Find two Laurent for z^-1(4-z)^-2 involving powers of z and state where they are valid.
Answer
Z^-1(4-z)^-2 = 1/16z + Sum(n=0 to infinity) of ((n+2)z^n)/(4^(n+3)) for |z|<4,
Z^-1(4-z)^-2 = Sum(n=0 to infinity) of (n(4)^(n-1))/(z^(n+2)) for |z|>4,
Answer
Z^-1(4-z)^-2 = 1/16z + Sum(n=0 to infinity) of ((n+2)z^n)/(4^(n+3)) for |z|<4,
Z^-1(4-z)^-2 = Sum(n=0 to infinity) of (n(4)^(n-1))/(z^(n+2)) for |z|>4,
-
d/dz 1/(4 - z) = 1/(4 - z)², and
1/(4 - z) = ¼/(1 - z/4) =
∞
Σ ¼(z/4)ⁿ for |z| < 1
n=0
Differentiate term by term, and multiply through by 1/z for 0 < |z| < 4 to get
1/[z(4 - z)²] = 1/z d/dz(1/4 + z/16 + z²/4^3 + z^3/4^4 + ....) =
= 1/(16z) + 2/4^3 + 3z/4^4 + 4z²/4^5 + ...) = 1/(16z) +
∞
Σ (n+2)zⁿ/4^(n+3), for 0 < |z| < 4
n=0
For |z| > 4, 1/(4 - z) = -(1/z)/(1 - 4/z) =
∞
Σ -4ⁿ/z^(n+1).
n=0
Differentiate term by term and multiply through by 1/z to get 1/(z(4 - z)²) =
∞
Σ (n+1)4ⁿ/z^(n+3) = <---reindex n + 1 -> n
n=0
∞
Σ n4^(n-1)/z^(n+2), for |z| > 4
n=1
Your solutions contain two errors. First, the top series is valid for 0 < |z| < 4. And second, the bottom series for the terms as you've written it should have the index starting at 1 not at 0.
1/(4 - z) = ¼/(1 - z/4) =
∞
Σ ¼(z/4)ⁿ for |z| < 1
n=0
Differentiate term by term, and multiply through by 1/z for 0 < |z| < 4 to get
1/[z(4 - z)²] = 1/z d/dz(1/4 + z/16 + z²/4^3 + z^3/4^4 + ....) =
= 1/(16z) + 2/4^3 + 3z/4^4 + 4z²/4^5 + ...) = 1/(16z) +
∞
Σ (n+2)zⁿ/4^(n+3), for 0 < |z| < 4
n=0
For |z| > 4, 1/(4 - z) = -(1/z)/(1 - 4/z) =
∞
Σ -4ⁿ/z^(n+1).
n=0
Differentiate term by term and multiply through by 1/z to get 1/(z(4 - z)²) =
∞
Σ (n+1)4ⁿ/z^(n+3) = <---reindex n + 1 -> n
n=0
∞
Σ n4^(n-1)/z^(n+2), for |z| > 4
n=1
Your solutions contain two errors. First, the top series is valid for 0 < |z| < 4. And second, the bottom series for the terms as you've written it should have the index starting at 1 not at 0.