scholarly journals An elementary proof for \sin x = x − \frac{x}{6} + o(x^3)

2016 ◽  
Vol 25 (2) ◽  
pp. 175-176
Author(s):  
RADU GOLOGAN ◽  
◽  
Keyword(s):  
Power Series ◽  
Elementary Proof ◽  
Series Development

Using only elementary trigonometrical calculations we prove the power series development for the sin and cos functions up to the terms of power three and four respectively.

Eigenvalues of the Barenblatt-Pattle similarity solution in nonlinear diffusion

1982 ◽  
Vol 383 (1784) ◽  
pp. 89-100 ◽  
Keyword(s):  
Power Series ◽  
Initial Data ◽  
Similarity Solution ◽  
Nonlinear Diffusion ◽  
Large Time ◽  
Infinite Sequence ◽  
Nonlinear Diffusion Equation ◽  
Time Behaviour ◽  
Series Development ◽  
Interactive Product

We consider the large-time behaviour of the nonlinear diffusion equation ∂ u /∂ t = r 1- μ ∂/∂ r ( r μ -1 u β ∂ u /∂ r ), u ≽ 0, β ≻ 0 for certain types of compact initial data. We show that the solution approaches the Barenblatt-Pattle similarity solution through an infinite sequence of negative real powers of t , which can be found in explicit form. These, together with their interactive product terms, determine the power-series development of u(r,t) as t → ∞.

Remarks on a paper of Olver

1991 ◽  
Vol 119 (3-4) ◽  
pp. 213-217 ◽  
Author(s):  
D. B. Fairlie
Keyword(s):  
Power Series ◽  
Topological Field Theories ◽  
Field Theories ◽  
Parallel Development ◽  
Associative Product ◽  
Topological Field ◽  
Series Development

SynopsisSome disparate ideas in the literature are drawn together. The work of P. J. Olver and his associates on Lagrangians which vanish for arbitrary variations, the so-called null Lagrangians, is viewed as a parallel development to Witten's study of topological field theories. A theorem of Olver, that all hyperjacobians are expressible as divergences, and are thus candidates for the construction of null Lagrangians, is shown to follow directly from the observation that such entities appear in a power series development of the general associative product, and this technique facilitates the construction of multi-dimensional examples.

Logarithmic growth and Frobenius filtrations for solutions of p-adic differential equations

2009 ◽  
Vol 8 (3) ◽  
pp. 465-505 ◽  
Author(s):  
Bruno Chiarellotto ◽  
Nobuo Tsuzuki
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Local Field ◽  
Differential Module ◽  
Frobenius Structures ◽  
Bounded Functions ◽  
Series Development ◽  
Rank 2 ◽  
Logarithmic Growth

AbstractFor a ∇-module M over the ring K[[x]]0 of bounded functions over a p-adic local field K we define the notion of special and generic log-growth filtrations on the base of the power series development of the solutions and horizontal sections. Moreover, if M also admits a Frobenius structure then it is endowed with generic and special Frobenius slope filtrations. We will show that in the case of M a ϕ–∇-module of rank 2, the Frobenius polygon for M and the log-growth polygon for its dual, Mv, coincide, this is proved by showing explicit relationships between the filtrations. This will lead us to formulate some conjectural links between the behaviours of the filtrations arising from the log-growth and Frobenius structures of the differential module. This coincidence between the two polygons was only known for the hypergeometric cases by Dwork.

scholarly journals Explicit Construction of the Inverse of an Analytic Real Function: Some Applications

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2154
Author(s):  
Joaquín Moreno ◽  
Miguel A. López ◽  
Raquel Martínez
Keyword(s):  
Power Series ◽  
Nonlinear Equations ◽  
Taylor Series ◽  
Real Function ◽  
General Procedure ◽  
Explicit Construction ◽  
Catalan Numbers ◽  
Analytical Function ◽  
Series Development

In this paper, we introduce a general procedure to construct the Taylor series development of the inverse of an analytical function; in other words, given y=f(x), we provide the power series that defines its inverse x=hf(y). We apply the obtained results to solve nonlinear equations in an analytic way, and generalize Catalan and Fuss–Catalan numbers.

A Power Series Development of the Convolution Theorem

1960 ◽  
Vol 67 (9) ◽  
pp. 893
Author(s):  
Louis C. Barrett ◽  
Carroll Wilde
Keyword(s):  
Power Series ◽  
Convolution Theorem ◽  
Series Development

AN ELEMENTARY PROOF OF A THEOREM OF LEE

1991 ◽  
Vol 11 (3) ◽  
pp. 356-360 ◽  
Author(s):  
Jia'an Yan
Keyword(s):  
Elementary Proof
Sign in / Sign up

Export Citation Format

Share Document