classification%20of%20singularities
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1: 20 Theta Functions
Chapter 20 Theta Functions
…2: 31.14 General Fuchsian Equation
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►The general second-order Fuchsian equation with regular singularities at , , and at , is given by
…The exponents at the finite singularities
are and those at are , where
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►The three sets of parameters comprise the singularity parameters
, the exponent parameters
, and the free accessory parameters
.
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31.14.3
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3: 16.21 Differential Equation
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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 .
When the only singularities of (16.21.1) are regular singularities at , , and .
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4: 31.12 Confluent Forms of Heun’s Equation
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►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity.
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►This has regular singularities at and , and an irregular singularity of rank 1 at .
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►This has irregular singularities at and , each of rank .
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►This has a regular singularity at , and an irregular singularity at of rank .
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►This has one singularity, an irregular singularity of rank at .
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5: 16.8 Differential Equations
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§16.8(i) Classification of Singularities
… ►All other singularities are irregular. … … ►In each case there are no other singularities. … ►§16.8(iii) Confluence of Singularities
…6: 31.2 Differential Equations
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31.2.1
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►This equation has regular singularities at , with corresponding exponents , , , , respectively (§2.7(i)).
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.
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7: 8 Incomplete Gamma and Related
Functions
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8: 28 Mathieu Functions and Hill’s Equation
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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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