Abel%E2%80%93Plana%20formula
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1: 1.15 Summability Methods
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Abel Summability
… ►Abel Means
… ► is a harmonic function in polar coordinates (1.9.27), and … ►Here is the Abel (or Poisson) sum of , and has the series representation … ►Abel Summability
…2: 22.18 Mathematical Applications
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§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem
… ►For any two points and on this curve, their sum , 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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(c)
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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.
4: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions.
Cambridge University Press, Cambridge.
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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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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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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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5: 1.2 Elementary Algebra
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§1.2(iv) Means
… ►where is defined by the Leibniz formula … ►Formula (1.2.77) is more generally valid for all square matrices , not necessarily non-defective, see Hall (2015, Thm 2.12).6: 1.13 Differential Equations
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►Then the following relation is known as Abel’s identity
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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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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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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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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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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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8: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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9: Bibliography
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Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions and , ,
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ACM Trans. Math. Software 3 (1), pp. 93–95.
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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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Special Functions.
Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge.
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Bernoulli’s power-sum formulas revisited.
Math. Gaz. 90 (518), pp. 276–279.
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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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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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A simple sum formula for Clebsch-Gordan coefficients.
Lett. Math. Phys. 5 (3), pp. 207–211.
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Theta functions and non-linear equations.
Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).
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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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Uniform asymptotic expansions for Charlier polynomials.
J. Approx. Theory 112 (1), pp. 93–133.
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