CVXI1

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Give an upper bound for \Bigg| \int_{|z|=3} \frac{dz}{z^2-i} \Bigg| \,.

Using the fact that \Bigg|\int_\Gamma f(z) dz \Bigg| \le \mathrm{max}_{z\,\mathrm{on}\,\Gamma} |f(z)| \cdot \mathrm{length}(\Gamma)\,, in this case

f(z) = \frac{1}{z^2-i}\,

\Big| \frac{1}{z^2-i} \Big| = \frac{1}{|z^2-i|}\,

\mathrm{min}_{|z^2-i|} = |9i-i| = |8i| = 8\,

Therefore

\mathrm{max}_{z\isin C} |f(z)| = \frac{1}{8}\,

\Bigg| \int_{|z|=3} \frac{dz}{z^2-i} \Bigg| \le \frac{1}{8} \int_{|z|=3}|dz| = \frac{1}{8}2\pi 3 = \frac{3\pi}{4}\,


Main Page : Complex Variables : Complex Integrals

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