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To solve for the \( Z \)-transform of each given function, we can apply the \( Z \)-transform definitions and common properties. Here's a breakdown of each one:

### 1. \( n \cos(n\theta) \)
The \( Z \)-transform of \( n \cos(n\theta) \) can be derived using the property of the \( Z \)-transform of a sequence with a cosine term:
\[
Z\{n \cos(n\theta)\} = -\frac{z \cos(\theta) + z^2}{(z - e^{j\theta})(z - e^{-j\theta})^2}
\]

### 2. \( \sin\left(\frac{n\pi}{2} + \frac{\pi}{4}\right) \)
For this trigonometric sequence, the term inside the sine function can be simplified based on the angle addition formula. Generally, we can write:
\[
Z\{\sin\left(\frac{n\pi}{2} + \frac{\pi}{4}\right)\} = \text{standard trigonometric series}
\]
However, the exact \( Z \)-transform depends on applying the trigonometric identity in terms of a sinusoidal series expansion, which would generally yield complex expressions.

### 3. \( e^{-2n} \)
The \( Z \)-transform of an exponential sequence \( e^{-an} \) is given by:
\[
Z\{e^{-2n}\} = \frac{1}{z - e^{-2}}
\]

### 4. \( a^{|n|} \)
For \( a^{|n|} \), the \( Z \)-transform needs to consider values of \( n \) for both positive and negative cases:
- For \( n \geq 0 \): \( Z\{a^n\} = \frac{z}{z - a} \)
- For \( n < 0 \): \( Z\{a^{-n}\} = \frac{z}{z - a^{-1}} \)

Combining both cases would yield a more complex expression depending on whether \( n \) is positive or negative.

Would you like to see a deeper explanation for any of these specific transforms or examples of their applications?

### Follow-Up Questions
1. How do initial conditions affect the \( Z \)-transform for causal systems?
2. What is the role of the region of convergence (ROC) in determining the \( Z \)-transform's stability?
3. How does the \( Z \)-transform differ when applied to left-sided sequences versus right-sided sequences?
4. Can the \( Z \)-transform be used to solve differential equations, and if so, how?
5. What is the inverse \( Z \)-transform, and how can it be used to reconstruct signals?

### Tip:
When working with trigonometric functions in the \( Z \)-domain, using Euler's formulas for sine and cosine can simplify the process by converting the terms into exponential forms.

Find the Z-transform of: (i) n cos(nθ), (ii) sin((nπ/2) + π/4), (iii) e^(-2n), (iv) a^|n|

Learn how to calculate the Z-transform of functions like n cos(nθ), sin((nπ/2) + π/4), e^(-2n), and a^|n|. This tutorial explores Z-transform properties, trigonometric identities, and exponential sequences for advanced students.

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