On some infinite integrals involving logarithmic exponential and powers

1992 ◽  
Vol 122 (1-2) ◽  
pp. 11-15 ◽  
Author(s):  
M. Aslam Chaudhry ◽  
Munir Ahmad
Keyword(s):  
Recurrence Relation ◽  
Integral Function ◽  
Special Cases ◽  
Infinite Integral ◽  
Image Position

SynopsisIn this paper we define an integral function Iμ(α; a, b) for non-negative integral values of μ byIt is proved that the function Iμ(α; a, b) satisfies a functional recurrence-relation which is then exploited to evaluate the infinite integralSome special cases of the result are also discussed.

scholarly journals On a family of logarithmic and exponential integrals occurring in probability and reliability theory

1994 ◽  
Vol 35 (4) ◽  
pp. 469-478 ◽  
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Reliability Theory ◽  
Integral Function ◽  
Integral Representations ◽  
Exponential Integral ◽  
Error Functions ◽  
Special Cases ◽  
Image Position ◽  
Form Evaluation

AbstractWe define an integral function Iμ(α, x; a, b) for non-negative integral values of μ byIt is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.

scholarly journals An Interpolation Series for Integral Functions

1953 ◽  
Vol 9 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Sheila Scott Macintyre
Keyword(s):  
Taylor Expansion ◽  
Maximum Modulus ◽  
Integral Function ◽  
The Best Possible Constant ◽  
Special Cases ◽  
Interpolation Series ◽  
Image Position

1. The Gontcharoff interpolation serieswherehas been studied in various special cases. For example, if an = a0 (all n), (1.0) reduces to the Taylor expansion of F(z). If an = (−1)n, J. M. Whittaker showed that the series (1.0) converges to F(z) provided F(z) is an integral function whose maximum modulus satisfiesthe constant ¼π being the “best possible”. In the case |an| ≤ 1, I have shown that the series converges to F(z) provided F(z) is an integral function whose maximum modulus satisfiesand that while ·7259 is not the “best possible” constant here, it cannot be replaced by a number as great as ·7378.

scholarly journals Positive solutions and eigenvalues of conjugate boundary value problems

1999 ◽  
Vol 42 (2) ◽  
pp. 349-374 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Martin Bohner ◽  
Patricia J. Y. Wong
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Lower Bounds ◽  
Boundary Value ◽  
Upper And Lower Bounds ◽  
Special Cases ◽  
Image Position

We consider the following boundary value problemwhere λ > 0 and 1 ≤ p ≤ n – 1 is fixed. The values of λ are characterized so that the boundary value problem has a positive solution. Further, for the case λ = 1 we offer criteria for the existence of two positive solutions of the boundary value problem. Upper and lower bounds for these positive solutions are also established for special cases. Several examples are included to dwell upon the importance of the results obtained.

Connection formulae for spectral functions associated with singular Dirac equations

2004 ◽  
Vol 134 (1) ◽  
pp. 215-223 ◽  
Author(s):  
S. M. Riehl
Keyword(s):  
Dirac Equation ◽  
Spectral Function ◽  
Limit Point ◽  
Initial Condition ◽  
Spectral Functions ◽  
Dirac Equations ◽  
Special Cases ◽  
Limit Point Case ◽  
Image Position ◽  
Connection Formulae

We consider the Dirac equation given by with initial condition y1 (0) cos α + y2(0) sin α = 0, α ε [0; π ) and suppose the equation is in the limit-point case at infinity. Using to denote the derivative of the corresponding spectral function, a formula for is given when is known and positive for three distinct values of α. In general, if is known and positive for only two distinct values of α, then is shown to be one of two possibilities. However, in special cases of the Dirac equation, can be uniquely determined given for only two values of α.

Some Properties of Generalized Euler Numbers

1981 ◽  
Vol 33 (3) ◽  
pp. 606-617 ◽  
Author(s):  
D. J. Leeming ◽  
R. A. Macleod
Keyword(s):  
Positive Integer ◽  
Roots Of Unity ◽  
Euler Numbers ◽  
Special Cases ◽  
Image Position

We define infinitely many sequences of integers one sequence for each positive integer k ≦ 2 by(1.1)where are the k-th roots of unity and (E(k))n is replaced by En(k) after multiplying out. An immediate consequence of (1.1) is(1.2)Therefore, we are interested in numbers of the form Esk(k) (s = 0, 1, 2, …; k = 2, 3, …).Some special cases have been considered in the literature. For k = 2, we obtain the Euler numbers (see e.g. [8]). The case k = 3 is considered briefly by D. H. Lehmer [7], and the case k = 4 by Leeming [6] and Carlitz ([1]and [2]).

A Note on Divergent Series

1952 ◽  
Vol 4 ◽  
pp. 445-454 ◽  
Author(s):  
G. M. Petersen
Keyword(s):  
Divergent Series ◽  
Matrix Methods ◽  
Special Cases ◽  
Image Position

In this note we shall discuss certain matrix methods of summation, though otherwise, §1 and §2 are not connected.In this section, we shall study some properties of the method (Bh) where we say the series ∑uv is summable (Bh) whenThe method (Bh) has been studied in special cases airsing from different values of h by Rogosinski [11; 12], Bernstein [2], and more recently by Karamata [3; 4].

scholarly journals An infinite integral formula

1939 ◽  
Vol 6 (2) ◽  
pp. 75-77
Author(s):  
C. G. Lambe
Keyword(s):  
Integral Formula ◽  
The Other ◽  
Infinite Integral ◽  
Image Position

§ 1. The object of this note is to discuss the formulathe integral being supposed convergent for certain ranges of values of x and z. The contour is such that the poles of Γ(– s)lie to its right and the other poles of the integrand to its left. It will be seen that all the Pincherle-Mellin-Barnes integrals are particular cases of this formula.

scholarly journals Some Extensions of the Hausdorff-Young and Paley Theorems

1961 ◽  
Vol 4 (2) ◽  
pp. 123-138
Author(s):  
P. S. Bullen
Keyword(s):  
Special Cases ◽  
Image Position

Orthonormal sequences, o.n. s., {ϕn} defined on [0,1] and satisfying1have been studied in [3] and [1]. One of the objects of this paper is to indicate that the methods used to study such o. n. s. can be used for a much wider class, and that, although there seems to be no super theorem to cover all cases, a knowledge of the results and methods of proof in some fairly broad special cases enables one to state and prove theorems for other classes of o. n. s.

scholarly journals Non-linear integral equations for Heun functions

1969 ◽  
Vol 16 (4) ◽  
pp. 281-289 ◽  
Author(s):  
B. D. Sleeman
Keyword(s):  
Integral Equations ◽  
Heun Equation ◽  
Mathieu Functions ◽  
Special Cases ◽  
Mathieu’S Equation ◽  
Heun Functions ◽  
Non Linear ◽  
Linear Integral ◽  
Image Position ◽  
Linear Integral Equations

Some years ago Lambe and Ward (1) and Erdélyi (2) obtained integral equations for Heun polynomials and Heun functions. The integral equations discussed by these authors were of the formFurther, as is well known, the Heun equation includes, among its special cases, Lamé's equation and Mathieu's equation and so (1.1) may be considered a generalisation of the integral equations satisfied by Lamé polynomials and Mathieu functions. However, integral equations of the type (1.1) are not the only ones satisfied by Lamé polynomials; Arscott (3) discussed a class of non- linear integral equations associated with these functions. This paper then is concerned with discussing the existence of non-linear integral equations satisfied by solutions of Heun's equation.

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 ◽  
Image Position

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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