By James R. Kirkwood

Presents creation to research of real-valued features of 1 variable. this article is for a student's first summary arithmetic direction. Writing kind is much less formal and fabric provided in a fashion such that the scholar can boost an instinct for the topic and procure a few event in developing proofs. The slower velocity of the topic and the eye given to examples are supposed to ease the student's transition from computational to theoretical arithmetic.

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**Example text**

If the integral has multiple stationary points, we can write it as multiple integrals of this form. The method used in the previous section does not work as the stationary point becomes a singularity when we attempt to integrate by parts. Fortunately, it is possible to remove the singularity (for simplicity we assume that r = 2): I[f ] = I[f − f (ξ)] + f (ξ)I[1] b 1 f (x) − f (ξ) d iω g(x) dx + f (ξ)I[1] e iω a g (x) dx 1 f (b) − f (ξ) iω g(b) f (a) − f (ξ) iω g(a) = e e − iω g (b) g (a) 1 d f (x) − f (ξ) − I + f (ξ)I[1] .

2 Asymptotics Whereas standard quadrature schemes are ineﬃcient, a straightforward alternative exists in the form of asymptotic expansions. Asymptotic expansions actually improve with accuracy as the frequency increases, and—assuming suﬃcient diﬀerentiability of f and g—to arbitrarily high order. Furthermore the number of operations required to produce such an expansion is independent of the frequency, and extraordinarily small. Even more surprising is that this is all obtained by only requiring knowledge of the function at very few critical points within the interval—the endpoints and stationary points—as well as its derivatives at these points if higher asymptotic orders are required.

Math. Phys 8, 241–249. [Bru03] O. P. Bruno (2003). Fast, high-order, high-frequency integral methods for computational acoustics and electromagnetics, in: Lecture Notes in Computational Science and Engineering 31: Topics in Computational Wave Propagation (Springer, Berlin). [DH08] A. Dea˜ no, & D. Huybrechs (2008). Complex Gaussian quadrature of oscillatory integrals. Technical Report 2008/NA04, DAMTP, University of Cambridge. [Eva94] G. A. Evans (1994). An alternative method for irregular oscillatory integrals over a ﬁnite range, International Journal of Computer Mathematics 52(3), 185–193.