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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods

Summer 2025 • Question 3

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods Summer 2025 • Question 3

Use residue calculus to show that \[ \int _0^\infty \frac {\log x}{x^3-1}\,dx=\frac {4\pi ^2}{27}. \] Hint: Use an appropriate branch of the function \((\log z)/(z^3-1)\) and a sector of aperture \(2\pi /3\).

Raw LaTeX

Use residue calculus to show that
\[
\int_0^\infty \frac{\log x}{x^3-1}\,dx=\frac{4\pi^2}{27}.
\]
\textbf{Hint:} Use an appropriate branch of the function \((\log z)/(z^3-1)\) and a sector of aperture
\(2\pi/3\).

Topic Tags

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