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Transform methods
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Consider the example f(t) = eat. The integral defining F(s) makes sense only for Re(s) > a. Yet the formula for F(s) makes sense for s ≠ a. This phenomenon is a good example of analytic continuation. We briefly discuss this concept. Given a piecewise continuous function on [0, ∞), the integral in (2) converges for s in some subset of ℂ. This region of convergence is often abbreviated ROC. When f grows faster than exponentially at infinity, the ROC is empty. The ROC will be non-empty when f is piecewise continuous and does not grow too fast at zero and infinity, the endpoints of the interval of integration. See Exercise 1.8. The function F(s) is often complex analytic in a region larger than one might expect. Analytic continuation is significant in number theory, where the zeta and Gamma functions are defined by formulas and then extended to larger sets by continuation.
Introduction
Published in Jan Awrejcewicz, Roman Starosta, Grażyna Sypniewska-Kamińska, Asymptotic Multiple Scale Method in Time Domain, 2022
Jan Awrejcewicz, Roman Starosta, Grażyna Sypniewska-Kamińska
Fikioris et al. [69] employed several asymptotic techniques useful to study research fields associated with electromagnetics and antennas. In particular, the idea of analytic continuation of functions of a single complex variable played a crucial role in the analysis. The application-oriented research was supplemented by material covering integration by parts and the Riemann-Lebesque lemma, the use of contour integration in conjunction with other methods, techniques related to Laplace's method and Watson's lemma, the asymptotic behavior of certain Fourier sine and cosine transforms, and the Poisson summation formula.
Power Series Solutions II: Generalizations and Theory
Published in Kenneth B. Howell, Ordinary Differential Equations, 2019
Analytic continuation is any procedure that “continues” an analytic function defined on one region so that it becomes defined on a larger region. Perhaps it would be better to call it “analytic extension” because what we are really doing is extending the domain of our original function by creating an analytic function with a larger domain that equals the original function over the original domain.
On the mother bodies of steady polygonal uniform vortices. Part I: numerical experiments
Published in Geophysical & Astrophysical Fluid Dynamics, 2022
The continued conjugate of the Cauchy term (7) can be now evaluated. The role of the inverse map in this calculation is rather subtle and can be appreciated in rewriting as a function of . In doing this, the branch of the square root has to be chosen, in terms of one of the forms (11, 12) of the inverse where if (), while if (). As a function of , has two different forms: the first inside , this is the one that directly comes from the analytic continuation of the conjugate of the restriction of on the vortex boundary (and it is the one relevant for the present analysis), and the second outside , that is obtained by means of the form (12) of the inverse.
Semi-infinite crack between two dissimilar orthotropic strips
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2022
Prasanta Basak, Subhadeep Naskar, Subhas C. Mandal
It can be noted that the left hand side of Eq. (83) is analytic in the upper half plane Max( and the right hand side is analytic in the lower half plane Min( for any arbitrary small positive value of ϵ. Since τ = 0 is the common line of analyticity, by using analytic continuation the whole equation (83) is analytic and single valued throughout the complex plane. This is the reason why we had considered the condition (16′) instead of the constant load τ0, otherwise the regions of analyticity of Eq. (83) would have been and and hence no common region of analyticity would exists. Since Eq. (83) is analytic throughout the complex ω plane, we assume that both sides of Eq. (83) are analytic and equal to an entire function
Resummation of the Rayleigh-Schrödinger perturbation series. Vibrational energy levels of the H2S molecule.
Published in Molecular Physics, 2021
As mentioned above, the matrix eigenvalues can be considered as different branches of a single function of the perturbation parameter λ defined on a multi-sheet Riemann surface [19]. Since the function is multi-valued, there exist quadratic branch points that connect two different branches, the so-called Katz branch points [8,20]. Since for real λ the eigenvalues of a Hermitian matrix are real, branch points can come only in conjugate pairs not lying on the real axis. Two branches at the Katz branch points coincide and are continuous, but their derivatives with respect to λ are undefined. As a result, at Katz branch point these two branches are not analytic. Going around this point, one passes from one sheet of the Riemann surface to another. In other words, an analytic continuation along a closed path that completely encircles a Katz branch point, allows the eigenvalue on one sheet to be found from the eigenvalue given on the other. Among possible approximate methods of analytic continuation, multivalued Hermite -Padé algebraic approximants, which reflect the suspected singularity structure, were found to be effective [8,9,14].