2754. A Repeated Integral

1958 ◽  
Vol 42 (339) ◽  
pp. 52 ◽  
Author(s):  
H. G. Apsimon
Keyword(s):  
Repeated Integral

XLI.—On Sommerfeld's Form of Fourier's Repeated Integrals

1912 ◽  
Vol 31 ◽  
pp. 587-603
Author(s):  
W. H. Young
Keyword(s):  
Fourier Series ◽  
Applied Mathematics ◽  
Lebesgue Integral ◽  
General Hypothesis ◽  
Differential Coefficient ◽  
Right Hand ◽  
Fourier Sine ◽  
Repeated Integral ◽  
The Right

§ 1. In his treatise on Fourier Series and Integrals Carslaw quotes without proof Sommerfeld's theorem thatwhen the limit on the right-hand side exists. In applied mathematics, he remarks, it is this limit, rather than the corresponding Fourier repeated integral which occurs.In the present paper I propose to extend this result in various ways. After proving Sommerfeld's result on the general hypothesis, not considered by him, that the integral is a Lebesgue integral, I show that the limit in question is whenever the origin is a point at which f(u) is the differential coefficient of its integral, and I obtain the corresponding results for In all their generality these statements are only true when the interval (0, p) is a finite one. I then show how, under a variety of hypotheses with respect to the nature of f(x) at infinity, they can be extended so as to be still true when p = + ∞ . These hypotheses correspond precisely to those which have been proved f to be sufficient for the corresponding statements as to the Fourier sine and cosine repeated integrals in their usual forms.

scholarly journals Fourier's Integral

1915 ◽  
Vol 34 ◽  
pp. 3-10 ◽  
Author(s):  
T. A. Brown
Keyword(s):  
Integral Formula ◽  
Fourier Integral ◽  
Cardinal Function ◽  
Repeated Integral

The purposes of the following note are these:—(1) To show the relation between Whittaker's Cardinal Function and Fourier's Repeated Integral; (2) to give a new derivation of Fourier's Integral Formula; and (3) to extend the notion of the Fourier Integral to the case in which the variables involved are complex.

scholarly journals How to execute repeated integral to generate hyper-exponential functions

2020 ◽  
Vol 3 (3) ◽  
Keyword(s):  
Exponential Functions ◽  
Repeated Integral ◽  
The Way

As mentioned above, we need repeated integral to generate hyperexponential functions of n-order. By the way, what is the way how to execute repeated integral by using a computer?

2754. A repeated integral

1958 ◽  
Vol 42 (339) ◽  
pp. 52 ◽  
Author(s):  
H. G. Apsimon
Keyword(s):  
Repeated Integral

A uniqueness theorem for the exponential series of Herglotz

1960 ◽  
Vol 56 (3) ◽  
pp. 220-232 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Uniqueness Theorem ◽  
Complex Numbers ◽  
Exponential Series ◽  
Integrable Functions ◽  
Uniformly Convergent ◽  
Repeated Integral ◽  
Image Position

If {λ;n}, {bn} are sequences of complex numbers, and we consider the series ∑bn exp (−λnx), given as convergent in (0, 1) (i.e. the open invertal (0,1)) to f(x)∈L, then, writing(if λn = 0 the corresponding term is ½bnx2) where the series is supposed is to be uniformly convergent in (0, 1), we havefor 0<h<h(x).If we know that the second member of (2) tends to f(x) as h → +0, it will follow that F(x) is a repeated integral of f(x) ((1), 671). If there is a sequence {φv(x)} of integrable functions with the property thatthen, on multiplying (1) by φv(x) and integrating over (0,1), we obtain a formula for bv in terms of F(x). On integrating by parts twice, bv will be expressed in terms of f(x), and this will constitute a uniqueness theorem for the series ∑bn exp (−λnx).

Differentiation of the Repeated Integral

1959 ◽  
Vol 66 (2) ◽  
pp. 127
Author(s):  
John G. Christiano
Keyword(s):  
Repeated Integral

2800. On Note 2754: A Repeated Integral

1958 ◽  
Vol 42 (342) ◽  
pp. 292 ◽  
Author(s):  
F. Garwood ◽  
J. C. Tanner
Keyword(s):  
Repeated Integral

Repeated Integral Inequalities

1984 ◽  
Vol 4 (1) ◽  
pp. 99-107 ◽  
Author(s):  
JENNIFER DIXON ◽  
SEAN MCKEE
Keyword(s):  
Integral Inequalities ◽  
Repeated Integral

Consolidation under embankment-type loads

1976 ◽  
Vol 13 (1) ◽  
pp. 72-77 ◽  
Author(s):  
N. Babu Shanker ◽  
K. S. Sarma ◽  
M. Venkataratnam
Keyword(s):  
Plane Strain ◽  
Integral Transformation ◽  
Infinite Strip ◽  
Clay Layer ◽  
Transformation Technique ◽  
Loading Pattern ◽  
Distributed Load ◽  
Uniformly Distributed Load ◽  
Repeated Integral ◽  
Line Loads

Plane strain problems of consolidation (or poro-elasticity) can be solved using the two displacement functions defined by McNamee and Gibson with the help of a repeated integral transformation technique. The problem of a semi-infinite clay layer whose surface is subjected to an embankment-type of normal trapezoidal pressure applied along an infinite strip is treated here. The general loading pattern selected easily degenerates into a rectangular (uniformly distributed) load for which NcNamee and Gibson gave the solutions, to the triangular loads and also to the line loads. Not only the settlements, but also the pore pressures have been evaluated under these types of loads when the surface is either pervious or impervious.The nondimensional solutions presented are useful to highway and embankment engineers. There is also an example of the use of these solutions.

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