written 2.3 years ago by |
Given,
$F(z)=\dfrac {7z-2}{z(z-2)(z+1)}$
$\implies F(z)=\dfrac {7z-2}{z(z-2)(z+1)}$
$\implies F(z)=\dfrac 1z-\dfrac 3{z+1}+\dfrac 2{z-2}$
Put U=z+1
$\implies \therefore\ z=U-1$
$\implies \therefore\ F(z)=\dfrac 1{U-1}-\dfrac 3{(U-1)+1}+\dfrac 2{(U-1)-2}$
$\implies F(z) = \dfrac 1{U-1}-\dfrac 3U+\dfrac 2{U-3}$
Case 1 : for $|U| \lt 1$
Obviously $|U| \lt 3$
$\therefore |U| \lt 1$ and $|\dfrac U3| \lt 1$
$$\therefore f(z)=\dfrac 1{1(1-U)}-\dfrac 3U+\dfrac 2{3(\frac U3)-1}$$
$=-(1-U)^{-1}-\dfrac 3U-\dfrac 23[1-\dfrac U3]^{-1}\\ =-(1+U+U^2+...) -\dfrac 3U-\dfrac 23[1+\dfrac U3+\dfrac {U^2}{3^2}+...]\\ =-[1+(z+1)+(z+1)^2+....] -\dfrac 3{(z+1)}-\dfrac 23[1+\dfrac {(z+1)}3+\dfrac {(z+1)^2}{3^2}+...]$
Case 2 : $1\lt |U|\lt 3$
$\therefore 1 \lt|U|$ and $|U|\lt 3$
$\therefore |\dfrac 1U| \lt1$ and $|\dfrac U3| \lt1$
$$\therefore f(z)=\dfrac 1{U(1-\dfrac 1U)}-\dfrac 3U+\dfrac 2{3(\dfrac U3-1)}$$
$=\dfrac 1U[1-\dfrac 1U]^{-1}-\dfrac 3U-\dfrac 23[1-\dfrac U3]^{-1}\\ =\dfrac 1U[1+\dfrac 1U+\dfrac 1{U^2}+...] -\dfrac 3U-\dfrac 23[1+\dfrac U3+\dfrac {U^2}{3^2}+....]\\ =[\dfrac 1U+\dfrac 1{U^2}+\dfrac 1{U^3}+...]-\dfrac 3U-\dfrac 23[1+\dfrac U3+\dfrac {U^2}{3^2}+...]\\ =\dfrac 1{(z+1)}+\dfrac 1{(z+1)^2}+\dfrac 1{(z+1)^3}+.... -\dfrac 3{(z+1)}-\dfrac 23(1+\dfrac {(z+1)}3+....)$
Case 3 : for $|U| \gt 3$
Obviously $|U|\gt1$
$\therefore 1\lt | U |$ and $3 \lt |U|$
$\therefore |\dfrac 1U| \lt 1$ and $|\dfrac 3U| \lt 1$
$$\therefore f(z)=\dfrac 1{U(1-\dfrac 1U)}-\dfrac 3U+\dfrac 2{U(1-\dfrac 3U)}$$
$=\dfrac 1U[1-\dfrac 1U]^{-1}-\dfrac 3U-\dfrac 2U[1-\dfrac 3U]^{-1}\\ =\dfrac 1U[1+\dfrac 1U+\dfrac 1{U^2}+\dfrac 1{U^3}+...]-\dfrac 3U+\dfrac 2U[1+\dfrac 3{U^2}+\dfrac {3^2}{U^3}+...]\\ =[\dfrac 1U+\dfrac 1{U^2}+\dfrac 1{U^3}+...]-\dfrac 3U+2[\dfrac 1U+\dfrac 3{U^2}+\dfrac {3^2}{U^3}+...] \\ =[\dfrac 1{U^2}+\dfrac 1{U^3}+\dfrac 1{U^4}+...]+2[\dfrac 3{U^2}+\dfrac {3^2}{U^3}+\dfrac {3^3}{U^4}+...]\\ =\dfrac 1{(z+1)^2}+\dfrac 1{(z+1)^3}+\dfrac 1{(z+1)^4} +.... +2[\dfrac 1{(z+1)^2}+\dfrac 3{(z+1)^3}+\dfrac {3^2}{(z+1)^4}+... ]$