Q3Advanced Engineering Mathematics

The problem provides a complex frequency-domain function $\bar{u}(z)=\frac{2z^2+3z+4}{(z-3{)}^3}$ and tasks us with extracting the specific time-domain discrete sequence values $u_1,u_2,u_3$. The foundational mathematical definition of the Z-transform is $U(z)={\sum }_{n=0}^{\infty }{u}_n{z}^{-n}$. Therefore, to extract the sequence coefficients $u_n$, we must forcefully expand the provided rational function into an infinite Laurent series arranged strictly in descending negative powers of $z$.

1. Algebraic Reformulation of the Function

First, we must algebraically restructure both the numerator and the denominator to facilitate a binomial expansion. We divide both by $z^3$ to force negative exponents.

2. Applying the Binomial Theorem

We now isolate the term $(1-3z^{-1}{)}^{-3}$ and expand it using the generalized infinite Binomial Theorem for negative fractional indices: $(1-x{)}^{-n}=1+nx+\frac{n(n+1)}{2!}{x}^2+\frac{n(n+1)(n+2)}{3!}{x}^3+\dots$. Substituting $n=3$ and $x=3z^{-1}$:

3. Polynomial Multiplication and Coefficient Extraction

We substitute this newly generated infinite expansion back into our fully restructured main equation and meticulously perform algebraic polynomial multiplication.

We carefully expand this, strictly tracking the coefficients for the first three negative powers of $z$. The $z^0$ inside the bracket: $2\times 1=2$. Multiplied by outer $z^{-1}⟹2z^{-1}$. The $z^{-1}$ inside the bracket: $(2\times 9)+(3\times 1)=18+3=21$. Multiplied by outer $z^{-1}⟹21z^{-2}$. The $z^{-2}$ inside the bracket: $(2\times 54)+(3\times 9)+(4\times 1)=108+27+4=139$. Multiplied by outer $z^{-1}⟹139z^{-3}$.

The final truncated series is mathematically $2z^{-1}+21z^{-2}+139z^{-3}+\dots$. By directly comparing these generated terms with the definitive Z-transform definition $\sum u_n{z}^{-n}$, we decisively extract the required sequence values: $u_1=2$, $u_2=21$, and $u_3=139$.