Using double integrals to solve single integrals

2016 ◽  
Vol 100 (548) ◽  
pp. 257-265
Author(s):  
G. J. O. Jameson
Keyword(s):  
Finite Limit ◽  
Double Integral ◽  
Image Position ◽  
Double Integrals

Consider the integralwhere b > a > 0. First, let us clarify why it even exists. Of course, convergence at infinity is ensured by the exponential terms, but the integrals of and e–ax/x and e–bx/x, taken separately, are divergent at 0, since these integrands equate asymptotically to 1/x as x → 0. However,so (e–ax – e–bx)/x tends to the finite limit b – a as x → 0 and there is no problem integrating it on intervals of the form [0, r].A neat way to evaluate I1 starts by expressing the integrand itself as an integral:(1)Inserting this into I1 converts it into a double integral.

On double integrals in the calculus of variations

1932 ◽  
Vol 28 (4) ◽  
pp. 442-454 ◽  
Author(s):  
R. P. Gillespie
Keyword(s):  
Calculus Of Variations ◽  
Double Integral ◽  
Image Position ◽  
Double Integrals

The problem of the double integral in the calculus of variations, when expressed in the parametric notation, was first fully discussed by Kobb. In this paper Kobb finds conditions for the minimising of the double integraltaken over the surface

Quadrature relations for finite-limit, principal-value integrals

1992 ◽  
Vol 70 (8) ◽  
pp. 656-666 ◽  
Author(s):  
K. T. R. Davies ◽  
M. L. Glasser ◽  
R. W. Davies
Keyword(s):  
Arbitrary Order ◽  
Finite Limit ◽  
Limit Case ◽  
Double Integral ◽  
Divergent Integral ◽  
Step Functions ◽  
Principal Value ◽  
Indefinite Integral ◽  
The Difference ◽  
Double Integrals

An intuitive expression is obtained for a general finite-limit, prinicipal-value (PV) integral containing a pole of arbitrary order. Such an integral is simply the difference between its "end-point quadratures," the quadrature being the result of the indefinite integral. This PV expression is fairly obvious for simple poles but requires careful justification for higher order poles. Moreover, quadratures are useful in evaluating related integrals that contain both poles and other singularities (e.g., step functions or logarithmic divergences). Then, for a certain type of finite-limit divergent integral, the PV is interpreted as its "convergent part." Also, for cases in which there are two PV's in a double integral, it is shown that the famous Poincaré–Bertrand theorem applies to finite-limit as well as infinite-limit integrals. Finally, an interesting quadrature relation is derived for such double integrals, and the validity of the Poincaré–Bertrand theorem is explicitly demonstrated for a simple finite-limit case.

The general double integral problem in the calculus of variations

1933 ◽  
Vol 29 (2) ◽  
pp. 207-211
Author(s):  
R. P. Gillespie
Keyword(s):  
Calculus Of Variations ◽  
General Problem ◽  
Necessary Condition ◽  
Double Integral ◽  
Integral Problem ◽  
Image Position

In a previous paper in these Proceedings the problem of the double integralwas discussed when the function F had the formwhereIt is proposed in the present paper to extend the method to the general problem, where F may have any form provided only that it satisfies the necessary condition of being homogeneous of the first degree in A, B, C.

Rate of growth and convergence factors for power methods of limitation

1974 ◽  
Vol 76 (1) ◽  
pp. 241-246 ◽  
Author(s):  
Abraham Ziv
Keyword(s):  
Radius Of Convergence ◽  
Complex Numbers ◽  
Finite Limit ◽  
Power Method ◽  
Real Constant ◽  
Rate Of Growth ◽  
Image Position

Let , where pk are complex numbers, have 0 < ρ ≤ ∞ for radius of convergence and assume that P(x) ≠ 0 for α ≤ x < ρ (α < ρ is some real constant). Assuming that is convergent for all (x ∈ [0, ρ), we define the P-limit of the sequence s = {sk} byThis, so called, power method of limitation (see (3), Definition 9 and (1) Definition 6) will be denoted by P. The best known power methods are Abel's (P(x) = 1/(1 – x), α = 0, ρ = 1) and Borel's (P(x) = ex, α = 0, ρ = ∞). By Cp we denote the set of all sequences, P-limitable to a finite limit and by the set of all sequences, P-limitable to zero.

scholarly journals Failing Cases of Fourier's Double-Integral Theorem

1884 ◽  
Vol 3 ◽  
pp. 12-18
Author(s):  
Peter Alexander
Keyword(s):  
Double Integral ◽  
Integral Theorem ◽  
Image Position

Fourier's Theorem, as usually stated, isMost writers give this without limitation, but De Morgan (Diff. and Int. Calc. pp. 618 &c.) directs attention to what he calls the apparent neglect by previous writers of the limitation of the theorem to functions which satisfy the condition

Absolute summability of infinite series on a scale of Abel type summability methods

1968 ◽  
Vol 64 (2) ◽  
pp. 377-387 ◽  
Author(s):  
Babban Prasad Mishra
Keyword(s):  
Infinite Series ◽  
Open Interval ◽  
Absolute Summability ◽  
Finite Limit ◽  
Summability Methods ◽  
Image Position

Suppose that λ > − 1 and thatIt is easy to show thatWith Borwein(1), we say that the sequence {sn} is summable Aλ to s, and write sn → s(Aλ), if the seriesis convergent for all x in the open interval (0, 1)and tends to a finite limit s as x → 1 in (0, 1). The A0 method is the ordinary Abel method.

On the summability of Fourier integrals

1970 ◽  
Vol 67 (1) ◽  
pp. 23-28
Author(s):  
M. K. Nayak
Keyword(s):  
Open Interval ◽  
Finite Limit ◽  
Image Position ◽  
Fourier Integrals

We say a series is summable L iftends to a finite limit s as x → 1 in the open interval (0, 1) where

Double integral involving G-function of two variables

2020 ◽  
pp. 333-338
Author(s):  
Shukla Vinay Kumar
Keyword(s):  
Integral Equation ◽  
Population Density ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Research Paper ◽  
Statistical Distribution ◽  
Double Integral ◽  
Expansion Formulae ◽  
Function Of Two Variables ◽  
Double Integrals

In the study of certain boundary value problems integrals are useful with their connections. To obtain expansion formulae it also helps. In the study of integral equation, probability and statistical distribution, integrals are also used. To measure population density within a certain area, we can also use integrals. With integrals we can analyzed anything that changes in time. The object of this research paper is to establish a double integrals involving G-Function of two variables.

scholarly journals Tauberian conditions under which convergence follows from Cesàro summability of double integrals over R2

Filomat ◽  
2019 ◽  
Vol 33 (11) ◽  
pp. 3425-3440
Author(s):  
Gökşen Fındık ◽  
İbrahim Çanak
Keyword(s):  
Continuous Function ◽  
The Other ◽  
Cesàro Summability ◽  
Double Integral ◽  
Tauberian Conditions ◽  
Summability Methods ◽  
Cesaro Summability ◽  
The Difference ◽  
Complex Valued ◽  
Double Integrals

For a real- or complex-valued continuous function f over R2+:= [0,1) x [0,1), we denote its integral over [0,u] x [0,v] by s(u,v) and its (C,1, 1) mean, the average of s(u,v) over [0,u] x [0,v], by ?(u,v). The other means (C,1,0) and (C; 0; 1) are defined analogously. We introduce the concepts of backward differences and the Kronecker identities in different senses for double integrals over R2+. We give onesided and two-sided Tauberian conditions based on the difference between double integral of s(u, v) and its means in different senses for Ces?ro summability methods of double integrals over [0,u] x [0,v] under which convergence of s(u,v) follows from integrability of s(u,v) in different senses.

A modular version of Jensen's formula

1984 ◽  
Vol 95 (1) ◽  
pp. 15-20 ◽  
Author(s):  
David E. Rohrlich
Keyword(s):  
Meromorphic Function ◽  
Modular Forms ◽  
Unit Disc ◽  
Function Theory ◽  
Modular Function ◽  
Fuchsian Groups ◽  
Double Integral ◽  
Kronecker Limit Formula ◽  
Classical Function ◽  
Image Position

A well-known result of classical function theory, Jensen's formula, expresses the integral around a circle of the log modulus of a meromorphic function in terms of the log modulus of the zeros and poles of that function lying inside the circle. Explicitly, if F is a meromorphic function on the unit disc {ω ε ℂ: |ω| < 1} and F(0) = 1, then, for 0 < r < 1,where ordωF is the order of F at ω. The purpose of this note is to observe that a formula analogous to (1) holds when F is replaced by a modular function for SL2(ℤ) and the integral by a suitable double integral over a fundamental domain. We shall derive this modular variant of Jensen's formula from the usual version by applying the Rankin-Selberg method and the first Kronecker limit formula. The argument admits some extension to Fuchsian groups other than SL2(ℤ), and to modular forms of weight other than zero; this point will be discussed later.

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