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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods

Fall 2023 • Question 4

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods Fall 2023 • Question 4

Evaluate the following integral by residue integration: \[ \int _0^\infty \frac {dx}{x^3+a^3},\qquad a>0. \] Clearly indicate the contour of integration and the treatment of the integral along each segment.

Raw LaTeX

Evaluate the following integral by residue integration:
\[
\int_0^\infty \frac{dx}{x^3+a^3},\qquad a>0.
\]
Clearly indicate the contour of integration and the treatment of the integral along each
segment.

Topic Tags

Cauchy Integral Formula and Integral Theorems Contour Integration and Residue Calculus Isolated Singularities and Residues Real Integral Evaluation by Complex Methods