https://file.aiquickdraw.com/mathbot/file/1725268334325m2mpcs7n.jpg

The function given in the image is:

\[
f(z) = \frac{z^2 - 1}{(z-1)(z-3)}
\]

To find the partial fraction decomposition of this function, we can express \(f(z)\) as:

\[
f(z) = \frac{A}{z - 1} + \frac{B}{z - 3}
\]

### Steps to find \(A\) and \(B\):

1. **Set up the equation:**

   \[
   \frac{z^2 - 1}{(z - 1)(z - 3)} = \frac{A}{z - 1} + \frac{B}{z - 3}
   \]

2. **Combine the fractions on the right-hand side:**

   \[
   \frac{A(z - 3) + B(z - 1)}{(z - 1)(z - 3)} = \frac{z^2 - 1}{(z - 1)(z - 3)}
   \]

   This simplifies to:

   \[
   A(z - 3) + B(z - 1) = z^2 - 1
   \]

3. **Expand and collect like terms:**

   \[
   Az - 3A + Bz - B = z^2 - 1
   \]

   Combine like terms:

   \[
   (A + B)z - (3A + B) = z^2 - 1
   \]

4. **Compare coefficients:**

   - Coefficient of \(z^2\): \(0 = 1\) (This suggests an error, so we recognize there's no \(z^2\) term here, meaning only linear comparison is needed).
   - Coefficient of \(z\): \(A + B = 0\)
   - Constant term: \(-3A - B = -1\)

5. **Solve the system of equations:**

   From \(A + B = 0\), we have \(B = -A\).

   Substitute \(B = -A\) into the second equation:

   \[
   -3A - (-A) = -1
   \]
   Simplifying:

   \[
   -2A = -1 \implies A = \frac{1}{2}
   \]

   Since \(B = -A\), we have \(B = -\frac{1}{2}\).

6. **Write the partial fraction decomposition:**

   \[
   f(z) = \frac{1/2}{z - 1} - \frac{1/2}{z - 3}
   \]

So the partial fraction decomposition of the given function is:

\[
f(z) = \frac{1/2}{z - 1} - \frac{1/2}{z - 3}
\]

Do you want details or have any questions?

Here are some related questions:
1. How can we apply partial fraction decomposition to solve integrals?
2. What happens if the denominator has repeated linear factors?
3. Can partial fraction decomposition be applied to irreducible quadratic factors?
4. How can we use partial fractions in solving differential equations?
5. What is the significance of partial fractions in control systems?

**Tip:** Always check the degree of the numerator relative to the denominator before starting partial fraction decomposition. If the numerator's degree is equal to or greater than the denominator, perform polynomial division first.

Learn how to perform partial fraction decomposition for the function (z^2 - 1) / ((z-1)(z-3)). This detailed solution covers the steps to find the partial fractions, including setting up equations, solving the system, and writing the decomposition. Suitable for advanced high school students and college-level learners.

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