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solve for f(s) or the laplace transform, step by step

The image you provided seems to describe a periodic function \( f(t) \), represented as a sum of shifted versions of another function \( g(t) \). Let's extract the key information and solve for the Laplace transform \( f(s) \) step by step.

### Step 1: Understanding the Periodic Function

From the whiteboard, the function \( f(t) \) is expressed as:

\[
f(t) = \sum_{n=0}^{\infty} g(t - nT)
\]

This is a periodic summation of \( g(t) \) where the period is \( T \). This suggests that \( f(t) \) is periodic with period \( T \), and the same function \( g(t) \) is repeated every \( T \) units of time.

### Step 2: Laplace Transform of a Periodic Function

The Laplace transform of a periodic function \( f(t) \) with period \( T \) is given by:

\[
\mathcal{L}\{f(t)\} = \frac{1}{1 - e^{-sT}} \mathcal{L}\{g(t)\}
\]

Here:
- \( g(t) \) is the function in one period.
- \( T \) is the period of the periodic function.
- \( \mathcal{L}\{g(t)\} \) represents the Laplace transform of the non-periodic part \( g(t) \).

### Step 3: Applying the Formula

To solve for \( f(s) \), we need the Laplace transform of \( g(t) \), denoted as \( G(s) \). The Laplace transform of the full periodic function \( f(t) \) is then:

\[
f(s) = \frac{G(s)}{1 - e^{-sT}}
\]

This is the general form for the Laplace transform of a periodic function \( f(t) \) that consists of repetitions of \( g(t) \) at intervals of \( T \).

### Final Answer:

\[
f(s) = \frac{G(s)}{1 - e^{-sT}}
\]

Where \( G(s) \) is the Laplace transform of the base function \( g(t) \).

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Would you like me to go deeper into any particular step, or explain how to compute the Laplace transform of a specific function \( g(t) \)? Feel free to ask.

Here are 5 related questions you might be interested in:
1. How do we calculate the Laplace transform of specific functions like \( g(t) \)?
2. What are the conditions for a function to be periodic in the Laplace domain?
3. How does this method change for different periods \( T \)?
4. Can we apply this method to non-periodic functions as well?
5. What is the physical interpretation of the Laplace transform of a periodic signal?

**Tip**: When dealing with periodic functions, knowing the Laplace transform of the base function \( g(t) \) can simplify many calculations, especially for systems with repetitive behavior.

Solve for f(s) or the Laplace transform, step by step.

Solve step-by-step for the Laplace transform of a periodic function f(t) represented as a summation of shifted versions of g(t). Learn the key mathematical concepts, formulas, and theorems involved in obtaining f(s) from g(t) and understanding the periodic function.

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