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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods

Summer 2023 • Question 5

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods Summer 2023 • Question 5

Compute the following asymptotic expansions:

  1. (5 pts) \(I(k)=\int _k^\infty e^{-t^3}\,dt\) as \(k\to 0^+\). You must compute the entire asymptotic expansion.
  2. (5 pts) \(I(k)=\int _0^1 e^{-k(t^2-t)}\,dt\) as \(k\to \infty \). You need only compute the leading order term in the expansion.

Raw LaTeX

Compute the following asymptotic expansions:
\begin{enumerate}
\item
(5 pts) \(I(k)=\int_k^\infty e^{-t^3}\,dt\) as \(k\to 0^+\). You must compute the entire asymptotic
expansion.
\item
(5 pts) \(I(k)=\int_0^1 e^{-k(t^2-t)}\,dt\) as \(k\to\infty\). You need only compute the leading order
term in the expansion.
\end{enumerate}

Topic Tags

Entire Functions and Liouville's Theorem Stationary Phase and Steepest Descent Watson's Lemma and Laplace's Method