Rydberg constant
(0.001 seconds)
11—20 of 435 matching pages
11: 5.13 Integrals
12: 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.
13: 30.6 Functions of Complex Argument
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►The solutions
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►of (30.2.1) with and are real when , and their principal values (§4.2(i)) are obtained by analytic continuation to .
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►with as in (30.11.4).
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14: 32.2 Differential Equations
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►with , , , and arbitrary constants.
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►In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions.
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►For arbitrary values of the parameters , , , and , the general solutions of – are transcendental, that is, they cannot be expressed in closed-form elementary functions.
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►If in , then set and , without loss of generality, by rescaling and if necessary.
…Lastly, if and , then set and , without loss of generality.
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15: 8.8 Recurrence Relations and Derivatives
16: 5.8 Infinite Products
17: 15.11 Riemann’s Differential Equation
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►The most general form is given by
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►Here , , are the exponent pairs at the points , , , respectively.
…Also, if any of , , , is at infinity, then we take the corresponding limit in (15.11.1).
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►where , , , are real or complex constants such that .
These constants can be chosen to map any two sets of three distinct points and onto each other.
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18: 30.4 Functions of the First Kind
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►The eigenfunctions of (30.2.1) that correspond to the eigenvalues are denoted by , .
…the sign of being when is even, and the sign of being when is odd.
►When
is the prolate angular spheroidal wave function, and when
is the oblate angular spheroidal wave function.
If , reduces to the Ferrers function :
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has exactly zeros in the interval .
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19: 8.2 Definitions and Basic Properties
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►The general values of the incomplete gamma functions
and are defined by
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►In this subsection the functions and have their general values.
►The function is entire in and .
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►If or , then
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►If , then
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20: 8.1 Special Notation
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►The functions treated in this chapter are the incomplete gamma functions , , , , and ; the incomplete beta functions and ; the generalized exponential integral ; the generalized sine and cosine integrals , , , and .
►Alternative notations include: Prym’s functions
, , Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); , , Dingle (1973); , , Magnus et al. (1966); , , Luke (1975).
real variable. | |
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arbitrary small positive constant. | |
gamma function (§5.2(i)). | |
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