Stieltjes%20constants
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21: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►For a Lebesgue–Stieltjes measure on let be the space of all Lebesgue–Stieltjes measurable complex-valued functions on which are square integrable with respect to ,
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1.18.11
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1.18.13
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►Note that the integral in (1.18.66) is not singular if approached separately from above, or below, the real axis: in fact analytic continuation from the upper half of the complex plane, across the cut, and onto higher Riemann Sheets can access complex poles with singularities at discrete energies corresponding to quantum resonances, or decaying quantum states with lifetimes proportional to .
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►Then is constant for and also constant for .
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22: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Stieltjes transforms and the Stokes phenomenon.
Proc. Roy. Soc. London Ser. A 429, pp. 227–246.
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23: 18.1 Notation
24: Bibliography M
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Rational approximations, software and test methods for sine and cosine integrals.
Numer. Algorithms 12 (3-4), pp. 259–272.
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Explicit error terms for asymptotic expansions of Stieltjes transforms.
J. Inst. Math. Appl. 22 (2), pp. 129–145.
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Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems.
Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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25: 5.22 Tables
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►Abramowitz and Stegun (1964, Chapter 6) tabulates , , , and for to 10D; and for to 10D; , , , , , , , and for to 8–11S; for to 20S.
Zhang and Jin (1996, pp. 67–69 and 72) tabulates , , , , , , , and for to 8D or 8S; for to 51S.
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►Abramov (1960) tabulates for () , () to 6D.
Abramowitz and Stegun (1964, Chapter 6) tabulates for () , () to 12D.
…Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of , , and for , to 8S.
26: Errata
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►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions.
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Subsection 25.2(ii) Other Infinite Series
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Section 1.14
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Chapters 8, 20, 36
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References
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There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.
Transform | New | Abbreviated | Old |
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Notation | Notation | Notation | |
Fourier | |||
Fourier Cosine | |||
Fourier Sine | |||
Laplace | |||
Mellin | |||
Hilbert | |||
Stieltjes |
Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.
27: 3.10 Continued Fractions
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Stieltjes Fractions
… ►is called a Stieltjes fraction (-fraction). … ►For the same function , the convergent of the Jacobi fraction (3.10.11) equals the convergent of the Stieltjes fraction (3.10.6). …28: 1.1 Special Notation
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real variables. | |
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the space of all Lebesgue–Stieltjes measurable functions on which are square integrable with respect to . | |
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29: 25.12 Polylogarithms
30: Bibliography H
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Formulae for growth factors in expanding universes containing matter and a cosmological constant.
Monthly Notices Roy. Astronom. Soc. 322 (2), pp. 419–425.
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Note on Dr. Vacca’s series for
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Quart. J. Math. 43, pp. 215–216.
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Definitions of Stieltjes Integrals of the Riemann Type.
Amer. Math. Monthly 45 (5), pp. 265–278.
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