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11: 31.4 Solutions Analytic at Two Singularities: Heun Functions
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►For an infinite set of discrete values , , of the accessory parameter
, the function is analytic at , and hence also throughout the disk .
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31.4.1
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►The eigenvalues satisfy the continued-fraction equation
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31.4.2
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31.4.3
,
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12: 20.16 Software
13: 31.2 Differential Equations
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►All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, , can be transformed into (31.2.1).
►The parameters play different roles: is the singularity parameter; are exponent parameters; is the accessory parameter.
The total number of free parameters is six.
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►Then (suppressing the parameter
)
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►Except for the identity automorphism, each alters the parameters.
14: 8.17 Incomplete Beta Functions
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►Throughout §§8.17 and 8.18 we assume that , , and .
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8.17.4
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►With , , and ,
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8.17.13
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8.17.16
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15: 9.14 Incomplete Airy Functions
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►Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter.
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16: 28.27 Addition Theorems
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►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems.
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17: 31.12 Confluent Forms of Heun’s Equation
18: 32.1 Special Notation
19: 34.14 Tables
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►Tables of exact values of the squares of the and symbols in which all parameters are are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of , and symbols on pp.
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►Tables of and symbols in which all parameters are are given in Appel (1968) to 6D.
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►In Varshalovich et al. (1988) algebraic expressions for the Clebsch–Gordan coefficients with all parameters
and numerical values for all parameters
are given on pp.
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