L{f(t)}=F(s)=∫ 
0
∞
​
 e 
−st
 f(t)dt how to read this formula?

The formula you provided is a **Laplace Transform**. Here's how to read it step by step:

**L{f(t)} = F(s) = ∫₀^∞ e⁻ˢᵗ f(t) dt**

- **L{f(t)}**: This represents the Laplace transform of a function \( f(t) \).
- **= F(s)**: The Laplace transform converts the function \( f(t) \), which is in the time domain, into \( F(s) \), which is in the frequency domain.
- **∫₀^∞ e⁻ˢᵗ f(t) dt**: This is the integral expression of the Laplace transform.
    - The integral runs from **0 to infinity**.
    - \( e⁻ˢᵗ \) is the exponential decay factor, where **\( s \)** is the Laplace variable and **\( t \)** is time.
    - \( f(t) \) is the original function in the time domain.
    - \( dt \) indicates that the integral is being taken with respect to \( t \).

In words, this equation tells you that the Laplace transform of a function \( f(t) \) is an integral that multiplies the function by an exponentially decaying factor \( e^{-st} \) and sums the result from 0 to infinity.

Would you like more details or have any further questions?

Here are 5 related questions to expand on this:

1. What is the significance of the Laplace transform in differential equations?
2. How does the Laplace transform help in solving linear systems?
3. What are some common functions and their Laplace transforms?
4. Can you explain how the \( s \)-domain is used in control systems?
5. How do we apply the inverse Laplace transform?

**Tip:** The Laplace transform is particularly useful for turning differential equations into algebraic equations, making them easier to solve.

L{f(t)}=F(s)=∫₀^∞ e⁻ˢᵗ f(t) dt how to read this formula?

This page explains how to read and understand the Laplace Transform formula: L{f(t)} = F(s) = ∫₀^∞ e⁻ˢᵗ f(t) dt. Learn about the core concepts of Laplace transforms, integral calculus, and how exponential functions come into play. Suitable for advanced high school or college-level calculus students.

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