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HW1 - of these equations(where 0< << 1(a z 2 2 ǫz-1...

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Math 401 (Sec. 502) Spring, 2009 Homework 1 (due January 27) Q1. Let f ( z ) = O ( g ( z )) and F ( z ) = O ( G ( z )) as z 0. (a) Prove that f ( z ) F ( z ) = O ( g ( z ) G ( z )). (b) Prove that f ( z ) + F ( z ) = O ( | g ( z ) | + | G ( z ) | ). Q2. Obtain two-term expansions for the solutions of the equation (where 0 < ǫ << 1) ( z - 1)( z - 2)( z - 3) + ǫ = 0 . Q3. Find the first three terms in the approximate solutions of all the roots
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Unformatted text preview: of these equations (where 0 < ǫ << 1) (a) z 2 + 2 ǫz-1 = 0. (b) ǫz 3-z + 1 = 0. Q4. Compute the ±rst two terms of all four roots of ǫ 2 z 4-2 ǫz 3 + z 2-2 z + 1 = 0 , where 0 < ǫ << 1....
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