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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2017 • Question 10

Question

Math 6420 • PDE Fall 2017 • Question 10

Consider the Dawson’s integral \[ D(x)=e^{-x^2}\int _0^x e^{t^2}\,dt \qquad \text {as }x\to \infty . \]

  1. Find the first non-constant term of the asymptotic expansion of \(D(x)\) and show that the relationship is asymptotic.
  2. Find the full asymptotic series of \(D(x)\).

HINT: Introduce a cutoff parameter.

Raw LaTeX

Consider the Dawson's integral
\[
  D(x)=e^{-x^2}\int_0^x e^{t^2}\,dt
  \qquad\text{as }x\to\infty.
\]
\begin{enumerate}
\item[(a)] Find the first non-constant term of the asymptotic expansion of \(D(x)\) and show
that the relationship is asymptotic.
\item[(b)] Find the full asymptotic series of \(D(x)\).
\end{enumerate}
HINT: Introduce a cutoff parameter.

Topic Tags

Cauchy Integral Formula and Integral Theorems Harmonic Functions and Poisson Integral Formula Linear Programming and Optimization Watson's Lemma and Laplace's Method