Stieltjes constants
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10 matching pages
1: 25.2 Definition and Expansions
2: 25.6 Integer Arguments
3: Errata
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Subsection 25.2(ii) Other Infinite Series
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4: 31.15 Stieltjes Polynomials
§31.15 Stieltjes Polynomials
… ►§31.15(ii) Zeros
… ►This is the Stieltjes electrostatic interpretation. … ►§31.15(iii) Products of Stieltjes Polynomials
…5: 1.14 Integral Transforms
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§1.14(vi) Stieltjes Transform
►The Stieltjes transform of a real-valued function is defined by … … ►Inversion
… ►Laplace Transform
…6: 9.10 Integrals
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9.10.15
,
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9.10.16
.
►For the confluent hypergeometric function and the incomplete gamma function see §§13.1, 13.2, and 8.2(i).
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9.10.17
.
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§9.10(vii) Stieltjes Transforms
…7: 18.1 Notation
8: 2.6 Distributional Methods
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§2.6(ii) Stieltjes Transform
… ►The Stieltjes transform of is defined by … ►For a more detailed discussion of the derivation of asymptotic expansions of Stieltjes transforms by the distribution method, see McClure and Wong (1978) and Wong (1989, Chapter 6). Corresponding results for the generalized Stieltjes transform …An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). …9: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►Let
or
or
or
be a (possibly infinite, or semi-infinite)
interval in
.
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
,
►
1.18.11
…
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1.18.13
,
…
►Note that the integral in (1.18.67) 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
. For this latter see Simon (1973), and Reinhardt (1982); wherein advantage is taken of the fact that although branch points are actual singularities of an analytic function, the location of the branch cuts are often at our disposal, as they are not singularities of the function, but simply arbitrary lines to keep a function single valued, and thus only singularities of a specific representation of that analytic function. This is accomplished by the variable change
, in
, which rotates the continuous spectrum
and the branch cut of (1.18.67) into the lower half complex plain by the angle
, with respect to the unmoved branch point at
; thus, providing access to resonances on the higher Riemann sheet should
be large enough to expose them. This dilatation transformation, which does require analyticity of
, or an analytic approximation thereto, leaves the poles, corresponding to the discrete spectrum, invariant, as they are, as is the branch point, actual singularities of
.
…
►Let
be a symmetric operator on a Hilbert space
, so
is dense in
and
.
For
let
be the
-eigenspace of
, i.e.,
is the linear subspace of
consisting of all
for which
. Then
is constant
for
and also constant for
. Put
(
) and
(
),
the deficiency indices for
. Then
has self-adjoint extensions iff
.
…
10: Bibliography C
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Calcolo delle funzioni speciali , , , , alle alte precisioni.
Atti Accad. Sci. Lett. Arti Palermo Ser. (5) 2(1981/82) (1), pp. 7–25 (Italian).
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On Stieltjes’ continued fraction for the gamma function.
Math. Comp. 34 (150), pp. 547–551.
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rays from an extranuclear direct capture process.
Nuclear Physics 24 (1), pp. 89–101.
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Über die Fälle, wenn die Reihe von der Form etc. ein Quadrat von der Form etc. hat.
J. Reine Angew. Math. 3, pp. 89–91.
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