Mathematical Methods in the Physical Sciences · Problem 11.23

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · contour-integration-verify-formula

Problem

Verify the formula by contour integration or as indicated. Assume a>0a > 0, m>0m > 0. ∫02πsin⁡θ dθa+bsin⁡θ=∫02πcos⁡θ dθa+bcos⁡θ=2πb(1−aa2−b2),∣b∣<a\int_0^{2\pi} \frac{\sin\theta\,d\theta}{a+b\sin\theta} = \int_0^{2\pi} \frac{\cos\theta\,d\theta}{a+b\cos\theta} = \frac{2\pi}{b}\left(1 - \frac{a}{\sqrt{a^2-b^2}}\right), \qquad |b| < a

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