Questions involving asymptotic analysis, including growth of functions, Big-O notation, Big-Omega and Big-Theta notations.

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Asymptotic expression of $\int_{- D}^{D} \frac{\text{tanh}(\xi)}{\xi -\omega}\mathrm{d}\xi$

How to derive the following asymptotic expression ($|\omega| \ll D $)? $$P.V.\int_{- D}^{D} d\xi \frac{\text{tanh}(\beta \xi)}{\xi -\omega} \approx 2 \text{ln}(\frac{D}{\sqrt{\omega^2+T^2}}),\ \ \ ...