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QUESTION


Let $C_R$ denote the upper half of the circle $|z|=R$, $(R>2)$,
taken in the counterclockwise direction. Show that
$$\Big|\int_{C_R}{{2z^2-1\over z^4+5z^2+4}dz}\Big|\le {\pi
R(2R^2+1)\over (R^2-1)(R^2-4)}.$$Then, by dividing the numerator
and denominator of the expression  on the right by $R^4$, show
that the value of the integral tends to zero as $R$ tends to
infinity.


ANSWER


Length of contour is $\pi R$. Also, $$|{2z^2-1\over
(z^2+1)(z^2+4)}|\le
|{2|z|^2|+1\over(|z|^2-1)(|z|^2-4)}|=|{2R^2+1\over
(R^2-1)(R^2-4)}|$$ Now just apply the Estimation Theorem to get
the result.




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