax polynomial and rational approximations for $E_1\left(x\right)+\ln x$, $x{\mathrm{e}}^x{E}_1\left(x\right)$, and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$. These are included in Abramowitz and Stegun (1964, Ch. 5).

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Cody and Thacher (1968) provides minimax rational approximations for $E_1\left(x\right)$, with accuracies up to 20S.

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Cody and Thacher (1969) provides minimax rational approximations for $\mathrm{Ei}\left(x\right)$, with accuracies up to 20S.

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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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Luke and Wimp (1963) covers $\mathrm{Ei}\left(x\right)$ for $x\le -4$ (20D), and $\mathrm{Si}\left(x\right)$ and $\mathrm{Ci}\left(x\right)$ for $x\ge 4$ (20D).

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18: 10.9 Integral Representations

§10.9 Integral Representations

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Neumann’s Integral

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§10.9(ii) Contour Integrals

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Hankel’s Integrals

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19: 20 Theta Functions

Chapter 20 Theta Functions

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20: 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

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