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The question is:

By using Rouché's theorem, calculate number of zeros $F(z)=(z^2 + 2)(z^2 > + 1) - iz(z^2 + 2)$

where $D={\Im(z)> 0}$.

How do I need to choose $h(z)$ and $g(z)$ s.t $F= h + g$ so I can apply the theorem and what about the condition $D={\Im(z)> 0}$?

I know the application of rouche theorem when we have condition on $|z|$, for example $1<|z|<2$ but I don't know what I need to do with the condition $\Im(z)> 0$.

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    To tackle domains like this, try using the curve $\gamma_R$, which we define for $R>0$ to be the union of the upper half circle of radius $R$ with the segment $-R on the real line. Apply Rouche's Theorem with this curve, and check what happens as $R\rightarrow\infty$.2012-06-02
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    ok I got the idea. but then I could not divide F into two parts :S2012-06-02
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    What is the meaning of $z^2>+1?$ for $z\in \mathbb{C}$? Is the polynomial $F(z)=(z^2+1)-iz(z^2+2)?$2015-02-09

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