Expansions of Oscillatory Integrals Sample Clauses

Expansions of Oscillatory Integrals. ‌ As mentioned above, by combining Lemma 4.5 and Lemma 4.6 one can obtain an expansion for the integral (4.2). Since our final goal is to obtain an asymptotic expansion for this integral, we need to be able to obtain a sharp estimate for the remainder R(M)(τ ) defined in the previous section. Because of the importance of the information contained in the remainder terms for our future purposes, we derive a precise expression for this factor. We will make use of these expressions in the next chapter. The following theorem is the central result of this chapter. It gives a prescription on how to obtain an expansion for (4.2). It can be regarded as a stationary phase formula for (4.2) and the similarity with Theorem 4.1 is evident. ρ,δ
Sample 1
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