homework20111128

homework20111128 - 6 How many roots of z 6 − 5 z 4 z 3...

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Math 417, Fall 2011 Exercise Set #9 corrected Turn in: November 28 For problems 1 to 5 below, use the method of residues to calculate the indicated integrals. Show all of your work. 1. Compute 0 dx x 2 + 9 . 2. Show that 0 dx ( x 2 + 4) 3 = 3 π 512 . 3. Compute the Cauchy Principal Value for lim R →∞ R R dx x 2 + 4 x + 5 . 4. Show that 0 dx ( x 2 + a 2 )( x 2 + b 2 ) = π 2 ab ( a + b ) where a, b > 0 and a = b . 5. Compute −∞ cos 5 x x 2 + 2 x + 5 dx . For problems 6 to 8 below, use Rouch´ e’s Theorem. Show all of your work.
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Unformatted text preview: 6. How many roots of z 6 − 5 z 4 + z 3 − 2 z lie inside the unit circle, counting multiplicities? 7. Show that if | C | > e , then the equation Cz n = e z has n solutions inside the unit circle, counting multiplicities. 8. Show that 2 z 5 + 8 z − 1 = 0 has exactly 4 solutions inside the annulus 1 < | z | < 2. 1...
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