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11: 8.12 Uniform Asymptotic Expansions for Large Parameter
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8.12.5
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►For error bounds for (8.12.7) see Paris (2002a).
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8.12.18
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►Higher coefficients , , up to , are given in Paris (2002b).
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►For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function see Paris (2002b) and Dunster (1996a).
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12: 10.59 Integrals
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10.59.1
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13: 4.35 Identities
14: 33.11 Asymptotic Expansions for Large
15: 14.23 Values on the Cut
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14.23.1
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14.23.2
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14.23.4
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14.23.6
►If cuts are introduced along the intervals and , then (14.23.4) and (14.23.6) could be used to extend the definitions of and to complex .
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16: 12.19 Tables
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Abramowitz and Stegun (1964, Chapter 19) includes and for , , 5S; for , , 4-5D or 4-5S.
Kireyeva and Karpov (1961) includes for , , and , , 7D.
Karpov and Čistova (1964) includes for , ; , , 6D.
Zhang and Jin (1996, pp. 455–473) includes , , , , and derivatives, , , , 8S; , , and derivatives, , and , , , 8S. Also, first zeros of , , and of derivatives, , 6D; first three zeros of and of derivative, , 6D; first three zeros of and of derivative, , 6D; real and imaginary parts of , , , , , 8S.
17: 10.27 Connection Formulas
18: 14.9 Connection Formulas
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§14.9(i) Connections Between , , ,
… ►§14.9(ii) Connections Between , ,
… ►§14.9(iii) Connections Between , , ,
…19: 15.19 Methods of Computation
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►For it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when .
This is because the linear transformations map the pair onto itself.
However, by appropriate choice of the constant in (15.15.1) we can obtain an infinite series that converges on a disk containing .
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►The representation (15.6.1) can be used to compute the hypergeometric function in the sector .
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►The relations in §15.5(ii) can be used to compute , provided that care is taken to apply these relations in a stable manner; see §3.6(ii).
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20: 4.23 Inverse Trigonometric Functions
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►In (4.23.1) and (4.23.2) the integration paths may not pass through either of the points .
…In (4.23.3) the integration path may not intersect .
… and have branch points at ; the other four functions have branch points at .
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►For the corresponding results for , , and , use (4.23.7)–(4.23.9).
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►where and in (4.23.34) and (4.23.35), and in (4.23.36).
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