- #1
tadf2
- 5
- 0
Homework Statement
f = [itex]\frac{1}{z(z-1)(z-2)}[/itex]
Homework Equations
Partial fraction
The Attempt at a Solution
R1 = 0 < z < 1
R2 = 1 < z < 2
R3 = z > 2
f = [itex]\frac{1}{z(z-1)(z-2)}[/itex] = [itex]\frac{1}{z}[/itex] * ([itex]\frac{A}{z-1}[/itex] + [itex]\frac{B}{z-2}[/itex])
Where A = -1 , B = 1.
f = [itex]\frac{1}{z}[/itex] * ([itex]\frac{1}{z-2}[/itex] + [itex]\frac{1}{1-z}[/itex])
My question is, why do you have to use partial fractions?
Can't you just leave the initial multiplication of three terms and expand them individually;
[itex]\frac{1}{z-1}[/itex] (dividing 1/z to numerator and denominator at R2,R3)
and
[itex]\frac{1}{z-2}[/itex] (dividing 1/z to numerator and denominator at R3)
so for example,
for R2,
f = [itex]\frac{1}{z}[/itex] * [itex]\frac{1}{z-2}[/itex] * (-[itex]\frac{1}{z}[/itex]*[itex]\frac{1}{1-z^{-1}}[/itex])
You can expand 1/(1-z) and 1/(z-2) using taylor series.
Should you always use partial fractions?
Or does expanding without using partial fractions still work?
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