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 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.

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 Integral Functions with Gap Power Series

1954 ◽  
Vol 10 (2) ◽  
pp. 62-70 ◽  
Author(s):  
P. Erdös ◽  
A. J. Macintyee
Keyword(s):  
Power Series ◽  
Maximum Modulus ◽  
Integral Function ◽  
Minimum Modulus ◽  
Maximum Term ◽  
Image Position

1. Letbe an integral function, λn being a strictly increasing sequence of nonnegative integers. We shall use the notationsdescribing M (r) as the maximum modulus, m (r) as the minimum modulus and μ(r) as the maximum term of f (z).

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.

Some generalizations of two-point expansions

1952 ◽  
Vol 48 (4) ◽  
pp. 583-586 ◽  
Author(s):  
Sheila Scott Macintyre
Keyword(s):  
Taylor Expansion ◽  
Image Position

Abel's series(1)may be regarded as a generalization of the Taylor expansionThis note generalizes the two-point series of Lidstone and Whittaker (see (9)) in a similar way. Alternatively, the series discussed might be regarded as generalizations of Abel's series.

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]).

On the Maximum Modulus and the Mean Modulus of an Entire Function

1969 ◽  
Vol 12 (6) ◽  
pp. 869-872 ◽  
Author(s):  
A.R. Reddy
Keyword(s):  
Entire Function ◽  
Maximum Modulus ◽  
The Mean ◽  
Image Position

Let be an entire function, but not a polynomial. As usual let,1

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].

Sign in / Sign up

Export Citation Format

Share Document