The image contains a mathematical problem related to the Laplace Transform and involves finding the region of convergence (ROC) for a given function. Here is the problem statement from the image:

Determine the ROC for the function:

$X(s) = \frac{1}{s^2 + 6s + 13}$

(Give your answer as an interval in the form $a < Re(s) < b$ or state it as all $Re(s) > a$ or all $Re(s) < b$, depending on whether the ROC is open or closed. Point out if the answer is empty. Explain the difference between the inverse transform and the bilateral equivalent.

Solution Outline:

1. Factor the denominator:

   • The denominator $s^2 + 6s + 13$ needs to be analyzed, typically by completing the square.

2. Determine the poles:

   • The poles are the values of $s$ that make the denominator zero.

3. Identify the ROC:

   • The ROC depends on whether we are dealing with a right-sided, left-sided, or two-sided signal.

Step-by-Step Solution:

1. Complete the square for the denominator: $s^2 + 6s + 13 = \left(s + 3\right)^2 + 4$

   This is in the form of $(s + 3)^2 + (2)^2$.

2. Find the poles:

   • The poles occur where the denominator is zero: $s = -3 \pm 2i$
   • These are complex poles located at $s = -3 + 2i$ and $s = -3 - 2i$.

3. Determine the ROC:

   • For the Laplace Transform, the ROC depends on the type of signal (causal, anti-causal, or two-sided):
     • Causal (right-sided signal): ROC is $Re(s) > -3$.
     • Anti-causal (left-sided signal): ROC is $Re(s) < -3$.
     • Two-sided (bilateral): ROC would be an empty set in this case because it would require the region between the poles, which is not possible since the real parts of the poles are the same.

Final Answer:

• ROC: Depending on the type of signal:
  • If it's a right-sided signal: $\text{ROC} = Re(s) > -3$.
  • If it's a left-sided signal: $\text{ROC} = Re(s) < -3$.
  • For a two-sided signal, the ROC would be an empty set.

Would you like a more detailed explanation, or do you have any questions?

Follow-up Questions:

1. What is the significance of the ROC in Laplace Transform?
2. How do the poles of a function relate to its stability?
3. Can you explain the difference between the inverse Laplace Transform and the bilateral Laplace Transform?
4. What does it mean if the ROC is an empty set?
5. How does the type of signal (causal, anti-causal) affect the ROC?

Tip: Always consider the nature of the signal (causal or anti-causal) when determining the ROC for Laplace Transforms, as it directly influences the region.