XVIII.—The Definite Integrals of Interpolation Theory

1931 ◽  
Vol 50 ◽  
pp. 220-224
Author(s):  
E. T. Copson
Keyword(s):  
Infinite Series ◽  
Interpolation Formula ◽  
Integral Function ◽  
Integral Representations ◽  
Definite Integral ◽  
Finite Region ◽  
Interpolation Theory ◽  
Definite Integrals ◽  
Uniformly Convergent ◽  
Image Position

It is well known that, if the infinite seriesis convergent for any non-integral value of z, it is uniformly convergent in any finite region of the z-plane and represents an integral function C(z), say, such that C(n) = an for n = 0, ± 1, ± 2, … It is called the Cardinal Function of the table of values, and is identical with Gauss's Interpolation Formula (suitably bracketed).The function C(z) defined by the series (1) has been given two different definite integral representations, due to Ferrar and Ogura respectively.

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.

2. On Methods in Definite Integrals

1880 ◽  
Vol 10 ◽  
pp. 271-271
Author(s):  
Tait
Keyword(s):  
Infinite Series ◽  
Definite Integrals ◽  
Great Ease ◽  
Definite Integration ◽  
Image Position

This paper deals with various formulæ of definite integration which are, in general, put into forms in which they enable us with great ease to sum a number of infinite series. As a simple example of such a formula the following may be given:—From this it is easy to deduce innumerable results, of which the annexed are some of the more immediate.

Analytic functions of finite dimensional linear transformations

1959 ◽  
Vol 55 (1) ◽  
pp. 51-61 ◽  
Author(s):  
S. N. Afriat
Keyword(s):  
Power Series ◽  
Integral Formula ◽  
Characteristic Root ◽  
Interpolation Formula ◽  
General Function ◽  
Linear Transformations ◽  
Multiple Characteristic ◽  
Characteristic Roots ◽  
Image Position ◽  
Necessary And Sufficient

Since the first introduction of the concept of a matrix, questions about functions of matrices have had the attention of many writers, starting with Cayley(i) in 1858, and Laguerre(2) in 1867. In 1883, Sylvester(3) defined a general function φ(a) of a matrix a with simple characteristic roots, by use of Lagrange's interpolation formula, and Buchheim (4), in 1886, extended his definition to the case of multiple characteristic roots. Then Weyr(5) showed in 1887 that, for a matrix a with characteristic roots lying inside the circle of convergence of a power series φ(ζ), the power series φ(a) is convergent; and in 1900 Poincaré (6) obtained the formulaefor the sum, where C is a circle lying in and concentric with the circle of convergence, and containing all the characteristic roots in its ulterior, such a formula having effectively been suggested by Frobenius(7) in 1896 for defining a general function of a matrix. Phillips (8), in 1919, discovered the analogue, for power series in matrices, of Taylor's theorem. In 1926 Hensel(9) completed the result of Weyr by showing that a necessary and sufficient condition for the convergence of φ(a) is the convergence of the derived series φ(r)(α) (0 ≼ r < mα; α) at each characteristic root α of a, of order r at most the multiplicity mα of α. In 1928 Giorgi(10) gave a definition, depending on the classical canonical decomposition of a matrix, which is equivalent to the contour integral formula, and Fantappie (11) developed the theory of this formula, and obtained the expressionfor the characteristic projectors.

scholarly journals Generalizations and applications of Young’s integral inequality by higher order derivatives

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Jun-Qing Wang ◽  
Bai-Ni Guo ◽  
Feng Qi
Keyword(s):  
Differential Geometry ◽  
Integral Inequality ◽  
Higher Order ◽  
Integral Inequalities ◽  
List Type ◽  
Definite Integral ◽  
Exponential Integral ◽  
Definite Integrals ◽  
Logarithmic Integral

Abstract In the paper, the authors generalize Young’s integral inequality via Taylor’s theorems in terms of higher order derivatives and their norms, and apply newly-established integral inequalities to estimate several concrete definite integrals, including a definite integral of a function which plays an indispensable role in differential geometry and has a connection with the Lah numbers in combinatorics, the exponential integral, and the logarithmic integral.

Representation Formulas for Integrable and Entire Functions of Exponential Type I

1988 ◽  
Vol 40 (04) ◽  
pp. 1010-1024 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Exponential Type ◽  
Entire Functions ◽  
Conjugate Function ◽  
Interpolation Formula ◽  
Type I ◽  
Additional Hypothesis ◽  
Period 2 ◽  
Representation Formulas ◽  
Functions Of Exponential Type ◽  
Image Position

Let Bτ denote the class of entire functions of exponential type τ (&gt;0) bounded on the real axis. For the function f ∊ Bτ we have the interpolation formula [1, p. 143] 1.1 where t, γ are real numbers and is the so called conjugate function of f. Let us put 1.2 The function Gγ,f is a periodic function of α, with period 2. For t = 0 (the general case is obtained by translation) the righthand member of (1) is 2τGγ,f (1). In the following paper we suppose that f satisfies an additional hypothesis of the form f(x) = O(|x|-ε), for some ε &gt; 0, as x → ±∞ and we give an integral representation of Gγ,f(α) which is valid for 0 ≦ α ≦ 2.

On the Absolute Nörlund Summability of a Fourier Series

1973 ◽  
Vol 16 (4) ◽  
pp. 599-602
Author(s):  
D. S. Goel ◽  
B. N. Sahney
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Partial Sums ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write(1.1)If(1.2)as n→∞, we say that the series is summable by the Nörlund method (N,pn) to σ. The series is said to be absolutely summable (N,pn) or summable |N,pn| if σn is of bounded variation, i.e.,(1.3)

scholarly journals A Determinantal Expansion for a Glass of Definite Integral. Part 1

1953 ◽  
Vol 9 (1) ◽  
pp. 44-52 ◽  
Author(s):  
L. R. Shenton
Keyword(s):  
Weight Function ◽  
Fourier Coefficients ◽  
Finite Interval ◽  
Orthonormal System ◽  
Definite Integral ◽  
Negative Weight ◽  
Image Position

1. Let w(x) be a non-negative weight function for the finite interval (a, b) such that exists and is positive, and let Tr(x), r = 0, 1, 2,…be the corresponding orthonormal system of polynomials. Then if F(x) is continuous on (a, b) and has “Fourier” coefficientsParseval's formula gives

scholarly journals A Recursive Formula for the Evaluation of Earth Return Impedance on Buried Cables

2015 ◽  
Vol 35 (3) ◽  
pp. 34-43
Author(s):  
Reynaldo Iracheta
Keyword(s):  
Infinite Series ◽  
Bessel Functions ◽  
Integration Method ◽  
Recursive Formula ◽  
Definite Integral ◽  
Mathematical Software ◽  
Font Size ◽  
Buried Cables ◽  
The Time Domain ◽  
Numerical Laplace Transform

<p class="Abstractandkeywordscontent"><span style="font-size: small;"><span style="font-family: Century Gothic;">This paper presents an alternative solution based on infinite series for the accurate and efficient evaluation of cable earth return impedances. This method uses Wedepohl and Wilcox’s transformation to decompose Pollaczek’s integral in a set of Bessel functions and a definite integral. The main feature of Bessel functions is that they are easy to compute in modern mathematical software tools such as Matlab. The main contributions of this paper are the approximation of the definite integral by an infinite series, since it does not have analytical solution; and its numerical solution by means of a recursive formula. The accuracy and efficiency of this recursive formula is compared against the numerical integration method for a broad range of frequencies and cable  configurations. Finally, the proposed method is used as a subroutine for cable parameter calculation in the inverse Numerical Laplace Transform (NLT) to obtain accurate transient responses in the time domain.</span></span></p>

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