We consider the asymptotic method designed by F. Olver [Olver, 1974] for linear differential equa... more We consider the asymptotic method designed by F. Olver [Olver, 1974] for linear differential equations of the second order containing a large (asymptotic) parameter Λ: x m y ′′ −Λ 2 y = g(x)y, with m ∈ Z and g continuous. Olver studies in detail the cases m = 2, specially the cases m = 0, ±1, giving the Poincaré-type asymptotic expansion of two independent solutions of the equation. The case m = 2 is different, as the behavior of the solutions for large Λ is not of exponential type, but of power type. In this case, Olver's theory does not give as many details as it gives in the cases m = 2. Then, we consider here the special case m = 2. We propose two different techniques to handle the problem: (i) a modification of Olver's method that replaces the role of the exponential approximations by power approximations and (ii) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter.
A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functio... more A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functions (or shape modes) to represent optical surfaces. The method is general so it can be applied to different initial shapes and different polynomials. Here we present results for surfaces with circular apertures when the first basis function (mode) is a conicoid. The system for aspheres with rotational symmetry is obtained applying an appropriate change of variables to Legendre polynomials, whereas the system for general freeform case is obtained applying a similar procedure to spherical harmonics. Numerical comparisons with standard systems, such as Forbes and Zernike polynomials, are performed and discussed.
Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertu... more Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit disk. In [Diaz et all, 2014] we introduced a new Zernike basis for elliptic and annular optical apertures based on an appropriate diffeomorphism between the unit disk and the ellipse and the annulus. Here, we present a generalization of this Zernike basis for a variety of important optical apertures, paying special attention to polygons and the polygonal facets present in segmented mirror telescopes. On the contrary to ad hoc solutions, most of them based on the Gram-Smith orthonormalization method, here we consider a piece-wise diffeomorphism that transforms the unit disk into the polygon under consideration. We use this mapping to define a Zernike-like orthonormal system over the polygon. We also consider ensembles of polygonal facets that are essential in the design of segmented mirror telescopes. This generalization, based on in-plane warping of the basis functions, provides a unique solution, and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both, the general form and the explicit expressions for a typical example of telescope optical aperture are provided. 2010 AMS Mathematics Subject Classification: 41A10 (Approximation by polynomials); 41A63 (Multidimensional problems); 78M99 (None of the above, but in this section: Optics, electromagnetic theory); 85-08 (Computational methods: Astronomy and astrophysics).
Electronic Journal of Qualitative Theory of Differential Equations, 2020
We consider the second-order linear differential equation (x 2 − 1)y + f (x)y + g(x)y = h(x) in t... more We consider the second-order linear differential equation (x 2 − 1)y + f (x)y + g(x)y = h(x) in the interval (−1, 1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet-Neumann). The functions f , g and h are analytic in a Cassini disk D r with foci at x = ±1 containing the interval [−1, 1]. Then, the two end points of the interval may be regular singular points of the differential equation. The two-point Taylor expansion of the solution y(x) at the end points ±1 is used to study the space of analytic solutions in D r of the differential equation, and to give a criterion for the existence and uniqueness of analytic solutions of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the analytic solutions when they exist.
Journal of Integral Equations and Applications, Mar 1, 2015
We consider the asymptotic method designed by Olver [6] for linear differential equations of seco... more We consider the asymptotic method designed by Olver [6] for linear differential equations of second order containing a large (asymptotic) parameter Λ, in particular, the second and third cases studied by Olver: differential equations with a turning point (second case) or a singular point (third case). It is well known that his method gives the Poincaré-type asymptotic expansion of two independent solutions of the equation in inverse powers of Λ. In this paper, we add initial conditions to the differential equation and consider the corresponding initial value problem. By using the Green's function of an auxiliary problem, we transform the initial value problem into a Volterra integral equation of the second kind. Then, using a fixed point theorem, we construct a sequence of functions that converges to the unique solution of the problem. This sequence also has the property of being an asymptotic expansion for large Λ (not of Poincaré-type) of the solution of the problem. Moreover, we show that the technique also works for nonlinear differential equations with a large parameter. 2010 AMS Mathematics subject classification. Primary 34A12, 34B27, 41A58, 41A60, 45D05. Keywords and phrases. Second order differential equations, turning points, regular singular points, Volterra integral equations of the second kind, asymptotic expansions, Green functions, fixed point theorems, airy functions, Bessel functions. The Dirección General de Ciencia y Tecnología (REF. MTM2010-21037) is acknowledged for its financial support.
Electronic Transactions on Numerical Analysis, 2018
We consider the incomplete beta function Bz(a, b) in the maximum domain of analyticity of its thr... more We consider the incomplete beta function Bz(a, b) in the maximum domain of analyticity of its three variables: a, b, z ∈ C, −a / ∈ N, z / ∈ [1, ∞). For b ≤ 1 we derive a convergent expansion of z −a Bz(a, b) in terms of the function (1 − z) b and of rational functions of z that is uniformly valid for z in any compact set in C \ [1, ∞). When −b ∈ N ∪ {0}, the expansion also contains a logarithmic term of the form log(1 − z). For b ≥ 1 we derive a convergent expansion of z −a (1 − z) b Bz(a, b) in terms of the function (1 − z) b and of rational functions of z that is uniformly valid for z in any compact set in the exterior of the circle |z − 1| = r for arbitrary r > 0. The expansions are accompanied by realistic error bounds. Some numerical experiments show the accuracy of the approximations.
Mediterranean Journal of Mathematics, Feb 17, 2023
We derive a new analytic representation of the error function erf z in the form of a convergent s... more We derive a new analytic representation of the error function erf z in the form of a convergent series whose terms are exponential and rational functions. The expansion holds uniformly in z in the double sector | arg(±z)| < π/4. The expansion is accompanied by realistic error bounds.
We consider the highly oscillatory integral () ∶= ∫ ∞ −∞ (+2 +) () for large positive values of ,... more We consider the highly oscillatory integral () ∶= ∫ ∞ −∞ (+2 +) () for large positive values of , This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.
A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functio... more A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functions (or shape modes) to represent optical surfaces. The method is general so it can be applied to different initial shapes and different polynomials. Here we present results for surfaces with circular apertures when the first basis function (mode) is a conicoid. The system for aspheres with rotational symmetry is obtained applying an appropriate change of variables to Legendre polynomials, whereas the system for general freeform case is obtained applying a similar procedure to spherical harmonics. Numerical comparisons with standard systems, such as Forbes and Zernike polynomials, are performed and discussed.
Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertu... more Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit disk. In [Diaz et all, 2014] we introduced a new Zernike basis for elliptic and annular optical apertures based on an appropriate diffeomorphism between the unit disk and the ellipse and the annulus. Here, we present a generalization of this Zernike basis for a variety of important optical apertures, paying special attention to polygons and the polygonal facets present in segmented mirror telescopes. On the contrary to ad hoc solutions, most of them based on the Gram-Smith orthonormalization method, here we consider a piece-wise diffeomorphism that transforms the unit disk into the polygon under consideration. We use this mapping to define a Zernike-like orthonormal system over the polygon. We also consider ensembles of polygonal facets that are essential in the design of segmented mirror telescopes. This generalization, based on in-plane warping of the basis functions, provides a unique solution, and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both, the general form and the explicit expressions for a typical example of telescope optical aperture are provided. 2010 AMS Mathematics Subject Classification: 41A10 (Approximation by polynomials); 41A63 (Multidimensional problems); 78M99 (None of the above, but in this section: Optics, electromagnetic theory); 85-08 (Computational methods: Astronomy and astrophysics).
We consider the asymptotic method designed by F. Olver [Olver, 1974] for linear differential equa... more We consider the asymptotic method designed by F. Olver [Olver, 1974] for linear differential equations of the second order containing a large (asymptotic) parameter Λ: x m y ′′ −Λ 2 y = g(x)y, with m ∈ Z and g continuous. Olver studies in detail the cases m = 2, specially the cases m = 0, ±1, giving the Poincaré-type asymptotic expansion of two independent solutions of the equation. The case m = 2 is different, as the behavior of the solutions for large Λ is not of exponential type, but of power type. In this case, Olver's theory does not give as many details as it gives in the cases m = 2. Then, we consider here the special case m = 2. We propose two different techniques to handle the problem: (i) a modification of Olver's method that replaces the role of the exponential approximations by power approximations and (ii) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter.
We consider initial value problems of the form..., where f : [−a, b] × U → Cn is a continuous fun... more We consider initial value problems of the form..., where f : [−a, b] × U → Cn is a continuous function in its variables and U ⊂ Cn is an open set. D(x) is an n × n diagonal matrix whose first n − m diagonal entries are 1 and the last m diagonal entries are x, with m = 0, 1, 2, . . . or n. This is an initial value problem where the initial condition is given at a regular singular point of the system of differential equations. The main result of this paper is an existence and uniqueness theorem for the solution of this initial value problem. It is shown that this problem has a unique solution and the Picard-Lindelof’s expansion converges to that solution if the function ¨ F(y, x) := xD−1 (x)f (x, y) is Lipschitz continuous in the variables y with Lipschitz constant L of the form L = N + Mxp for a certain p > 0, M > 0 and 0 ≤ N < 1. When we add the condition y (s) ∈ C[−a, b], s ∈ N, to the formulation of the problem and the Taylor polynomial of y at x = 0 and degree s − 1 is a...
We consider the swallowtail integral Ψ(x, y, z) := ∞ −∞ e i(t 5 +xt 3 +yt 2 +zt) dt for large val... more We consider the swallowtail integral Ψ(x, y, z) := ∞ −∞ e i(t 5 +xt 3 +yt 2 +zt) dt for large values of |x| and bounded values of |y| and |z|. The integrand of the swallowtail integral oscillates wildly in this region and the asymptotic analysis is subtle. The standard saddle point method is complicated and then we use the simplified saddle point method introduced in [López et al., 2009]. The analysis is more straightforward with this method and it is possible to derive complete asymptotic expansions of Ψ(x, y, z) for large |x| and fixed y and z. The asymptotic analysis requires the study of three different regions for arg x separated by three Stokes lines. The expansion is given in terms of inverse powers of x 1 3 and x 1 2 and the coefficients are elementary functions of y and z. The accuracy and the asymptotic character of the approximations is illustrated with some numerical experiments.
ETNA - Electronic Transactions on Numerical Analysis, 2020
We analyze the asymptotic behavior of the swallowtail integral ∞ −∞ e i(t 5 +xt 3 +yt 2 +zt) dt f... more We analyze the asymptotic behavior of the swallowtail integral ∞ −∞ e i(t 5 +xt 3 +yt 2 +zt) dt for large values of |y| and bounded values of |x| and |z|. We use the simplified saddle point method introduced in [López et al., J. Math. Anal. Appl., 354 (2009), pp. 347-359]. With this method, the analysis is more straightforward than with the standard saddle point method, and it is possible to derive complete asymptotic expansions of the integral for large |y| and fixed x and z. There are four Stokes lines in the sector (−π, π] that divide the complex y-plane into four sectors in which the swallowtail integral behaves differently when |y| is large. The asymptotic approximation is the sum of two asymptotic series whose terms are elementary functions of x, y, and z. One of them is of Poincaré type and is given in terms of inverse powers of y 1/2. The other one is given in terms of an asymptotic sequence whose terms are of the order of inverse powers of y 1/9 when |y| → ∞, and it is multiplied by an exponential factor that behaves differently in the four mentioned sectors. Some numerical experiments illustrate the accuracy of the approximation.
Journal of Mathematical Analysis and Applications, 2018
We consider the second-order linear differential equation (x + 1)y ′′ + f (x)y ′ + g(x)y = h(x) i... more We consider the second-order linear differential equation (x + 1)y ′′ + f (x)y ′ + g(x)y = h(x) in the interval (−1, 1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet-Neumann). The functions f (x), g(x) and h(x) are analytic in a Cassini disk D r with foci at x = ±1 containing the interval [−1, 1]. Then, the end point of the interval x = −1 may be a regular singular point of the differential equation. The two-point Taylor expansion of the solution y(x) at the end points ±1 is used to study the space of analytic solutions in D r of the differential equation, and to give a criterion for the existence and uniqueness of analytic solutions of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the analytic solutions when they exist.
We revise Laplace's and Steepest Descents methods of asymptotic expansions of integrals. The main... more We revise Laplace's and Steepest Descents methods of asymptotic expansions of integrals. The main difficulties in these methods are originated by a change of variables and an eventual deformation of the integration contour. We present a simplification of these methods that only requires an expansion of the integrand at the critical point(s). In this way, the calculation of the coefficients of the asymptotic expansion is simpler. The simplification in the case of several relevant critical points is even more significant and requires multi-point Taylor expansions. The new method that we present here unifies Laplace's and Steepest Descents methods in one unique formulation. Uniformity properties of the method are discussed. Asymptotic expansions of the Bernoulli and Jacobi polynomials and of a generalized confluent hypergeometric function are given as illustration.
We consider the asymptotic method designed by F. Olver [Olver, 1974] for linear differential equa... more We consider the asymptotic method designed by F. Olver [Olver, 1974] for linear differential equations of the second order containing a large (asymptotic) parameter Λ: x m y ′′ −Λ 2 y = g(x)y, with m ∈ Z and g continuous. Olver studies in detail the cases m = 2, specially the cases m = 0, ±1, giving the Poincaré-type asymptotic expansion of two independent solutions of the equation. The case m = 2 is different, as the behavior of the solutions for large Λ is not of exponential type, but of power type. In this case, Olver's theory does not give as many details as it gives in the cases m = 2. Then, we consider here the special case m = 2. We propose two different techniques to handle the problem: (i) a modification of Olver's method that replaces the role of the exponential approximations by power approximations and (ii) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter.
A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functio... more A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functions (or shape modes) to represent optical surfaces. The method is general so it can be applied to different initial shapes and different polynomials. Here we present results for surfaces with circular apertures when the first basis function (mode) is a conicoid. The system for aspheres with rotational symmetry is obtained applying an appropriate change of variables to Legendre polynomials, whereas the system for general freeform case is obtained applying a similar procedure to spherical harmonics. Numerical comparisons with standard systems, such as Forbes and Zernike polynomials, are performed and discussed.
Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertu... more Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit disk. In [Diaz et all, 2014] we introduced a new Zernike basis for elliptic and annular optical apertures based on an appropriate diffeomorphism between the unit disk and the ellipse and the annulus. Here, we present a generalization of this Zernike basis for a variety of important optical apertures, paying special attention to polygons and the polygonal facets present in segmented mirror telescopes. On the contrary to ad hoc solutions, most of them based on the Gram-Smith orthonormalization method, here we consider a piece-wise diffeomorphism that transforms the unit disk into the polygon under consideration. We use this mapping to define a Zernike-like orthonormal system over the polygon. We also consider ensembles of polygonal facets that are essential in the design of segmented mirror telescopes. This generalization, based on in-plane warping of the basis functions, provides a unique solution, and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both, the general form and the explicit expressions for a typical example of telescope optical aperture are provided. 2010 AMS Mathematics Subject Classification: 41A10 (Approximation by polynomials); 41A63 (Multidimensional problems); 78M99 (None of the above, but in this section: Optics, electromagnetic theory); 85-08 (Computational methods: Astronomy and astrophysics).
Electronic Journal of Qualitative Theory of Differential Equations, 2020
We consider the second-order linear differential equation (x 2 − 1)y + f (x)y + g(x)y = h(x) in t... more We consider the second-order linear differential equation (x 2 − 1)y + f (x)y + g(x)y = h(x) in the interval (−1, 1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet-Neumann). The functions f , g and h are analytic in a Cassini disk D r with foci at x = ±1 containing the interval [−1, 1]. Then, the two end points of the interval may be regular singular points of the differential equation. The two-point Taylor expansion of the solution y(x) at the end points ±1 is used to study the space of analytic solutions in D r of the differential equation, and to give a criterion for the existence and uniqueness of analytic solutions of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the analytic solutions when they exist.
Journal of Integral Equations and Applications, Mar 1, 2015
We consider the asymptotic method designed by Olver [6] for linear differential equations of seco... more We consider the asymptotic method designed by Olver [6] for linear differential equations of second order containing a large (asymptotic) parameter Λ, in particular, the second and third cases studied by Olver: differential equations with a turning point (second case) or a singular point (third case). It is well known that his method gives the Poincaré-type asymptotic expansion of two independent solutions of the equation in inverse powers of Λ. In this paper, we add initial conditions to the differential equation and consider the corresponding initial value problem. By using the Green's function of an auxiliary problem, we transform the initial value problem into a Volterra integral equation of the second kind. Then, using a fixed point theorem, we construct a sequence of functions that converges to the unique solution of the problem. This sequence also has the property of being an asymptotic expansion for large Λ (not of Poincaré-type) of the solution of the problem. Moreover, we show that the technique also works for nonlinear differential equations with a large parameter. 2010 AMS Mathematics subject classification. Primary 34A12, 34B27, 41A58, 41A60, 45D05. Keywords and phrases. Second order differential equations, turning points, regular singular points, Volterra integral equations of the second kind, asymptotic expansions, Green functions, fixed point theorems, airy functions, Bessel functions. The Dirección General de Ciencia y Tecnología (REF. MTM2010-21037) is acknowledged for its financial support.
Electronic Transactions on Numerical Analysis, 2018
We consider the incomplete beta function Bz(a, b) in the maximum domain of analyticity of its thr... more We consider the incomplete beta function Bz(a, b) in the maximum domain of analyticity of its three variables: a, b, z ∈ C, −a / ∈ N, z / ∈ [1, ∞). For b ≤ 1 we derive a convergent expansion of z −a Bz(a, b) in terms of the function (1 − z) b and of rational functions of z that is uniformly valid for z in any compact set in C \ [1, ∞). When −b ∈ N ∪ {0}, the expansion also contains a logarithmic term of the form log(1 − z). For b ≥ 1 we derive a convergent expansion of z −a (1 − z) b Bz(a, b) in terms of the function (1 − z) b and of rational functions of z that is uniformly valid for z in any compact set in the exterior of the circle |z − 1| = r for arbitrary r > 0. The expansions are accompanied by realistic error bounds. Some numerical experiments show the accuracy of the approximations.
Mediterranean Journal of Mathematics, Feb 17, 2023
We derive a new analytic representation of the error function erf z in the form of a convergent s... more We derive a new analytic representation of the error function erf z in the form of a convergent series whose terms are exponential and rational functions. The expansion holds uniformly in z in the double sector | arg(±z)| < π/4. The expansion is accompanied by realistic error bounds.
We consider the highly oscillatory integral () ∶= ∫ ∞ −∞ (+2 +) () for large positive values of ,... more We consider the highly oscillatory integral () ∶= ∫ ∞ −∞ (+2 +) () for large positive values of , This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.
A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functio... more A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functions (or shape modes) to represent optical surfaces. The method is general so it can be applied to different initial shapes and different polynomials. Here we present results for surfaces with circular apertures when the first basis function (mode) is a conicoid. The system for aspheres with rotational symmetry is obtained applying an appropriate change of variables to Legendre polynomials, whereas the system for general freeform case is obtained applying a similar procedure to spherical harmonics. Numerical comparisons with standard systems, such as Forbes and Zernike polynomials, are performed and discussed.
Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertu... more Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit disk. In [Diaz et all, 2014] we introduced a new Zernike basis for elliptic and annular optical apertures based on an appropriate diffeomorphism between the unit disk and the ellipse and the annulus. Here, we present a generalization of this Zernike basis for a variety of important optical apertures, paying special attention to polygons and the polygonal facets present in segmented mirror telescopes. On the contrary to ad hoc solutions, most of them based on the Gram-Smith orthonormalization method, here we consider a piece-wise diffeomorphism that transforms the unit disk into the polygon under consideration. We use this mapping to define a Zernike-like orthonormal system over the polygon. We also consider ensembles of polygonal facets that are essential in the design of segmented mirror telescopes. This generalization, based on in-plane warping of the basis functions, provides a unique solution, and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both, the general form and the explicit expressions for a typical example of telescope optical aperture are provided. 2010 AMS Mathematics Subject Classification: 41A10 (Approximation by polynomials); 41A63 (Multidimensional problems); 78M99 (None of the above, but in this section: Optics, electromagnetic theory); 85-08 (Computational methods: Astronomy and astrophysics).
We consider the asymptotic method designed by F. Olver [Olver, 1974] for linear differential equa... more We consider the asymptotic method designed by F. Olver [Olver, 1974] for linear differential equations of the second order containing a large (asymptotic) parameter Λ: x m y ′′ −Λ 2 y = g(x)y, with m ∈ Z and g continuous. Olver studies in detail the cases m = 2, specially the cases m = 0, ±1, giving the Poincaré-type asymptotic expansion of two independent solutions of the equation. The case m = 2 is different, as the behavior of the solutions for large Λ is not of exponential type, but of power type. In this case, Olver's theory does not give as many details as it gives in the cases m = 2. Then, we consider here the special case m = 2. We propose two different techniques to handle the problem: (i) a modification of Olver's method that replaces the role of the exponential approximations by power approximations and (ii) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter.
We consider initial value problems of the form..., where f : [−a, b] × U → Cn is a continuous fun... more We consider initial value problems of the form..., where f : [−a, b] × U → Cn is a continuous function in its variables and U ⊂ Cn is an open set. D(x) is an n × n diagonal matrix whose first n − m diagonal entries are 1 and the last m diagonal entries are x, with m = 0, 1, 2, . . . or n. This is an initial value problem where the initial condition is given at a regular singular point of the system of differential equations. The main result of this paper is an existence and uniqueness theorem for the solution of this initial value problem. It is shown that this problem has a unique solution and the Picard-Lindelof’s expansion converges to that solution if the function ¨ F(y, x) := xD−1 (x)f (x, y) is Lipschitz continuous in the variables y with Lipschitz constant L of the form L = N + Mxp for a certain p > 0, M > 0 and 0 ≤ N < 1. When we add the condition y (s) ∈ C[−a, b], s ∈ N, to the formulation of the problem and the Taylor polynomial of y at x = 0 and degree s − 1 is a...
We consider the swallowtail integral Ψ(x, y, z) := ∞ −∞ e i(t 5 +xt 3 +yt 2 +zt) dt for large val... more We consider the swallowtail integral Ψ(x, y, z) := ∞ −∞ e i(t 5 +xt 3 +yt 2 +zt) dt for large values of |x| and bounded values of |y| and |z|. The integrand of the swallowtail integral oscillates wildly in this region and the asymptotic analysis is subtle. The standard saddle point method is complicated and then we use the simplified saddle point method introduced in [López et al., 2009]. The analysis is more straightforward with this method and it is possible to derive complete asymptotic expansions of Ψ(x, y, z) for large |x| and fixed y and z. The asymptotic analysis requires the study of three different regions for arg x separated by three Stokes lines. The expansion is given in terms of inverse powers of x 1 3 and x 1 2 and the coefficients are elementary functions of y and z. The accuracy and the asymptotic character of the approximations is illustrated with some numerical experiments.
ETNA - Electronic Transactions on Numerical Analysis, 2020
We analyze the asymptotic behavior of the swallowtail integral ∞ −∞ e i(t 5 +xt 3 +yt 2 +zt) dt f... more We analyze the asymptotic behavior of the swallowtail integral ∞ −∞ e i(t 5 +xt 3 +yt 2 +zt) dt for large values of |y| and bounded values of |x| and |z|. We use the simplified saddle point method introduced in [López et al., J. Math. Anal. Appl., 354 (2009), pp. 347-359]. With this method, the analysis is more straightforward than with the standard saddle point method, and it is possible to derive complete asymptotic expansions of the integral for large |y| and fixed x and z. There are four Stokes lines in the sector (−π, π] that divide the complex y-plane into four sectors in which the swallowtail integral behaves differently when |y| is large. The asymptotic approximation is the sum of two asymptotic series whose terms are elementary functions of x, y, and z. One of them is of Poincaré type and is given in terms of inverse powers of y 1/2. The other one is given in terms of an asymptotic sequence whose terms are of the order of inverse powers of y 1/9 when |y| → ∞, and it is multiplied by an exponential factor that behaves differently in the four mentioned sectors. Some numerical experiments illustrate the accuracy of the approximation.
Journal of Mathematical Analysis and Applications, 2018
We consider the second-order linear differential equation (x + 1)y ′′ + f (x)y ′ + g(x)y = h(x) i... more We consider the second-order linear differential equation (x + 1)y ′′ + f (x)y ′ + g(x)y = h(x) in the interval (−1, 1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet-Neumann). The functions f (x), g(x) and h(x) are analytic in a Cassini disk D r with foci at x = ±1 containing the interval [−1, 1]. Then, the end point of the interval x = −1 may be a regular singular point of the differential equation. The two-point Taylor expansion of the solution y(x) at the end points ±1 is used to study the space of analytic solutions in D r of the differential equation, and to give a criterion for the existence and uniqueness of analytic solutions of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the analytic solutions when they exist.
We revise Laplace's and Steepest Descents methods of asymptotic expansions of integrals. The main... more We revise Laplace's and Steepest Descents methods of asymptotic expansions of integrals. The main difficulties in these methods are originated by a change of variables and an eventual deformation of the integration contour. We present a simplification of these methods that only requires an expansion of the integrand at the critical point(s). In this way, the calculation of the coefficients of the asymptotic expansion is simpler. The simplification in the case of several relevant critical points is even more significant and requires multi-point Taylor expansions. The new method that we present here unifies Laplace's and Steepest Descents methods in one unique formulation. Uniformity properties of the method are discussed. Asymptotic expansions of the Bernoulli and Jacobi polynomials and of a generalized confluent hypergeometric function are given as illustration.
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