measure
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11—20 of 30 matching pages
11: 35.1 Special Notation
12: 20.15 Tables
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►Tables of Neville’s theta functions , , , (see §20.1) and their logarithmic -derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for , where (in radian measure) , and is defined by (20.15.1).
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13: Publications
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►
R. F. Boisvert and D. W. Lozier (2001)
Handbook of Mathematical Functions,
in A Century of Excellence in Measurements Standards and Technology
(D. R. Lide, ed.), CRC Press, pp. 135–139.
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14: 18.33 Polynomials Orthogonal on the Unit Circle
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►After a quadratic transformation (18.2.23) this would express OP’s on with an even orthogonality measure in terms of the .
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►Let be a probability measure on the unit circle of which the support is an infinite set.
A system of monic polynomials , , where is of proper degree , is orthogonal on the unit circle with respect
to the measure
if
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►If the measure
is absolutely continuous, i.
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►For as in (18.33.19) (or more generally as the weight function of the absolutely continuous part of the measure
in (18.33.17)) and with the Verblunsky coefficients in (18.33.23), (18.33.24), Szegő’s theorem states that
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15: 22.18 Mathematical Applications
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►The arc length in the first quadrant, measured from , is
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►The arc length , measured from , is
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16: 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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, see §1.4(v), where the nonnegative weight function is Lebesgue measurable on .
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►
1.18.13
,
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►For fixed angular momentum the appropriate self-adjoint extension of the above operator may have both a discrete spectrum of negative eigenvalues , with corresponding eigenfunctions , and also a continuous spectrum , with Dirac-delta normalized eigenfunctions , also with measure
.
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17: 18.39 Applications in the Physical Sciences
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►Orthogonality and normalization of eigenfunctions of this form is respect to the measure
.
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►where the orthogonality measure is now ,
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►Orthogonality, with measure
for , for fixed
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►thus recapitulating, for , line 11 of Table 18.8.1, now shown with explicit normalization for the measure
.
…normalized with measure
, .
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18: DLMF Project News
error generating summary19: 36.7 Zeros
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►, ), the number of rings in the th row, measured from the origin and before the transition to hairpins, is given by
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