https://file.aiquickdraw.com/mathbot/file/1726355621828ow6gicwk.jpg

The given image shows a piecewise waveform \( f(t) \) that has a constant value \( c \) between \( t = a \) and \( t = b \), and is zero elsewhere.

To find the Laplace transform of this waveform, we can represent the function using step functions (Heaviside functions) as follows:

\[
f(t) = c \cdot [ u(t - a) - u(t - b) ]
\]

Here:
- \( u(t - a) \) is the Heaviside step function that "turns on" at \( t = a \),
- \( u(t - b) \) is the Heaviside step function that "turns off" at \( t = b \).

Now, let's find the Laplace transform of this piecewise function.

### Step 1: Laplace Transform of the Step Function
The Laplace transform of \( u(t - a) \cdot f(t - a) \) is given by:

\[
\mathcal{L}\{ u(t - a) \cdot f(t - a) \} = e^{-as} \mathcal{L}\{ f(t) \}
\]

### Step 2: Apply to the Waveform
The Laplace transform of \( f(t) = c \cdot [ u(t - a) - u(t - b) ] \) becomes:

\[
\mathcal{L}\{ f(t) \} = c \cdot \left[ \mathcal{L}\{ u(t - a) \} - \mathcal{L}\{ u(t - b) \} \right]
\]

Using the Laplace transform of the Heaviside function:

\[
\mathcal{L}\{ u(t - a) \} = \frac{e^{-as}}{s}, \quad \mathcal{L}\{ u(t - b) \} = \frac{e^{-bs}}{s}
\]

### Step 3: Final Laplace Transform
Thus, the Laplace transform of the given waveform is:

\[
\mathcal{L}\{ f(t) \} = c \left( \frac{e^{-as}}{s} - \frac{e^{-bs}}{s} \right)
\]

\[
\mathcal{L}\{ f(t) \} = \frac{c}{s} \left( e^{-as} - e^{-bs} \right)
\]

This is the Laplace transform of the given waveform.

Would you like further clarification or details on this process? Here are five related questions for deeper understanding:

1. How does the Heaviside step function help in solving piecewise functions using Laplace transforms?
2. What is the physical interpretation of the Laplace transform in the context of this waveform?
3. How would the Laplace transform change if the waveform had a different amplitude or different intervals?
4. Can we find the inverse Laplace transform from this expression to recover the original function?
5. How can the Laplace transform be applied to analyze electrical circuits with step inputs?

**Tip:** When dealing with piecewise functions, always break them down using step functions, as it simplifies the process of finding the Laplace transform.

Find the Laplace Transform for the waveform shown.

Discover how to find the Laplace Transform of a waveform with constant value 'c' between t = a and t = b, using Heaviside step functions. This example covers step functions and piecewise functions for engineering and applied mathematics students.

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