---
title: "Z-transform property proof"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:z-transform-property-proof-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/z-transform-property-proof-two
site: BCT Engineering
---

# Z-transform property proof

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2081 Chaitra (new course), 2 marks
- Video: [Properties of Z-Transform (Part 1)](https://www.youtube.com/watch?v=RYXlHkqqdh8) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/z-transform-property-proof-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

If Z[x(k)] = X(z), show that x(1) = lim_{z→∞} z[X(z) − x(0)], where the symbols have their usual meanings.

## Short answer

Expand X(z)=x(0)+x(1)z^{-1}+x(2)z^{-2}+... . Then z[X(z)-x(0)]= x(1)+x(2)z^{-1}+... . As z→∞, all terms with negative powers vanish, leaving x(1).

## Answer

**Proof.** By definition,

```text
X(z) = Σ_{k=0}^∞ x(k) z^(−k) = x(0) + x(1) z⁻¹ + x(2) z⁻² + x(3) z⁻³ + ...
```

Subtract x(0) and multiply by z:

```text
z [X(z) − x(0)] = x(1) + x(2) z⁻¹ + x(3) z⁻² + ...
```

Now let z → ∞. Every term containing a negative power of z tends to 0, so

```text
lim_{z→∞} z [X(z) − x(0)] = x(1)
```

**Hence x(1) = lim_{z→∞} z[X(z) − x(0)].** ∎

(Here x(0) = lim_{z→∞} X(z) is the initial value theorem; the result is its next-order extension.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The proof starts from
   A. X(z) = x(0) + x(1)z + x(2)z^2 + ...
   B. X(z) = x(0) + x(1)z^-1 + x(2)z^-2 + ...
   C. X(z) = integral of x(t)
   D. X(z) = z x(0)
   Answer: B. Definition of the Z-transform. (Key point: Start)

2. z[X(z) - x(0)] =
   A. x(1) + x(2)z + ...
   B. x(0) + x(1)z^-1 + ...
   C. x(1) + x(2)z^-1 + x(3)z^-2 + ...
   D. x(1)z^-1 + x(2)z^-2 + ...
   Answer: C. Subtract x(0) and multiply by z. (Key point: Manipulation)

3. As z to infinity, the remaining terms
   A. go to infinity
   B. equal x(0)
   C. cancel x(1)
   D. with negative powers of z go to 0, leaving x(1)
   Answer: D. Only the constant x(1) survives. (Key point: Limit)

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