2
$\begingroup$

How would you evaluate $\oint_C \ e^{2z}(z+1)^{-1} \, \mathrm dz$ where $C=\{z\in \mathbb{C}: |z|=2 \}$?

  • 0
    In my experience "$\oint$" means a line integral, so in "$\oint_C$" we need $C$ to be a curve.2011-10-23

1 Answers 1

4

Recall the Cauchy's integral formula. Use it with $f(z) = \mathrm{e}^{2 z}$ and $a = -1$ and integration contour $\gamma = C$. This gives

$ \int_C \frac{\mathrm{e}^{2z}}{z+1} \mathrm{d} z = 2 \pi i \mathrm{e}^{2 a} = 2 \pi i \mathrm{e}^{-2} $

  • 0
    Thank you greatly, this is far more clear now!2011-10-24