scholarly journals VII. A theory of asymptotic series

1912 ◽  
Vol 211 (471-483) ◽  
pp. 279-313 ◽  
Keyword(s):  
Power Series ◽  
Mathematical Analysis ◽  
Asymptotic Series ◽  
Convergent Series ◽  
Divergent Series ◽  
Consistent Theory

In their efforts to place mathematical analysis on the firm est possible foundations, Abel and Cauchy found it necessary to banish non-convergent series from their work ; from that time until a quarter of a century ago the theory o f divergent series was, in general, neglected by mathematicians. A consistent theory of divergent series was, however, developed by Poincaré in 1886, and, ten years later, Borel enunciated his theory of summability in connection with oscillating series. So far as diverging power series are concerned, the theory of Borel is more precise than that of Poincaré.

Summation of slowly convergent series

1950 ◽  
Vol 46 (3) ◽  
pp. 436-449 ◽  
Author(s):  
T. M. Cherry
Keyword(s):  
Infinite Series ◽  
Main Part ◽  
Asymptotic Series ◽  
Convergent Series ◽  
Divergent Series ◽  
Finite Form ◽  
Sum Formula ◽  
New Form ◽  
Practical Sense ◽  
Direct Summation

Let ∑un be a convergent infinite series which is not summable in finite form. In principle its sum can be found, to within any preassigned error ε, by adding numerically a sufficient number of terms; but if the series is slowly convergent, the ‘sufficient number’ of terms may be prohibitively large. A plan to deal with this case is to separate the series into a ‘main part’ u0+u1+ … +un−1 and a ‘remainder’ Rn = un+un+1+…; the main part is evaluated by direct summation, while the remainder is transformed analytically into a series which is more rapidly ‘convergent’, in the practical sense, and so evaluated. For example, the Euler-Maclaurin sum-formula gives such a transformation. It commonly happens that the new form of the remainder Rn is a divergent series, but that it represents Rn asymptotically as n ˜ ∞. It is for this reason that the transformation is applied to Rn instead of to the whole series; for practical use we have to choose n sufficiently large for the error inherent in the use of the asymptotic series to be below the preassigned bound ε.

Note on a Divergent Series

1941 ◽  
Vol 37 (1) ◽  
pp. 1-8 ◽  
Author(s):  
G. H. Hardy
Keyword(s):  
Power Series ◽  
Asymptotic Series ◽  
Radius Of Convergence ◽  
Divergent Series ◽  
Image Position

1. The seriesis the simplest and most familiar power series whose radius of convergence is zero. It is natural to regard it as a development of the function G(z) defined, whenbyForsay;andoraccording as x is positive or negative. Thus the series (1·1) is an asymptotic series for G(z) in the sense of Poincaré.

scholarly journals Solutions of first level of meromorphic differential equations

1982 ◽  
Vol 25 (2) ◽  
pp. 183-207 ◽  
Author(s):  
W. Balser
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Differential Equations ◽  
Formal Power Series ◽  
Convergent Series ◽  
Constant Matrix ◽  
Image Position ◽  
Formal Power ◽  
Meromorphic Differential Equation ◽  
Meromorphic Differential

Let a meromorphic differential equationbe given, where r is an integer, and the series converges for |z| sufficiently large. Then it is well known that (0.1) is formally satisfied by an expressionwhere F( z) is a formal power series in z–1 times an integer power of z, and F( z) has an inverse of the same kind, L is a constant matrix, andis a diagonal matrix of polynomials qj( z) in a root of z, 1≦ j≦ n. If, for example, all the polynomials in Q( z) are equal, then F( z) can be seen to be a convergent series (see Section 1), whereas if not, then generally the coefficients in F( z) grow so rapidly that F( z) diverges for every (finite) z.

A family of hyper-Bessel functions and convergent series in them

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Jordanka Paneva-Konovska
Keyword(s):  
Power Series ◽  
Bessel Function ◽  
Classical Theory ◽  
Bessel Functions ◽  
Differential Operators ◽  
Arbitrary Order ◽  
Convergent Series ◽  
Multi Index ◽  
Fatou Theorem ◽  
Convergence Of Series

AbstractThe Delerue hyper-Bessel functions that appeared as a multi-index generalizations of the Bessel function of the first type, are closely related to the hyper-Bessel differential operators of arbitrary order, introduced by Dimovski. In this work we consider an enumerable family of hyper-Bessel functions and study the convergence of series in such a kind of functions. The obtained results are analogues to the ones in the classical theory of the widely used power series, like Cauchy-Hadamard, Abel and Fatou theorem.

scholarly journals Solving Fuzzy Differential Equations by Using Power Series

2020 ◽  
pp. 92-107
Author(s):  
Rasha H. Ibraheem
Keyword(s):  
Approximate Solution ◽  
Power Series ◽  
Differential Equations ◽  
Taylor Expansion ◽  
Series Solution ◽  
Convergent Series ◽  
Initial Value ◽  
Third Order ◽  
Technical Accuracy ◽  
Made In

In this paper, the series solution is applied to solve third order fuzzy differential equations with a fuzzy initial value. The proposed method applies Taylor expansion in solving the system and the approximate solution of the problem which is calculated in the form of a rapid convergent series; some definitions and theorems are reviewed as a basis in solving fuzzy differential equations. An example is applied to illustrate the proposed technical accuracy. Also, a comparison between the obtained results is made, in addition to the application of the crisp solution, when the-level equals one.

A Bernstein–Walsh Type Inequality and Applications

2006 ◽  
Vol 49 (2) ◽  
pp. 256-264 ◽  
Author(s):  
Tejinder Neelon
Keyword(s):  
Power Series ◽  
Type Inequality ◽  
Divergent Series ◽  
Transfinite Diameter ◽  
Several Variables ◽  
Real Analytic ◽  
Positive Capacity ◽  
Double Power Series

AbstractA Bernstein–Walsh type inequality forC∞functions of several variables is derived, which then is applied to obtain analogs and generalizations of the following classical theorems: (1) Bochnak– Siciak theorem: aC∞function on ℝnthat is real analytic on every line is real analytic; (2) Zorn–Lelong theorem: if a double power seriesF(x,y) converges on a set of lines of positive capacity thenF(x,y) is convergent; (3) Abhyankar–Moh–Sathaye theorem: the transfinite diameter of the convergence set of a divergent series is zero.

scholarly journals On interpreting the sums of asymptotic series of positive terms

1982 ◽  
Vol 24 (1) ◽  
pp. 28-39 ◽  
Author(s):  
J. E. Drummond
Keyword(s):  
Quantum Mechanics ◽  
Asymptotic Series ◽  
Divergent Series ◽  
Integral Model ◽  
Complex Numbers ◽  
Binomial Model ◽  
Exponential Integral ◽  
The Third

AbstractFour different kinds of positive asymptotic series are identified by the limiting ratio of successive terms. When the limiting ratio is 1 the series is unsummable. When the ratio tends rapidly to a constant, whether greater or less than 1, the series is easily summed. When the ratio tends slowly to a constant not equal to 1 the series is compared with a binomial model which is then used to speed the convergence. When the ratio increases linearly, a limiting binomial and an exponential integral model are both used to speed convergence. The two resulting model sums are consistent and in this case are complex numbers. Truncation at the smallest term is found to be unreliable in the second case, invalid in the third case, and the exponential integral is used to produce a significantly improved truncation in the third case. A divergent series from quantum mechanics is also examined.

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