MA 513 HW 7

MA 513 HW 7 - z + 1) + ( z + 1) 2 ) =-2 + ( z + 1) + 1 z +...

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MA 513 Homework 7 Solutions p.239: #1, 2; p.243: #1, 2; p.248: #1,3. p. 239: #1: (a) 1 z + z 2 = 1 z ( z + 1) .Res = 1 . (b) z cos(1 /z ) = z (1 - (1 /z ) 2 / 2 + ... ) = - 1 / 2 z + series with other powers . Res = - 1 / 2 . (c) z - sin z z = 1 z ( z - ( z - z 3 / 6+ ... )) , so there is no 1 /z term in the Laurent series. Res = 0 . (d) cotan z = cos z/ sin z has a simple pole at z = 0 , so cotan z = cos z/ sin z = (1 - z 2 / 2 + z 4 / 4! - ... ) / ( z - z 3 / 6+ .. ) = 1 z ( a 0 + a 1 z + a 2 z 2 + a 3 z 3 + a 4 z 4 + ... ) . Thus, (1 - z 2 / 2+ z 4 / 4! - ... ) = (1 - z 2 / 6 + z 4 / 5! + .. )( a 0 + a 1 z + a 2 z 2 + a 3 z 3 + a 4 z 4 + ... ) . Multiplying out, and comparing terms, we get a 0 = 1 , a 1 = 0 , - 1 / 2 = - a 0 / 6+ a 2 , a 3 = 0 , 1 / 4! = a 0 / 5! - a 2 / 6+ a 4 . Therefore, a 2 = - 1 / 2 + 1 / 6 = - 1 / 3; a 4 = - 1 / 45 , which is the residue. #2: (a) Double pole at z = 0; Res = - 1 . (b) Double pole at z = 1; Res = - 1 /e. (c) Simple pole at z = 0; z 2 e 1 /z = z 2 (1 + 1 /z + 1 / 2 z 2 + 1 / 6 z 3 + ... ); Res = 1 / 6 . (d) simple poles at z = 0 , 2; Residues - 1 / 2 , 3 / 2 . Thus, integral = 2 πi ( - 1 / 2 + 3 / 2) = 2 πi. p. 243: #1: (a) ze 1 /z = z (1 + 1 /z + 1 / 2 z 2 + 1 / 6 z 3 + ... ) = 1 + z + n =1 1 ( n + 1)! z - n ; essential singular point. (b) z 2 1 + z = ( z + 1 - 1) 2 1 + z = 1 z + 1 (1 - 2(
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Unformatted text preview: z + 1) + ( z + 1) 2 ) =-2 + ( z + 1) + 1 z + 1 . Simple pole at z =-1 . (c) sin z/z : Removable singularity at z = 0 . No singular (principal) part of the expansion. (d) cos z/z : Simple pole at z = 0 . Principal part is 1 /z. (e) 1 (2-z ) 3 =-1 ( z-2) 3 Already a Laurent series about pole of order 3 at z = 2 . #2: Answers given in book; expand numerator in (a), (b); diFerentiate it at z = 1 to get the residue in case (c). p. 248: #1: Answers given in book. #3: Answers given in book. (a): Singular points (simple poles) at z = 1; (b): Singular points (simple poles) at z = 1 , z = 3 i ;...
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This note was uploaded on 06/11/2010 for the course MA 513 taught by Professor Staff during the Spring '08 term at N.C. State.

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