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Let $\gamma$ be a closed and continuously differentiable path in the upper half plane
$\{z \in \mathbb{C} : z = x + iy,\; x, y \in\mathbb{ R}, \;y > 0 \}$ not passing through the point $i$. Describe the set of all possible values of the integral $$(1/2\pi i)∫_\gamma \frac{2i}{z^2+ 1}\,\mathrm dz.$$

How can I be able to solve this problem because here $\gamma$ is not properly mentioned. So I am stuck.

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Are you familiar with Cauchy's integral theorem or the residue theorem? – Ron Gordon Jan 12 at 9:34
yes, I know. here the pole are $i$ & $-i$.but how do I know that they $i$are inside or outside the curve.$-i$ is deinitely outside the curve. – user56997 Jan 12 at 9:39
That's all you can say: the value of the integral is $1/(i 2 \pi)$ if $\gamma$ encloses $i$, 0 otherwise. Knowing whether $\gamma$ encloses $z=i$ is another problem in which you must know $\gamma$. – Ron Gordon Jan 12 at 9:56
@rlgordonma Don't forget winding numbers. – mrf Jan 12 at 9:58
@mrf: good point. And the integral is equal to the winding number $n$, not $n/(i 2 \ pi)$, when $\gamma$ encloses $i$. – Ron Gordon Jan 12 at 10:16

1 Answer

up vote 1 down vote

The only thing that matters here is whether the point $z=i$ lies inside or outside $\gamma$.

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agree. if outside then the integral becomes 0. but if inside then it becomes 1. but given answer is all integers. which made me confused] – user56997 Jan 12 at 9:44
@user56997 The curve can wind around $z=i$ many times. – mrf Jan 12 at 9:53

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