Related papers: On Singular Points and Oscillatory Integrals
In this paper, we first generalize the Fresnel integrals by changing of a path for integration in the proof of the Fresnel integrals by Cauchy's integral theorem. Next, according to oscillatory integral, we also obtain further…
As to methods for expanding an oscillatory integral into an asymptotic series with respect to the parameter, the method of stationary phase for the non-degenerate phases and the method of using resolution of singularities for degenerate…
We prove a sharp asymptotic formula for certain oscillatory integrals that may be approached using the stationary phase method. The estimates are uniform in terms of auxiliary parameters, which is crucial for application in analytic number…
In this note, by using the result in one variable, we obtain asymptotic expansions of oscillatory integrals for certain multivariable phase functions with {\bf degenerate} singular points. Moreover by using this result, we have asymptotic…
We consider the oscillatory integrals with parameter-dependent phases. We decompose the integrals into a leading term and a remainder term. Instead of the pointwise estimate, we use some $L^p$-estimate for the remainder term and get various…
We describe an elementary method for bounding a one-dimensional oscillatory integral in terms of an associated non-oscillatory integral. The bounds obtained are efficient in an appropriate sense and behave well under perturbations of the…
We propose a new method of estimating oscillatory integrals, which we call a stationary set method. We use it to obtain the sharp convergence exponents of Tarry's problems in dimension two for every degree $k\ge 2$. As a consequence, we…
We develop a theory of oscillatory integrals whose phase is given by the trace of a polynomial over an algebraic number field. We present an application to the singular integral for a version of Tarry's problem for algebraic integers.
The purpose of this article is to describe the singularities of one-dimensional oscillatory integrals, whose phases have a certain singularity, in the form of an asymptotic expansion. In the case of the Laplace integral, an analogous result…
In this paper we consider the problem on uniform estimates for generalized oscillatory integrals given by Mittag- Leffler functions with the homogeneous polynomial phase. We obtain a variant of Ricci-Stein Lemma and invariant estimates for…
An $n$th-order first derivative test for oscillatoric integrals is established. When the phase has a single stationary point, an $n$th-order asymptotic expansion of a weighted stationary phase integral is proved for arbitrary $n\geq1$. This…
We investigate estimating scalar oscillatory integrals by integrating by parts in directions based on $(x_1 \partial_{x_1} f(x) ,..., x_n \partial_{x_n}f(x))$, where $f(x)$ is the phase function. We prove a theorem which provides estimates…
The definition of infinite dimensional Fresnel integrals is generalized to the case of polynomial phase functions of any degree and applied to the construction of a functional integral representation of the solution of a general class of…
We consider saddle point integrals in d variables whose phase function is neither real nor purely imaginary. Results analogous to those for Laplace (real phase) and Fourier (imaginary phase) integrals hold whenever the phase function is…
The method of expansion of integrals in external parameters is suggested. It is quite universal and works for Feynman integrals both in Euclidean and Minkowski regions of momenta.
The two Fresnel Integrals are real and imaginary part of the integral over complex-valued exp(ix^2) as a function of the upper limit. They are special cases of the integrals over x^m*exp(i*x^n) for integer powers m and n, which are…
We present a methodology for numerically integrating ordinary differential equations containing rapidly oscillatory terms. This challenge is distinct from that for differential equations which have rapidly oscillatory solutions: here the…
A generalized notion of oscillatory integrals that allows for inhomogeneous phase functions of arbitrary positive order is introduced. The wave front set of the resulting distributions is characterized in a way that generalizes the…
We consider a fractional generalization of gradient systems. We use differential forms and exterior derivatives of fractional orders. Examples of fractional gradient systems are considered. We describe the stationary states of these…
In this paper we develop a theory for oscillatory integrals with complex phases. When $f:{\mathbb C}^n \to {\mathbb C}$, we evaluate this phase function on the basic character ${\rm e}(z) := e^{2\pi i x} e^{2\pi i y}$ of ${\mathbb C} \simeq…