solnsA1b - C&O 330 - SOLUTIONS #1 PROFESSOR D.M....

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: C&O 330 - SOLUTIONS #1 PROFESSOR D.M. JACKSON (1) (20 points) Give an expression for [ x n ] G as an explicit function of n of each of the following two functional equations. (a) (10 points) G = F, where F satisfies the functional equation F = x (1- F ) m , where m is a positive integer. Solution : By Lagrange’s (Implicit Function) Theorem, G = F = X n ≥ 1 x n n λ n- 1 (1- λ )- mn = X n ≥ 1 x n n ( m + 1) n- 2 n- 1 . (b) (10 points) G = e T where T satisfies the functional equation T = xe T . Solution: By Lagrange’s (Implicit Function) Theorem, G = e T = G = [ x n ] e T = 1 + X n ≥ 1 x n n λ n- 1 e λ e nλ = 1 + X n ≥ 1 x n n λ n- 1 e ( n +1) λ = 1 + X n ≥ 1 x n n ! ( n + 1) n- 1 . Comment: Other solutions : i) By Lagrange’s Theorem, T ( x ) = X n ≥ 1 x n n λ n- 1 e nλ = X n ≥ 1 n n- 1 x n n ! = x X n ≥ 1 n n- 1 x n- 1 n ! , so xe T = x X n ≥ 1 n n- 1 x n- 1 n ! . Then, by the Cancellation Law, or by observing that x e T- X n ≥ 1 n n- 1 x n- 1 n ! = 0 1 2 PROFESSOR D.M. JACKSON and that C [[ x ]] has no zero divisors, we conclude that e T- X n ≥ 1 n n- 1 x n- 1 n ! = 0 so e T = X n ≥ 1 n n- 1 x n- 1 n ! = X n ≥ ( n + 1) n- 1 x n n ! . Zero divisors are mentioned in Math 235, in connexion with the ring of square matrices. ii) Alternatively, x- 1 can be used as long as it is clear that the ring being used is the ring of Laurent series over C . This ring was mentioned in the lectures. Half credit should be given for a ‘solution’ in C...
View Full Document

This note was uploaded on 11/18/2010 for the course CO 330 taught by Professor R.metzger during the Spring '05 term at Waterloo.

Page1 / 5

solnsA1b - C&O 330 - SOLUTIONS #1 PROFESSOR D.M....

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online