classification%20of%20parameters
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1: 20 Theta Functions
Chapter 20 Theta Functions
…2: 31.14 General Fuchsian Equation
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►The three sets of parameters comprise the singularity parameters
, the exponent parameters
, and the free accessory parameters
.
With and the total number of free parameters is .
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31.14.1
.
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31.14.3
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3: 16.21 Differential Equation
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16.21.1
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►With the classification of §16.8(i), when the only singularities of (16.21.1) are a regular singularity at and an irregular singularity at .
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16.21.2
.
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4: 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.
5: 31.12 Confluent Forms of Heun’s Equation
6: 8 Incomplete Gamma and Related
Functions
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7: 28 Mathieu Functions and Hill’s Equation
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8: 16.8 Differential Equations
9: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
Abramowitz 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.
10: 23 Weierstrass Elliptic and Modular
Functions
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