By Borwein J.M., Luke D.R.

We examine numerous generalizaions of the AGM endured fraction of Ramanujan encouraged by way of a chain of contemporary articles within which the validity of the AGM relation and the area of convergence of the ongoing fraction have been decided for definite complicated parameters [2, three, 4]. A examine of the AGM persevered fraction is reminiscent of an research of the convergence of convinced distinction equations and the soundness of dynamical structures. utilizing the matrix analytical instruments constructed in [4]. we make certain the convergence homes of deterministic, and stochastic distinction equations and so divergence in their corresponding endured fractions.

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The classical procedure for nding the nite part of divergent integrals appears ad hoc and leads to questions about the validity of the procedure. First, could the appearance of divergent integrals in applications be the result of errors in modeling the physics of the problem? Second, will the method lead to a unique analytical expression or do di erent analytical expressions lead to equivalent numerical results? The generalized function theory clearly answers these questions. Let us rst examine the function f (x) = lnjxj , which is locally integrable.

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The momentum equation gives @ @t (ui ) + @ 0  ui uj @xj 1 + Pij = Pij nj  (f ) (4:60) where Pij = E ij + (p 0 p0)ij is the compressive stress tensor and Eij is the viscous stress tensor. 60), subtract the latter from the former, and nally subtract c 2 @ =@ xi2 from both sides to get 2 0 p = @ 2 @ xi @ xj 2 3 Ti j h(f ) 2 0 @@x Pij nj  (f ) i 3 + @ @t [0 vn (f )] (4:61) where p0 = c2 ( 0 0). Here Ti j is the Lighthill stress tensor. Now we have added h(f ), the Heaviside function, on the right side to indicate that Tij 6= 0 outside the surface f = 0.

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