Euler constant
(0.012 seconds)
11—20 of 339 matching pages
11: 5.8 Infinite Products
12: 8.7 Series Expansions
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8.7.1
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8.7.2
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8.7.3
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8.7.4
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►For an expansion for in series of Bessel functions that converges rapidly when and () is small or moderate in magnitude see Barakat (1961).
13: 30.8 Expansions in Series of Ferrers Functions
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►Then the set of coefficients , is the solution of the difference equation
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30.8.7
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30.8.8
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►The coefficients satisfy (30.8.4) for all when we set for .
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30.8.10
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14: 5.3 Graphics
15: 16.13 Appell Functions
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16.13.1
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16.13.2
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16.13.3
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16.13.4
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►Here and elsewhere it is assumed that neither of the bottom parameters and is a nonpositive integer.
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16: 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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gamma function (§5.2(i)). | |
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17: 5.6 Inequalities
18: 8.3 Graphics
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►Some monotonicity properties of and in the four quadrants of the ()-plane in Figure 8.3.6 are given in Erdélyi et al. (1953b, §9.6).
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19: 8.14 Integrals
20: 30.16 Methods of Computation
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►For , , ,
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►If is large, then we can use the asymptotic expansions referred to in §30.9 to approximate .
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►If is known, then can be found by summing (30.8.1).
The coefficients are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5).
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►The coefficients calculated in §30.16(ii) can be used to compute , from (30.11.3) as well as the connection coefficients from (30.11.10) and (30.11.11).
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