scholarly journals Recurrence Formulae for the Functions which represent Solutions of the Differential Equation:

1914 ◽  
Vol 33 ◽  
pp. 107-117
Author(s):  
H. T. Flint
Keyword(s):  
Differential Equation ◽  
Positive Integer ◽  
In Series ◽  
Recurrence Formulae ◽  
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The contents of this paper were suggested by a discussion of the equation:in a paper by Glaisher which appears in the Philosophical Transactions, 1881, Part III.The solutions in series of (1) are:and in the paper referred to it is shewn that the coefficients of hp+1 in the expansions of and of satisfy equation (1) when p is a positive integer.

An integral equation

1932 ◽  
Vol 28 (2) ◽  
pp. 165-173
Author(s):  
G. H. Hardy ◽  
E. C. Titchmarsh
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Positive Integer ◽  
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1. The integral equationwhere x, f (x), and λ are real and α positive, may be regarded as a differential equation of order α. Suppose for example that α is a positive integer p, that f (x) tends to 0, when x → ∞, with sufficient rapidity, and thatThen, if we integrate repeatedly by parts, and write z for fp (x), (1·1) becomesThe only solutions are finite combinations of exponentials.

scholarly journals On the Linear Differential Equation of the Second Order

1915 ◽  
Vol 34 ◽  
pp. 45-60 ◽  
Author(s):  
S. Brodetsky
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
General Equation ◽  
Second Order ◽  
General Properties ◽  
In Series ◽  
Linear Differential ◽  
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The linear differential equation of the second orderis not in general integrable by any method at present available. At the same time, several equations of this type have been integrated, either in terms of finite functions or by means of expansions in series. Some properties of the integrals of the general equation have also been obtained. It is the object of this paper to develop some general properties of these integrals, which throw some light on the nature of the solutions, even if not obtainable in explicit terms.

scholarly journals On the Equation of the Parabolic Cylinder Functions

1913 ◽  
Vol 32 ◽  
pp. 2-14 ◽  
Author(s):  
Arch Milne
Keyword(s):  
Differential Equation ◽  
Positive Integer ◽  
Arbitrary Function ◽  
Parabolic Cylinder ◽  
Orthogonal Functions ◽  
Parabolic Cylinder Functions ◽  
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Hermite, in 1864 (Comptes Rendus, vol. 58) introduced into analysis the polynomials defined by the relationwhere n is a positive integer. He showed that they satisfied the differential equationthat they were orthogonal functions, and that an arbitrary function f(x) could be expanded in the form

scholarly journals Expansions in Series of Spherical Harmonics

1923 ◽  
Vol 42 ◽  
pp. 88-92
Author(s):  
T. M. MacRobert
Keyword(s):  
Positive Integer ◽  
Spherical Harmonics ◽  
Recurrence Formula ◽  
Summation Formula ◽  
In Series ◽  
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A Generalisation of Christoffel's Summation Formula. The Recurrence Formulais valid for all values of n and m; but, if m is a positive integer, , hence, when n = m, (1) becomes

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 ◽  
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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.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
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Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

scholarly journals XXV.—The Generating Function of the Reciprocal of a Determinant

1905 ◽  
Vol 40 (3) ◽  
pp. 615-629
Author(s):  
Thomas Muir
Keyword(s):  
Generating Function ◽  
Specific Character ◽  
Partial Fraction ◽  
Homogeneous Functions ◽  
Series Form ◽  
General Proposition ◽  
In Series ◽  
Image Position ◽  
The Given

(1) This is a subject to which very little study has been directed. The first to enunciate any proposition regarding it was Jacobi; but the solitary result which he reached received no attention from mathematicians,—certainly no fruitful attention,—during seventy years following the publication of it.Jacobi was concerned with a problem regarding the partition of a fraction with composite denominator (u1 − t1) (u2 − t2) … into other fractions whose denominators are factors of the original, where u1, u2, … are linear homogeneous functions of one and the same set of variables. The specific character of the partition was only definable by viewing the given fraction (u1−t1)−1 (u2−t2)−1…as expanded in series form, it being required that each partial fraction should be the aggregate of a certain set of terms in this series. Of course the question of the order of the terms in each factor of the original denominator had to be attended to at the outset, since the expansion for (a1x+b1y+c1z−t)−1 is not the same as for (b1y+c1z+a1x−t)−1. Now one general proposition to which Jacobi was led in the course of this investigation was that the coefficient ofx1−1x2−1x3−1…in the expansion ofy1−1u2−1u3−1…, whereis |a1b2c3…|−1, provided that in energy case the first term of uris that containing xr.

On a singular eigenvalue problem

1968 ◽  
Vol 64 (2) ◽  
pp. 439-446 ◽  
Author(s):  
D. Naylor ◽  
S. C. R. Dennis
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Bessel Functions ◽  
Asymptotic Condition ◽  
Real Constant ◽  
Limit Circle ◽  
Expansion In Eigenfunctions ◽  
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Sears and Titchmarsh (1) have formulated an expansion in eigenfunctions which requires a knowledge of the s-zeros of the equationHere ka > 0 is supposed given and β is a real constant such that 0 ≤ β < π. The above equation is encountered when one seeks the eigenfunctions of the differential equationon the interval 0 < α ≤ r < ∞ subject to the condition of vanishing at r = α. Solutions of (2) are the Bessel functions J±is(kr) and every solution w of (2) is such that r−½w(r) belongs to L2 (α, ∞). Since the problem is of the limit circle type at infinity it is necessary to prescribe a suitable asymptotic condition there to make the eigenfunctions determinate. In the present instance this condition is

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

scholarly journals On Runge-Kutta processes of high order

1964 ◽  
Vol 4 (2) ◽  
pp. 179-194 ◽  
Author(s):  
J. C. Butcher
Keyword(s):  
Differential Equation ◽  
High Order ◽  
Runge Kutta ◽  
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An (explicit) Runge-Kutta process is a means of numerically solving the differential equation , at the point x = x0+h, where y, f may be vectors.

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