Abel%E2%80%93Plana%20formula

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1: 1.15 Summability Methods

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Abel Summability

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Abel Means

… ► $A(r,\theta )$ is a harmonic function in polar coordinates (1.9.27), and … ►Here $u(x,y)=A(r,\theta )$ is the Abel (or Poisson) sum of $f(\theta )$, and $v(x,y)$ has the series representation … ►

Abel Summability

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2: 22.18 Mathematical Applications

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§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

… ►For any two points $(x_1,y_1)$ and $(x_2,y_2)$ on this curve, their sum $(x_3,y_3)$, always a third point on the curve, is defined by the Jacobi–Abel addition law …a construction due to Abel; see Whittaker and Watson (1927, pp. 442, 496–497). …

3: 2.10 Sums and Sequences

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§2.10(i) Euler–Maclaurin Formula

… ►This is the Euler–Maclaurin formula. Another version is the Abel–Plana formula: … ►

The first infinite integral in (2.10.2) converges.

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4: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

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Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.

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Notes:

Reprint of the 1897 original, with a foreword by Igor Krichever.

External Links:

ISBN 0-521-49877-5, MathReview (John B. Little), ZentralBlatt

Referenced by:

§20.9(ii).

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Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.

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ISSN 0080-4614, FullText, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

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Document, Code

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1st item, 4th item.

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Methods of calculation of radial wave functions and new tables of Coulomb functions. Physical Rev. (2) 80, pp. 553–560.

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Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§33.7.

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5: 1.2 Elementary Algebra

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§1.2(iv) Means

… ►where $det(𝐀)$ is defined by the Leibniz formula … ►Formula (1.2.77) is more generally valid for all square matrices $𝐀$, not necessarily non-defective, see Hall (2015, Thm 2.12).

6: 1.13 Differential Equations

… ►Then the following relation is known as Abel’s identity …

7: Bibliography H

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The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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MathReview (L. Berg)

Referenced by:

§7.14(iii).

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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

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Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

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Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:

MathReview (M. Lawrence Glasser)

Referenced by:

§7.14(iii).

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Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.

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ZentralBlatt

Referenced by:

§7.18(iii), §7.18(iv), §7.18(v), §7.18(vi).

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8: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►

Table 27.2.1: Primes.

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$n$	$p_n$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
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Table 27.2.2: Functions related to division.

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$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
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5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
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7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
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11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
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9: Bibliography

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Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions $I_{\nu }(x)$ and $J_{\nu }(x)$, $x\ge 0$, $\nu \ge 0$ . ACM Trans. Math. Software 3 (1), pp. 93–95.

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Notes:

Single-precision Fortran, maximum accuracy 16S.

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ISSN 0098-3500, Document, Companion paper. Erratum. GAMS (module 511; package TOMS)

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1st item, 1st item.

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Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.

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Notes:

Double- and single-precision Fortran, maximum accuracy 18S or 14S.

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ISSN 0098-3500, Document, GAMS (module 683; package TOMS), MathReview, ZentralBlatt

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1st item, 1st item.

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Special Functions. Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-62321-9, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§1.10(vii), §1.15(iii), §1.15(vi), §1.15(vii), §10.22(vi), §13.6(v), §13.9(i), Ch.13, §14.3(iii), §14.30(iv), §15.10(i), §15.11(i), §15.11(ii), §15.14, §15.17(iv), §15.2(i), §15.4(ii), §15.5(i), §15.5(ii), §15.6, §15.7, §15.8(iv), §15.8(iii), Ch.15, §16.4(ii), §16.4(iii), §16.4(v), §16.4(v), Ch.16, §17.12, Ch.17, §18.10(ii), §18.10(iv), §18.11(i), §18.12, (18.17.41_5), §18.17(iv), §18.17(viii), §18.18(iv), §18.18(iv), §18.18(v), §18.18(vii), §18.2(i), §18.2(iv), §18.2(vi), §18.20(ii), §18.22(i), §18.25(ii), §18.26(i), Table 18.3.1, §18.38(ii), §18.5(iii), (18.7.25), Ch.18, §26.19, §4.10(i), §4.10(ii), §5.13, §5.14, §5.16, §5.16, §5.18(i), §5.18(ii), §5.18(iii), §5.20, §5.4(ii), §5.5(iv), Ch.5, Ch.6, Profile Richard A. Askey, Profile Ranjan Roy, Erratum (V1.0.28) for Table 18.3.1.

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Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

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Referenced by:

§24.4(iii).

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Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.

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ISSN 0022-247X, Document, Link, MathReview (Vivek Sahai), ZentralBlatt

Referenced by:

§18.15(vi), §18.15(vi).

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10: Bibliography D

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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.

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ISSN 0022-314X, Document, MathReview (T. Metsänkylä)

Referenced by:

§24.12(iii).

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A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.

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ISSN 0377-9017, Document, MathReview, ZentralBlatt

Referenced by:

§14.17(iv).

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Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).

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Notes:

With an appendix by I. M. Krichever. In Russian. English translation: Russian Math. Surveys 36(1981), no. 2, pp. 11–92 (1982)

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ISSN 0042-1316, Document, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§21.1, §21.6(ii), §21.7(i), Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2, Ch.21, Notations T.

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Error analysis in a uniform asymptotic expansion for the generalised exponential integral. J. Comput. Appl. Math. 80 (1), pp. 127–161.

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ISSN 0377-0427, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§2.11(iii), §8.20(ii).

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Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.

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ISSN 0021-9045, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.24.

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