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Let $\gamma:[0,1] \to \mathbb{C}$ be the curve $\gamma(t) = e^{2\pi i t}, 0 \le t \le 1$. Find, giving justifications, the value of the contour integral $$ \int_\gamma \frac{dz}{4z^2-1}. $$

I know the Cauchy residue theorem and how to apply it. But I couldn't the let part in the question.

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  • $\begingroup$ Hint: Which poles are in the circle? $\endgroup$ Jul 27, 2017 at 19:18
  • $\begingroup$ Is the given curve unit circle? $\endgroup$
    – Tortoise
    Jul 27, 2017 at 19:19
  • $\begingroup$ the ''let'' part simply define a circle of radius $1$ centerd at the origin. $\endgroup$ Jul 27, 2017 at 19:19

1 Answer 1

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The curve described parametrically by $\gamma(t)=e^{i2\pi t}$, $0\le t\le 1$ is simply the unit circle centered at $z=0$ (i.e., $|z|=1$).

Hence, we have

$$\begin{align} \int_\gamma \frac{1}{4z^2-1}\,dz&= \oint_{|z|=1}\frac{1}{4z^2-1}\,dz\\\\ &=2\pi i \text{Res}\left(\frac{1}{4z^2-1}, z=\pm 1/2\right)\\\\ &=2\pi i \left(\frac14-\frac14 \right)\\\\ &=0 \end{align}$$

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