Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2001 • Question 10

Question

Math 6420 • PDE Fall 2001 • Question 10

  1. Find the leading order behavior of \[ G(x)=\int _0^\infty \cos \left (x\left (\frac {t^3}{3}-t\right )\right )\,dt \] for large \(x\).
  2. Outline the method to find more terms in the asymptotic expansion of \(G(x)\).

Useful information: \[ \int _{-\infty }^{\infty }\cos (ax^2)\,dx =\int _{-\infty }^{\infty }\sin (ax^2)\,dx =\sqrt {\frac {\pi }{2a}}. \]

Raw LaTeX

\begin{enumerate}
\item[(a)] Find the leading order behavior of
\[
  G(x)=\int_0^\infty \cos\left(x\left(\frac{t^3}{3}-t\right)\right)\,dt
\]
for large \(x\).
\item[(b)] Outline the method to find more terms in the asymptotic expansion of \(G(x)\).
\end{enumerate}

Useful information:
\[
  \int_{-\infty}^{\infty}\cos(ax^2)\,dx
  =\int_{-\infty}^{\infty}\sin(ax^2)\,dx
  =\sqrt{\frac{\pi}{2a}}.
\]

Topic Tags

Linear Programming and Optimization Real Integral Evaluation by Complex Methods Stationary Phase and Steepest Descent Watson's Lemma and Laplace's Method