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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2008 • Question 10

Question

Math 6420 • PDE Fall 2008 • Question 10

Find the leading term of the asymptotic expansion of the integral \[ I(s)=\int _{-\infty }^{\infty } e^{is\left (t+\frac {t^3}{3}\right )}\,dt, \qquad s\text { is real and }s\to +\infty , \] as well as a rigorous estimate of the error.

Raw LaTeX

Find the leading term of the asymptotic expansion of the integral
\[
  I(s)=\int_{-\infty}^{\infty} e^{is\left(t+\frac{t^3}{3}\right)}\,dt,
  \qquad s\text{ is real and }s\to+\infty,
\]
as well as a rigorous estimate of the error.

Topic Tags

Cauchy Integral Formula and Integral Theorems Linear Programming and Optimization Real Integral Evaluation by Complex Methods Watson's Lemma and Laplace's Method