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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2019 • Question 10

Question

Math 6420 • PDE Spring 2019 • Question 10

Consider integral \[ I(s)=\int _C \frac {e^{s z^2}}{z^2-1}\,dz, \] \(C\) is the vertical line \(\operatorname {Re}z=2\), from \(z=2-i\infty \) to \(z=2+i\infty \), with large positive parameter \(s\). Find the three-term asymptotic expansion (approximation with three non-zero terms) of this integral as \(s\to +\infty \).

Raw LaTeX

Consider integral
\[
  I(s)=\int_C \frac{e^{s z^2}}{z^2-1}\,dz,
\]
\(C\) is the vertical line \(\operatorname{Re}z=2\), from \(z=2-i\infty\) to
\(z=2+i\infty\), with large positive parameter \(s\). Find the three-term asymptotic
expansion (approximation with three non-zero terms) of this integral as \(s\to+\infty\).

Topic Tags

Cauchy Integral Formula and Integral Theorems Linear Programming and Optimization Low-Rank Approximation Watson's Lemma and Laplace's Method