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11: 15.10 Hypergeometric Differential Equation
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15.10.1
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Singularity
… βΊSingularity
… βΊSingularity
… βΊThe connection formulas for the principal branches of Kummer’s solutions are: …12: 30.2 Differential Equations
§30.2 Differential Equations
βΊ§30.2(i) Spheroidal Differential Equation
… βΊ … βΊThe Liouville normal form of equation (30.2.1) is … βΊIf , Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).13: 36.2 Catastrophes and Canonical Integrals
§36.2 Catastrophes and Canonical Integrals
… βΊCanonical Integrals
… βΊ is related to the Airy function (§9.2): … … βΊ§36.2(iii) Symmetries
…14: 28.2 Definitions and Basic Properties
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28.2.1
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βΊThis is the characteristic equation of Mathieu’s equation (28.2.1).
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§28.2(iv) Floquet Solutions
… βΊ§28.2(vi) Eigenfunctions
… βΊFor simple roots of the corresponding equations (28.2.21) and (28.2.22), the functions are made unique by the normalizations …15: 32.2 Differential Equations
§32.2 Differential Equations
… βΊThe six Painlevé equations – are as follows: … βΊThe fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of –. … βΊ§32.2(ii) Renormalizations
… βΊ …16: 28.20 Definitions and Basic Properties
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§28.20(i) Modified Mathieu’s Equation
βΊWhen is replaced by , (28.2.1) becomes the modified Mathieu’s equation: βΊ
28.20.1
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βΊThen from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to as in the respective sectors , being an arbitrary small positive constant.
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§28.20(iv) Radial Mathieu Functions ,
…17: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
§8.26(iv) Generalized Exponential Integral
βΊAbramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.
18: 36 Integrals with Coalescing Saddles
Chapter 36 Integrals with Coalescing Saddles
…19: 20 Theta Functions
Chapter 20 Theta Functions
…20: 6.19 Tables
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§6.19(ii) Real Variables
βΊAbramowitz and Stegun (1964, Chapter 5) includes , , , , ; , , , , ; , , , , ; , , , , ; , , . Accuracy varies but is within the range 8S–11S.
Zhang and Jin (1996, pp. 652, 689) includes , , , 8D; , , , 8S.
Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of , , , 6D; , , , 6D; , , , 6D.
Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of , , , 8S.