scholarly journals On certain infinite integrals involving Struve functions and parabolic cylinder functions

1946 ◽  
Vol 7 (4) ◽  
pp. 171-173 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Present Note ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Struve Functions ◽  
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The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved thatFrom (1)follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.

scholarly journals On the squares of Weber's parabolic cylinder functions and certain integrals connected with them

1934 ◽  
Vol 4 (1) ◽  
pp. 27-32 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
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The parabolic cylinder functions Dn(x) and D−(n+1) (± ix) are defined byfor all values of n and x.

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

Error bounds for the Liouville–Green (or WKB) approximation

1961 ◽  
Vol 57 (4) ◽  
pp. 790-810 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Error Bounds ◽  
Bessel Functions ◽  
Approximate Solutions ◽  
Parabolic Cylinder ◽  
Real Interval ◽  
Positive Parameter ◽  
Modified Bessel Functions ◽  
Elementary Functions ◽  
Parabolic Cylinder Functions ◽  
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ABSTRACTError bounds are derived and examined for approximate solutions in terms of elementary functions of the differential equationsin which u is a positive parameter, the functions f and p are free from singularities and p does not vanish. Bounds are also obtained for the remainder terms in the asymptotic expansions of the solutions in descending powers of u. The variable x ranges over a real interval, finite or infinite or over a region of the complex plane, bounded or unbounded.Applications are made to parabolic cylinder functions of large orders, and modified Bessel functions of large orders.

A method for the determination of converging factors, applied to the asymptotic expansions for the parabolic cylinder functions

1952 ◽  
Vol 48 (2) ◽  
pp. 243-254 ◽  
Author(s):  
J. C. P. Miller
Keyword(s):  
Infinite Series ◽  
Asymptotic Expansions ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
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1. The method of converging factors, for hastening the convergence of slowly convergentseries and improving the accuracy of asymptotic expansions, was introduced by J. R. Airey and is well known to computers (see Airey(1) and Rosser(2)). The principle is as follows. It is required to compute a quantity which is expressed as an infinite seriesThe series may be either convergent or asymptotic and divergent.

scholarly journals Expansions in terms of Parabolic Cylinder Functions

1948 ◽  
Vol 8 (2) ◽  
pp. 50-65 ◽  
Author(s):  
T. M. Cherry
Keyword(s):  
Wave Equation ◽  
Positive Integer ◽  
Plane Wave ◽  
Arbitrary Function ◽  
Hermite Polynomials ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
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When the plane wave equation is expressed in terms of parabolic co-ordinates x, y, the variables are separable, and the elementary solutions have the formwhere x, y, μ are real. In this context, therefore, the functions Dν (z) which are directly significant are those where amp z = ± π/4 and ν + ½ is purely imaginary, rather than those where z is real and ν is a positive integer. The expansion of an arbitrary function in terms of the latter sort of D-function (substantially, in terms of Hermite polynomials) is well known. This paper is concerned with the expansion in terms of the former sort of D-function.

Two inequalities for parabolic cylinder functions

1961 ◽  
Vol 57 (4) ◽  
pp. 811-822 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Complex Variable ◽  
Asymptotic Theory ◽  
Upper Bounds ◽  
Parabolic Cylinder ◽  
Principal Solution ◽  
Second Order Differential Equations ◽  
Parabolic Cylinder Functions ◽  
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In this paper upper bounds are established for the principal solution of the differential equationand its derivative, for unrestricted values of the complex variable t and the complex parameter μ. The results may have little interest in their own right, but they are of great value in developing the asymptotic theory of linear second-order differential equations in a domain containing two turning points. Equation (1·1) is the simplest example of a differential equation of this type.

Sophocles, ‘Electra’ 1205–10

1973 ◽  
Vol 19 ◽  
pp. 45-46
Author(s):  
R. D. Dawe
Keyword(s):  
Present Note ◽  
Greek Tragedy ◽  
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The attribution of lines to different speakers in Greek tragedy is a matter on which MSS have notoriously little authority. As for Electra itself, there are at least three places where the name of the heroine has been incorrectly added in some or all MSS. In my Studies in the Text of Sophocles, I, 198, I list these places and suggest that the same error has happened at a fourth place, viz. 1323. The purpose of the present note is to suggest that at El. 1205–10 the same mistake has happened yet again.The situation is that Electra is holding the urn which she falsely believes to contain the ashes of her dead brother, Orestes. But Orestes is alive, and before her at this very moment. He is trying to persuade her to give up the urn. If the text before us had been preserved in a MS devoid of ascriptions to speakers, no one would have been so perverse as to do what all MSS and editors do in fact do, namely attribute the words οὔ φημ᾿ ἐάσειν to Orestes.

scholarly journals On general curves lying on a quadric

1927 ◽  
Vol 1 (1) ◽  
pp. 19-30 ◽  
Author(s):  
H. F. Baker
Keyword(s):  
Present Note ◽  
Rational Curves ◽  
Stationary Points ◽  
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Introduction. The present note, though in continuation of the preceding one dealing with rational curves, is written so as to be independent of this. It is concerned to prove that if a curve of order n, and genus p, with k cusps, or stationary points, lying on a quadric, Ω, in space of any number of dimensions, is such that itself, its tangents, its osculating planes, … , and finally its osculating (h – 1)-folds, all lie on the quadric Ω, then the number of its osculating h-folds which lie on the quadric isTwo proofs of this result are given, in §§ 4 and 5.

scholarly journals On the application of the operational calculus to the expansion of a function in a series of Legendre's functions of the second kind

1942 ◽  
Vol 7 (1) ◽  
pp. 1-2
Author(s):  
D. P. Banerjee
Keyword(s):  
Analytic Function ◽  
Present Note ◽  
Operational Calculus ◽  
Integral Function ◽  
Legendre Functions ◽  
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In the present note we shall obtain the expansion in a series of Legendre functions of the second kind of an integral function φ (ω) represented by Laplace's integralwhere f (x) is an analytic function of x, regular in the circlewhere an are constants and qn (ω) = in+1Qn (iω).

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