Multi-point Taylor expansions of analytic functions

Authors:
José L. López and Nico M. Temme

Journal:
Trans. Amer. Math. Soc. **356** (2004), 4323-4342

MSC (2000):
Primary 30B10, 30E20; Secondary 40A30

Published electronically:
May 28, 2004

MathSciNet review:
2067121

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.

**1.**C. Chester, B. Friedman, and F. Ursell,*An extension of the method of steepest descents*, Proc. Cambridge Philos. Soc.**53**(1957), 599–611. MR**0090690****2.**Kathy A. Driver and Nico M. Temme,*On polynomials related with Hermite-Padé approximations to the exponential function*, J. Approx. Theory**95**(1998), no. 1, 101–122. MR**1645979**, 10.1006/jath.1998.3195**3.**A. J. E. M. Janssen,*On the asymptotics of some Pearcey-type integrals*, J. Phys. A**25**(1992), no. 13, L823–L831. MR**1172064****4.**N. P. Kirk, J. N. L. Connor, P. R. Curtis, and C. A. Hobbs,*Theory of axially symmetric cusped focusing: numerical evaluation of a Bessoid integral by an adaptive contour algorithm*, J. Phys. A**33**(2000), no. 26, 4797–4808. MR**1792900**, 10.1088/0305-4470/33/26/306**5.**José L. Lopez and Nico M. Temme,*Asymptotic expansions of Charlier, Laguerre and Jacobi polynomials.*Accepted for publication in The Proceedings of the Royal Society of Edinburgh A (Mathematics).**6.**José L. López and Nico M. Temme,*Two-point Taylor expansions of analytic functions*, Stud. Appl. Math.**109**(2002), no. 4, 297–311. MR**1934653**, 10.1111/1467-9590.00225**7.**Raimundas Vidunas and Nico M. Temme,*Symbolic evaluation of coefficients in Airy-type asymptotic expansions*, J. Math. Anal. Appl.**269**(2002), no. 1, 317–331. MR**1907888**, 10.1016/S0022-247X(02)00026-4**8.**J. L. Walsh,*Interpolation and approximation by rational functions in the complex domain*, Third edition. American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1960. MR**0218587****9.**R. Wong,*Asymptotic approximations of integrals*, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1989. MR**1016818**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
30B10,
30E20,
40A30

Retrieve articles in all journals with MSC (2000): 30B10, 30E20, 40A30

Additional Information

**José L. López**

Affiliation:
Departamento de Matématica e Informática, Universidad Pública de Navarra, 31006-Pamplona, Spain

Email:
jl.lopez@unavarra.es

**Nico M. Temme**

Affiliation:
CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

Email:
nicot@cwi.nl

DOI:
https://doi.org/10.1090/S0002-9947-04-03619-0

Keywords:
Multi-point Taylor expansions,
Cauchy's theorem,
analytic functions,
multi-point Laurent expansions,
uniform asymptotic expansions of integrals

Received by editor(s):
November 14, 2002

Published electronically:
May 28, 2004

Additional Notes:
The first author thanks the saving bank Caja Rural de Navarra for its financial support. He also acknowledges the scientific and financial support of CWI in Amsterdam

The authors thank the referee for the comments on the first version of the paper

Article copyright:
© Copyright 2004
American Mathematical Society