XVII.—On the Asymptotic Expansion of the Characteristic Numbers of the Mathieu Equation

1930 ◽  
Vol 49 ◽  
pp. 210-223 ◽  
Author(s):  
Sydney Goldstein
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Mathieu Equation ◽  
Characteristic Number ◽  
Numerical Approximations ◽  
Characteristic Numbers ◽  
The Difference ◽  
Image Position ◽  
Integral Order

An asymptotic formula has recently been given for the characteristic numbers of the Mathieu equation From tabular values, it will be seen that the formula provides good numerical approximations to the characteristic numbers of integral order; but as pointed out by Ince, it provides better approximations to the characteristic numbers of order (m + ½), where m is a positive integer or zero. In this paper we shall first attempt to find out why this should be so, and then go on to show that the formula is probably an asymptotic expansion, in the Poincaré sense, for any characteristic number. A new asymptotic formula is then found for the difference between two characteristic numbers.

The Circle Problem in an Arithmetic Progression

1968 ◽  
Vol 11 (2) ◽  
pp. 175-184 ◽  
Author(s):  
R.A. Smith
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Arithmetic Progression ◽  
Error Term ◽  
Circle Problem ◽  
Image Position

In following a suggestion of S. Chowla to apply a method of C. Hooley [3] to obtain an asymptotic formula for the sum ∑ r(n)r(n+a), where r(n) denotes the number of representations of n≤xn as the sum of two squares and is positive integer, we have had to obtain non-trivial estimates for the error term in the asymptotic expansion of1

scholarly journals A note on Mathieu functions

1957 ◽  
Vol 3 (3) ◽  
pp. 132-134 ◽  
Author(s):  
M. Bell
Keyword(s):  
Positive Integer ◽  
General Formula ◽  
Algebraic Equation ◽  
Explicit Construction ◽  
High Order ◽  
Mathieu Functions ◽  
Unstable Zone ◽  
Leading Term ◽  
Image Position ◽  
Integral Order

The Mathieu functions of integral order [1] are the solutions with period π or 2π of the equationThe eigenvalues associated with the functions ceN and seN, where N is a positive integer, denoted by aN and bN respectively, reduce toaN = bN = N2when q is zero. The quantities aN and bN can be expanded in powers of q, but the explicit construction of high order coefficients is very tedious. In some applications the quantity of most interest is aN – bN, which may be called the “width of the unstable zone“. It is the object of this note to derive a general formula for the leading term in the expansion of this quantity, namelySuppose first that N is an odd integer. Then there is an expansionwhereThese functions π satisfyandOn Substituting (3) in (1), one obtains the algebraic equationwhereExplicitly,{11} = q{lm} = 0 otherwise.

XXVIII.—Researches into the Characteristic Numbers of the Mathieu Equation—(Second Paper)

1927 ◽  
Vol 46 ◽  
pp. 316-322 ◽  
Author(s):  
E. L. Ince
Keyword(s):  
Present Author ◽  
Mathieu Equation ◽  
London Mathematical Society ◽  
Mathematical Society ◽  
Characteristic Numbers ◽  
Image Position

In a paper recently communicated to the London Mathematical Society the present author showed that the characteristic numbers an and bn of the Mathieu equation—may be developed, for large positive values of q, in the forms

scholarly journals Asymptotic expansion of a series of Ramanujan

1992 ◽  
Vol 35 (2) ◽  
pp. 189-199
Author(s):  
Bruce C. Berndt ◽  
Ronald J. Evans
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Image Position

An asymptotic expansion is given for the seriesas x→∞ in the sector |Argx|≦π/2–δ. Here δ, Re(a), and Re(s) are positive and r is a positive integer. In the case a = r = s = 1, this yields the nontrivial resultstated by Ramanujan in his notebooks [6].

scholarly journals On a generalisation of a result of Ramanujan connected with the exponential series

1981 ◽  
Vol 24 (3) ◽  
pp. 179-195
Author(s):  
R. B. Paris
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Exponential Series ◽  
Large Positive Integer ◽  
The Many ◽  
Image Position

One of the many interesting problems discussed by Ramanujan is an approximation related to the exponential series for en, when n assumes large positive integer values. If the number θn is defined byRamanujan (9) showed that when n is large, θn possesses the asymptotic expansionThe first demonstrations that θn lies between ½ and and that θn decreases monotoni-cally to the value as n increases, were given by Szegö (12) and Watson (13). Analogous results were shown to exist for the function e−n, for positive integer values of n, by Copson (4). If φn is defined bythen πn lies between 1 and ½ and tends monotonically to the value ½ as n increases, with the asymptotic expansionA generalisation of these results was considered by Buckholtz (2) who defined, in a slightly different notation, for complex z and positive integer n, the function φn(z) by

IV.—Researches into the Characteristic Numbers of the Mathieu Equation

1927 ◽  
Vol 46 ◽  
pp. 20-29 ◽  
Author(s):  
E. L. Ince
Keyword(s):  
Fourier Series ◽  
Periodic Solutions ◽  
Mathieu Equation ◽  
Mathieu Functions ◽  
Characteristic Numbers ◽  
Image Position

The characteristic numbers of the Mathieu equationare those values of a for which, when q is given, the equation admits of a solution of period π or 2π. The periodic solutions, or Mathieu functions, may be developed as a Fourier-series convergent for all values of q,multiplied by one or other of the factors

Fermat and the difference of two squares

2012 ◽  
Vol 96 (537) ◽  
pp. 480-491 ◽  
Author(s):  
Stan Dolan
Keyword(s):  
Positive Integer ◽  
Powerful Technique ◽  
Previous Note ◽  
Solving Equations ◽  
Integer Solutions ◽  
The Difference ◽  
Image Position

In a previous note [1], Fermat's method of descente infinie was used to prove that the equations.have no positive integer solutions. The geometrically based proof of [1] masked the underlying use of the difference of two squares. In the proofs of this article we shall make its use explicit, just as Fermat did [2, pp. 293-294].We shall use the elementary idea of the difference of two squares to develop a powerful technique for solving equations of the form ax4 + bx2y2 + cy4 = z2. This will then be applied to three problems of historical interest.

Asymptotic formulae for solutions of linear second-order differential equations with a large parameter

1960 ◽  
Vol 1 (4) ◽  
pp. 439-464 ◽  
Author(s):  
R. C. Thorne
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Positive Integer ◽  
Complex Variable ◽  
Regular Function ◽  
Arbitrary Parameter ◽  
Large Parameter ◽  
Second Order Differential Equations ◽  
Asymptotic Formulae ◽  
Image Position

AbstractUniform asymptotic formulae are obtained for solutions of the differential equation for large positive values of the parameter u. Here p is a positive integer, θ an arbitrary parameter and z a complex variable whose domain of variation may be unbounded. The function ƒ (u, θ, z) is a regular function of ζ having an asymptotic expansion of the form for large u.The results obtained include and extend those of earlier writers which are applicable to this equation.

Some arithmetical properties of m-ary partitions

1970 ◽  
Vol 68 (2) ◽  
pp. 447-453 ◽  
Author(s):  
Öystein Rödseth
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Fixed Integer ◽  
De Bruijn ◽  
Image Position

We denote by tm(n) the number of partitions of the positive integer n into non-decreasing parts which are positive or zero powers of a fixed integer m > 1 and we call tm(n) ‘the m-ary partition function’. Mahler(1) obtained an asymptotic formula for tm(n), the first term of which isMahler's result was later improved by de Bruijn (2).

scholarly journals THE LEAST COMMON MULTIPLE OF CONSECUTIVE TERMS IN A QUADRATIC PROGRESSION

2012 ◽  
Vol 86 (3) ◽  
pp. 389-404 ◽  
Author(s):  
GUOYOU QIAN ◽  
QIANRONG TAN ◽  
SHAOFANG HONG
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Periodic Function ◽  
Arithmetic Function ◽  
Local Analysis ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Image Position

AbstractLet k be any given positive integer. We define the arithmetic function gk for any positive integer n by We first show that gk is periodic. Subsequently, we provide a detailed local analysis of the periodic function gk, and determine its smallest period. We also obtain an asymptotic formula for log lcm0≤i≤k {(n+i)2+1}.

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