Properties of Inverse Laplace Transform
The Inverse Laplace Transform is a mathematical operation used to find the original function in the time domain from its Laplace Transform in the frequency domain. It involves several properties and formulas that simplify the calculation process. These properties include:
1. Linearity Property
According to this property states that if you have two constants, Ca and Cb, and their respective Laplace Transforms Fa(s) and Fb(s) for functions fa(t) and fb(t), then the Inverse Laplace Transform of Ca Fa(s)+Cb Fb(s) is equal to Ca a times the Inverse Laplace Transform of Fa(s) plus Cb times the Inverse Laplace Transform of Fb(s). In mathematical terms, we can write as
L−1{Ca Fa(s)+Cb Fb(s)}=Ca L−1{Fa(s)}+Cb L−1{Fb(s)},
OR
Ca fa(t)+Cb fb (t)
2. Shifting Property (First Translation)
If the Laplace Transform of eat f(t) is F(s−a), then the Inverse Laplace Transform of F(s−a) is equal to eat ⋅ f(t). Similarly, if it’s F(s−b), then it’s ebt ⋅ f(t).
3. Second Shifting Property (Second Translation)
If L-1{F(s)}=f(t), then L-1{e(-as)F(s)} is equal to g(t), where g(t) =
4. Change of Scale Property
If L-1{F(s)}=f(t), then, L-1{F(as)} is equal to
5. Property of Inverse Laplace Transform of Derivatives
If L−1{F(s)}=f(t), then the Inverse Laplace Transform of dF(s)/ds is equal to −tf(t), and the Inverse Laplace Transform of d2F(s)/ds2 is equal to (−1)2t2f(t).
6. Property of Inverse Laplace Transform of Integrals
If L−1{F(s)}=f(t), then the Inverse Laplace Transform of F(u)d(u) is equal to f(t)/t
7. Property of Multiplication by the Powers of s
According to this property, if L−1{F(s)}=f(t)), then L-1{sF(s)-f(0)}=f(t)) , which is the derivative of f(t). If f(0)equals 0, then L-1{sF(s)} is also equal to f(t).
8. Convolution Theorem
For two functions f(t) and g(t) with Inverse Laplace Transforms f(t) and g(t) respectively, the Inverse Laplace Transform of their product F(s)G(s) is equal to the convolution of f(t) and g(t). This convolution, denoted as f(t)⋅g(t), is defined as the integral from -∞ to ∞ of f(u)g(t-u)du.
Inverse Laplace Transform
In this Article, We will be going through the Inverse Laplace transform, We will start our Article with an introduction to the basics of the Laplace Transform, Then we will go through the Inverse Laplace Transform, will see its Basic Properties, Inverse Laplace Table for some Functions, We will also see the Difference between Laplace Transform and Inverse Laplace Transform, At last, we will conclude our Article with Some examples of inverse Laplace Transform, Applications of inverse Laplace and Some FAQs.
Table of Content
- Inverse Laplace Transform
- Inverse Laplace Transform Theorem
- Inverse Laplace Transform Table
- Laplace Transform Vs Inverse Laplace Transform
- Properties
- Advantages and Disadvantages
- Applications
- Examples