This tutorial covers the fundamentals of the Laplace transform, inverse Laplace transform, properties, partial fraction expansion, and their applications in solving differential equations.
1. Laplace Transform: Definition and Conditions
Definition
The Laplace Transform of a time-domain function $f(t)$ is given by:
$$
L[f(t)]=F(s)= \int_{0^-}^\infty f(t) e^{-st} dt
$$
- $s$: A complex variable $s=\sigma + j\omega$.
- Purpose: Converts time-domain problems into the domain for easier manipulation.
Conditions for Existence
For $L[f(t)]$] to exist:
- Existence: $f(t)$ is defined and finite for $t \geq 0$.
- Absolute Integrability: $f(t)$ must decay exponentially:
This ensures:
$$
\exists \, c, M, T > 0 \; \text{s.t.} \; |f(t)| \leq Me^{ct}, \, \forall t > T.
$$
- Piecewise Continuity: $f(t)$ is continuous except for a finite number of discontinuities in any finite interval.
2. Intuition Behind Laplace Transform
- Converts time functions (e.g., signals) into their frequency equivalents.
- Helps analyze systems (especially linear time-invariant systems) in the frequency domain by simplifying convolution and differential equations into algebraic equations.
3. Inverse Laplace Transform