共查询到20条相似文献,搜索用时 328 毫秒
1.
《Journal of Computational and Applied Mathematics》2002,147(1):215-231
We derive simple, explicit error bounds for the uniform asymptotic expansion of the incomplete gamma function Γ(a,z) valid for complex values of a and z as |a|→∞. Their evaluation depends on numerically pre-computed bounds for the coefficients ck(η) in the expansion of Γ(a,z) taken along rays in the complex η plane, where η is a variable related to z/a. The bounds are compared with numerical computations of the remainder in the truncated expansion. 相似文献
2.
N. M. Temme 《Constructive Approximation》1986,2(1):369-376
An asymptotic expansion including error bounds is given for polynomials {P n, Qn} that are biorthogonal on the unit circle with respect to the weight function (1?eiθ)α+β(1?e?iθ)α?β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function2 F 1(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion. 相似文献
3.
José Luis López 《Journal of Mathematical Analysis and Applications》2011,377(1):30-42
Asymptotic expansions are given for large values of n of the generalized Bessel polynomials . The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in the z-plane. New forms of expansions in terms of elementary functions valid in sectors not containing the turning points z=±i/n are derived, and a new expansion in terms of modified Bessel functions is given. Earlier asymptotic expansions of the generalized Bessel polynomials by Wong and Zhang (1997) and Dunster (2001) are discussed. 相似文献
4.
Chelo Ferreira 《Journal of Mathematical Analysis and Applications》2004,298(1):210-224
The Hurwitz-Lerch zeta function Φ(z,s,a) is considered for large and small values of a∈C, and for large values of z∈C, with |Arg(a)|<π, z∉[1,∞) and s∈C. This function is originally defined as a power series in z, convergent for |z|<1, s∈C and 1−a∉N. An integral representation is obtained for Φ(z,s,a) which define the analytical continuation of the Hurwitz-Lerch zeta function to the cut complex z-plane C?[1,∞). From this integral we derive three complete asymptotic expansions for either large or small a and large z. These expansions are accompanied by error bounds at any order of the approximation. Numerical experiments show that these bounds are very accurate for real values of the asymptotic variables. 相似文献
5.
The Gauss hypergeometric function 2 F 1(a,b,c;z) can be computed by using the power series in powers of $z, z/(z-1), 1-z, 1/z, 1/(1-z),~\textrm{and}~(z-1)/z$ . With these expansions, 2 F 1(a,b,c;z) is not completely computable for all complex values of z. As pointed out in Gil et al. (2007, §2.3), the points z?=?e ±iπ/3 are always excluded from the domains of convergence of these expansions. Bühring (SIAM J Math Anal 18:884–889, 1987) has given a power series expansion that allows computation at and near these points. But, when b???a is an integer, the coefficients of that expansion become indeterminate and its computation requires a nontrivial limiting process. Moreover, the convergence becomes slower and slower in that case. In this paper, we obtain new expansions of the Gauss hypergeometric function in terms of rational functions of z for which the points z?=?e ±iπ/3 are well inside their domains of convergence. In addition, these expansions are well defined when b???a is an integer and no limits are needed in that case. Numerical computations show that these expansions converge faster than Bühring’s expansion for z in the neighborhood of the points e ±iπ/3, especially when b???a is close to an integer number. 相似文献
6.
The reproducing kernel function of a weighted Bergman space over domains in Cd is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of (complex) codimension 1, which also leads to a series expansion of the reproducing kernel in terms of reproducing kernels defined on the subvariety. The problem of finding the reproducing kernel is thus reduced to the same kind of problem when one of the two entries is on the subvariety. A complete expansion of the reproducing kernel may be achieved in this manner. We carry this out in dimension d=2 for certain classes of weighted Bergman spaces over the bidisk (with the diagonal z1=z2 as subvariety) and the ball (with z2=0 as subvariety), as well as for a weighted Bargmann-Fock space over C2 (with the diagonal z1=z2 as subvariety). 相似文献
7.
R.B. Paris 《Journal of Computational and Applied Mathematics》2010,234(2):488-504
We consider exponentially small expansions present in the asymptotics of the generalised hypergeometric function, or Wright function, pΨq(z) for large |z| that have not been considered in the existing theory. Our interest is principally with those functions of this class that possess either a finite algebraic expansion or no such expansion and with parameter values that produce exponentially small expansions in the neighbourhood of the negative real z axis. Numerical examples are presented to demonstrate the presence of these exponentially small expansions. 相似文献
8.
The Barnes double gamma function G(z) is considered for large argument z. A new integral representation is obtained for log G(z). An asymptotic expansion in decreasing powers of z and uniformly valid for |Arg z|<π is derived from this integral. The expansion is accompanied by an error bound at any order of the approximation. Numerical experiments show that this bound is very accurate for real z. The accuracy of the error bound decreases for increasing Arg z. 相似文献
9.
Andrzej Daniluk 《Integral Equations and Operator Theory》2011,69(3):365-372
We show that if an operator-valued analytic function f of a complex variable attains its maximum modulus at z 0, then the coefficients of the nonconstant terms in the power series expansion about z 0 cannot be invertible, provided a complex uniform convexity condition holds. One application is that the norm of the resolvent of an operator on a complex uniformly convex space cannot have a local maximum. 相似文献
10.
11.
Richard B. Paris 《Lithuanian Mathematical Journal》2014,54(1):82-105
We investigate a particular aspect of the asymptotic expansion of the Wright function pΨq(z) for large |z|. In the case p?=?1, q ? 0, we establish the form of the exponentially small expansion of this function on certain rays in the z-plane (known as Stokes lines). The importance of such exponentially small terms is encountered in analytic probability theory and in the theory of generalised linear models. In addition, the transition of the Stokes multiplier connected with the subdominant exponential expansion across the Stokes lines is shown to obey the familiar error-function smoothing law expressed in terms of an appropriately scaled variable. Some numerical examples which confirm the accuracy of the expansion are given. 相似文献
12.
Let f be a nonconstant entire function and let a be a meromorphic function satisfying T(r,a)=S(r,f) and a?a′. If f(z)=a(z)⇔f′(z)=a(z) and f(z)=a(z)⇒f″(z)=a(z), then f≡f′, and a?a′ is necessary. This extended a result due to Jank, Mues and Volkmann. 相似文献
13.
Olga M. Katkova 《Journal of Mathematical Analysis and Applications》2008,347(1):81-89
A real polynomial is called Hurwitz (stable) if all its zeros have negative real parts. For a given n∈N we find the smallest possible constant dn>0 such that if the coefficients of F(z)=a0+a1z+?+anzn are positive and satisfy the inequalities akak+1>dnak−1ak+2 for k=1,2,…,n−2, then F(z) is Hurwitz. 相似文献
14.
We prove that, for positive integers a, b, c and d with c≠d, a>1, b>1, the number of simultaneous solutions in positive integers to ax2−cz2=1, by2−dz2=1 is at most two. This result is the best possible one. We prove a similar result for the system of equations x2−ay2=1, z2−bx2=1. 相似文献
15.
The standard saddle point method of asymptotic expansions of integrals requires to show the existence of the steepest descent paths of the phase function and the computation of the coefficients of the expansion from a function implicitly defined by solving an inversion problem. This means that the method is not systematic because the steepest descent paths depend on the phase function on hand and there is not a general and explicit formula for the coefficients of the expansion (like in Watson's Lemma for example). We propose a more systematic variant of the method in which the computation of the steepest descent paths is trivial and almost universal: it only depends on the location and the order of the saddle points of the phase function. Moreover, this variant of the method generates an asymptotic expansion given in terms of a generalized (and universal) asymptotic sequence that avoids the computation of the standard coefficients, giving an explicit and systematic formula for the expansion that may be easily implemented on a symbolic manipulation program. As an illustrative example, the well-known asymptotic expansion of the Airy function is rederived almost trivially using this method. New asymptotic expansions of the Hankel function Hn(z) for large n and z are given as non-trivial examples. 相似文献
16.
The main difficulties in the Laplace’s method of asymptotic expansions of integrals are originated by a change of variables. We propose a variant of the method which avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler. On the other hand, the asymptotic sequence is as simple as in the standard Laplace’s method: inverse powers of the asymptotic variable. New asymptotic expansions of the Gamma function Γ(z) for large z and the Gauss hypergeometric function 2F1(a,b,c;z) for large b and c are given as illustrations. An explicit formula for the coefficients of the classical Stirling expansion of Γ(z) is also given. 相似文献
17.
《Journal of Computational and Applied Mathematics》2003,155(1):153-162
The classical Eneström–Kakeya Theorem states that if p(z)=∑v=0navzv is a polynomial satisfying 0⩽a0⩽a1⩽⋯⩽an, then all of the zeros of p(z) lie in the region |z|⩽1 in the complex plane. Many generalizations of the Eneström–Kakeya theorem exist which put various conditions on the coefficients of the polynomial (such as monotonicity of the moduli of the coefficients). We will introduce several results which put conditions on the coefficients of even powers of z and on the coefficients of odd powers of z. As a consequence, our results will be applicable to some polynomials to which these related results are not applicable. 相似文献
18.
Christopher Hammond 《Journal of Mathematical Analysis and Applications》2005,303(2):499-508
We obtain a representation for the norm of a composition operator on the Dirichlet space induced by a map of the form φ(z)=az+b. We compare this result to an upper bound for ‖Cφ‖ that is valid whenever φ is univalent. Our work relies heavily on an adjoint formula recently discovered by Gallardo-Gutiérrez and Montes-Rodríguez. 相似文献
19.
D. S. Lubinsky 《Constructive Approximation》1987,3(1):307-330
Given a formal power seriesf(z)?∑ j=0 ∞ a j z j for which the quantitya j ?1a j +1/a j 2 has a prescribed asymptotic behavior asj→∞, we obtain the asymptotic behavior of poles of rows of the Padé table, and the associated Toeplitz determinants. In particular, we can show for large classes of entire functions of zero, finite, and infinite order (including the Mittag-Leffler functions) and forn=1,2,3,..., that the poles of [m/n](z) approach ∞ with ratea m /a m+1 asm→∞. 相似文献
20.
In this paper, by studying the properties of meromorphic functions which have few zeros and poles, we find all the entire functions f(z) which share a small and finite order meromorphic function a(z) with its derivative, and f(n)(z)−a(z)=0 whenever f(z)−a(z)=0 (n?2). This result is a generalization of several previous results. 相似文献