lemniscate%20constants
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11: 32.8 Rational Solutions
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►with , , and arbitrary constants.
►In the general case assume , so that as in §32.2(ii) we may set and .
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►with and arbitrary constants.
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(c)
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►with and arbitrary constants.
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, , and , with even.
12: 25.20 Approximations
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Cody et al. (1971) gives rational approximations for in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are , , , . Precision is varied, with a maximum of 20S.
13: 30.9 Asymptotic Approximations and Expansions
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§30.9(i) Prolate Spheroidal Wave Functions
►As , with , … ►The asymptotic behavior of and as in descending powers of is derived in Meixner (1944). …The asymptotic behavior of and as is given in Erdélyi et al. (1955, p. 151). The behavior of for complex and large is investigated in Hunter and Guerrieri (1982). …14: 8 Incomplete Gamma and Related
Functions
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15: 28 Mathieu Functions and Hill’s Equation
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16: 23 Weierstrass Elliptic and Modular
Functions
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17: 36.5 Stokes Sets
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►where denotes a real critical point (36.4.1) or (36.4.2), and denotes a critical point with complex or , connected with by a steepest-descent path (that is, a path where ) in complex or space.
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36.5.4
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36.5.7
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18: 11.6 Asymptotic Expansions
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11.6.1
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►where is an arbitrary small positive constant.
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11.6.2
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►where is Euler’s constant (§5.2(ii)).
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19: 5.11 Asymptotic Expansions
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►The scaled gamma function is defined in (5.11.3) and its main property is as in the sector .
Wrench (1968) gives exact values of up to .
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►In this subsection , , and are real or complex constants.
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5.11.12
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5.11.19
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