Isolated Singularities of Holomorphic Functions (Continued). Characterisation of an Isolated Singularity via the Laurent Series Expansion. Orders of Poles and Zeroes. Casorati-Weierstrass’ Theorem. Isolated Singularities of Holomorphic Functions at ∞ and their Characterisation via Laurent Series Expansions

2017 ◽  
pp. 117-125
Author(s):  
Alexander Isaev
Keyword(s):  
Series Expansion ◽  
Laurent Series ◽  
Holomorphic Functions ◽  
Series Expansions ◽  
Isolated Singularity ◽  
Weierstrass Theorem ◽  
Isolated Singularities

8.—On a Class of Series Expansions in the Theory of Emden's Equation

1973 ◽  
Vol 71 (2) ◽  
pp. 95-110 ◽  
Author(s):  
Einar Hille
Keyword(s):  
General Theory ◽  
Differential Equations ◽  
Series Expansion ◽  
Laurent Series ◽  
Second Order ◽  
Series Expansions ◽  
Section 8 ◽  
Second Order Differential Equations ◽  
Bona Fide ◽  
Image Position

SynopsisThis paper deals with the nature of movable singularities of solutions of Emden's equationat which the solution becomes infinite. If m = 1 + 2/p with p > 1 an integer, then the solution becomes infinite at a given point x = c asBy the general theory of P. Painlevé on movable poles of solutions of non-linear second order differential equations this ‘pseudo-pole’ cannot actually be a pole of order p. Instead of a bona fide Laurent series at x = c we obtain a series expansion of the formwhere Pn(t) is a polynomial in t of degree at most [n/(2p + 2)]. The object of this paper is to derive these series and to prove convergence for p = 2. In this case deg [P6m] is strictly equal to m. For other values of p, see Section 8, Addenda.

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