Mathematics Tripos Papers

  • Part IA
  • Part IB
  • Part II
  • FAQ

1.II.13D

Complex Analysis or Complex Methods | Part IB, 2006

By integrating round the contour involving the real axis and the line Im⁡(z)=2π\operatorname{Im}(z)=2 \piIm(z)=2π, or otherwise, evaluate

∫−∞∞eax1+exdx,0<a<1.\int_{-\infty}^{\infty} \frac{e^{a x}}{1+e^{x}} d x, \quad 0<a<1 .∫−∞∞​1+exeax​dx,0<a<1.

Explain why the given restriction on the value aaa is necessary.

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