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21: 2.8 Differential Equations with a Parameter
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►in which is a real or complex parameter, and asymptotic solutions are needed for large that are uniform with respect to in a point set
in or .
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►Solutions are Bessel functions, or modified Bessel functions, of order (§§10.2, 10.25).
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►In both cases uniform asymptotic approximations are obtained in terms of Bessel functions of order .
More generally, can have a simple or double pole at .
(In the case of the double pole the order of the approximating Bessel functions is fixed but no longer .)
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22: 19.23 Integral Representations
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19.23.1
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19.23.2
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19.23.3
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19.23.10
; ;
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►For generalizations of (19.23.6_5) and (19.23.8) see Carlson (1964, (6.2), (6.12), and (6.1)).
23: 29.2 Differential Equations
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►This equation has regular singularities at the points , where , and , are the complete elliptic integrals of the first kind with moduli , , respectively; see §19.2(ii).
In general, at each singularity each solution of (29.2.1) has a branch point (§2.7(i)).
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29.2.8
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24: 2.3 Integrals of a Real Variable
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►For an extension with more general
-powers see Bleistein and Handelsman (1975, §4.1).
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►Another extension is to more general factors than the exponential function.
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►Without loss of generality, we assume that this minimum is at the left endpoint .
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►In general
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►For the more general integral (2.3.19) we assume, without loss of generality, that the stationary point (if any) is at the left endpoint.
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25: 32.2 Differential Equations
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►If in , then set
and , without loss of generality, by rescaling and if necessary.
If and in , then set
and , without loss of generality.
Lastly, if and , then set
and , without loss of generality.
►If in , then set
, without loss of generality.
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26: 8.19 Generalized Exponential Integral
27: 31.2 Differential Equations
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►where and with are generators of the lattice for .
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28: 19.31 Probability Distributions
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►More generally, let () and () be real positive-definite matrices with rows and columns, and let be the eigenvalues of .
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19.31.2
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►§19.16(iii) shows that for the incomplete cases of and occur when and , respectively, while their complete cases occur when .
►For (19.31.2) and generalizations see Carlson (1972b).
29: 8.1 Special Notation
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►Unless otherwise indicated, primes denote derivatives with respect to the argument.
►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).
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