# Uniform Asymptotic Methods for Integrals

@article{Temme2013UniformAM, title={Uniform Asymptotic Methods for Integrals}, author={Nico M. Temme}, journal={Indagationes Mathematicae}, year={2013}, volume={24}, pages={739-765} }

We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson’s lemma, Laplace’s method, the saddle point method, and the method of stationary phase. Certain developments in the field of asymptotic analysis will be compared with De Bruijn’s book Asymptotic Methods in Analysis. The classical methods can be modified for obtaining expansions that hold uniformly with respect to additional parameters. We give an overview of examples in which special…

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## References

SHOWING 1-10 OF 49 REFERENCES

Uniform Airy-type expansions of integrals

- Mathematics, Physics
- 1990

A new method for representing the remainder and coefficients in Airy-type expansions of integrals is given. The quantities are written in terms of Cauchy-type integrals and are natural…

Large degree asymptotics of generalized Bernoulli and Euler polynomials

- Mathematics
- 2010

Abstract Asymptotic expansions are given for large values of n of the generalized Bernoulli polynomials B n μ ( z ) and Euler polynomials E n μ ( z ) . In a previous paper Lopez and Temme (1999)…

Uniform asymptotic expansions of confluent hypergeometric functions

- Mathematics
- 1978

IN A RECENT paper (Temme, 1975) new asymptotic expansions for the incomplete gamma functions y(oc, x) and J(oc, x) were derived. The methods immediately lead to results for the confluent…

Asymptotic expansions for second-order linear difference equations with a turning point

- Computer Science, MathematicsNumerische Mathematik
- 2003

Summary. A turning-point theory is developed for the second-order difference equation where the coefficients An and Bn have asymptotic expansions of the form θ≠0 being a real number. In particular,…

Asymptotic Expansions of Mellin Convolution Integrals

- Computer Science, MathematicsSIAM Rev.
- 2008

A new method for deriving asymptotic expansions of f(t)h(xt)dt for small x, which is a very general technique that unifies a certain set of asymPTotic methods.

Integrals with a large parameter: Legendre functions of large degree and fixed order

- Mathematics
- 1984

Suppose that a function f ( z , n ) depends on a large parameter n. A proposed expression g ( z, n ) is an asymptotic approximation for f ( z , n ) if it can be shown that the error (i.e. the…

Asymptotic approximations of integrals

- Mathematics, Computer ScienceClassics in applied mathematics
- 2001

The basic concepts of asymptotic expansions, Mellin transform techniques, and the distributional approach are explained.

Two-point Taylor expansions of analytic functions

- Mathematics
- 2002

textabstractTaylor 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…

Asymptotics and Mellin-Barnes Integrals

- Mathematics
- 2001

Asymptotics and Mellin-Barnes Integrals provides an account of the use and properties of a type of complex integral representation that arises frequently in the study of special functions typically…

The uniform asymptotic expansion of a class of integrals related to cumulative distribution functions

- Mathematics
- 1981

An asymptotic expansion is given for a class of integrals for large values of a parameter, which corresponds with the degrees of freedom in a certain type of cumulative distribution functions. The…