Quick Math Inverse Laplace Transform
Enter expression in this form
Example
Enter \( F(s) = \frac{s}{s^2 + 1} \) as s/(s^2 + 1)
Inverse Laplace Transform
The Inverse Laplace Transform is the process of finding the original time-domain function \( f(t) \) from its Laplace transform \( F(s) \). This operation is denoted as \( \mathcal{L}^{-1}\{F(s)\} \). The inverse transform allows us to revert a function from the complex frequency domain back to the time domain, which is particularly useful for solving differential equations and analyzing systems in engineering and physics.
Definition
Given a function \( F(s) \) which is the Laplace transform of \( f(t) \), the inverse Laplace transform is defined as:
\[ \mathcal{L}^{-1}\{F(s)\} = f(t) \]
Properties
– Linearity: \( \mathcal{L}^{-1}\{a F(s) + b G(s)\} = a \mathcal{L}^{-1}\{F(s)\} + b \mathcal{L}^{-1}\{G(s)\} \)
– First Shifting Theorem: If \( F(s) = \mathcal{L}\{f(t)\} \), then \( \mathcal{L}^{-1}\{F(s-a)\} = e^{at} f(t) \)
– Second Shifting Theorem: If \( F(s) = \mathcal{L}\{f(t)\} \), then \( \mathcal{L}^{-1}\{e^{-as} F(s)\} = u(t-a) f(t-a) \), where \( u(t-a) \) is the Heaviside step function.
Example
Example: Inverse Laplace Transform of \( \frac{s}{s^2 + 1} \)
Problem: Find the inverse Laplace transform of \( F(s) = \frac{s}{s^2 + 1} \).
Solution:
1. Identify the form: Recognize that \( \frac{s}{s^2 + 1} \) matches the standard Laplace transform of the derivative of the sine function.
2. Reference Standard Transform: The Laplace transform of \( \cos(t) \) is \( \mathcal{L}\{\cos(t)\} = \frac{s}{s^2 + 1} \).
Thus:
\[ \mathcal{L}^{-1}\left\{\frac{s}{s^2 + 1}\right\} = \cos(t) \]
Answer: The inverse Laplace transform of \( \frac{s}{s^2 + 1} \) is \( \cos(t) \).
The inverse Laplace transform is a crucial tool for translating functions from the frequency domain back to the time domain. By utilizing standard Laplace transform pairs and properties, we can effectively find the original time-domain functions. In the given example, the inverse Laplace transform of \( \frac{s}{s^2 + 1} \) was found to be \( \cos(t) \), demonstrating the process of identifying and applying known transform pairs.
