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21: 31.6 Path-Multiplicative Solutions
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►This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the -plane that encircles and once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor .
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22: 14.8 Behavior at Singularities
23: 31.4 Solutions Analytic at Two Singularities: Heun Functions
§31.4 Solutions Analytic at Two Singularities: Heun Functions
…24: 36.6 Scaling Relations
25: 29.2 Differential Equations
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►This equation has regular singularities at the points , where , and , are the complete elliptic integrals of the first kind with moduli , , respectively; see §19.2(ii).
In general, at each singularity each solution of (29.2.1) has a branch point (§2.7(i)).
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26: 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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27: 10.25 Definitions
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►This equation is obtained from Bessel’s equation (10.2.1) on replacing by , and it has the same kinds of singularities.
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28: 10.2 Definitions
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§10.2(i) Bessel’s Equation
… ►This differential equation has a regular singularity at with indices , and an irregular singularity at of rank ; compare §§2.7(i) and 2.7(ii). …29: 31.3 Basic Solutions
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►In general, one of them has a logarithmic singularity at .
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§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities
… ►Solutions (31.3.1) and (31.3.5)–(31.3.11) comprise a set of 8 local solutions of (31.2.1): 2 per singular point. …30: 32.2 Differential Equations
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►be a nonlinear second-order differential equation in which is a rational function of and , and is locally analytic in , that is, analytic except for isolated singularities in .
In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions.
An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however.
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