Laplace Transform Formulas & Theory
Reference guide for Laplace transform formulas and theoretical background
Laplace Transform Formulas
| Function Name | Time Domain f(t) | Laplace Transform F(s) | Convergence Condition |
|---|---|---|---|
| Constant | c | c/s | s > 0 |
| Power Function | t^n | n!/s^(n+1) | s > 0, n > -1 |
| Exponential Function | e^(at) | 1/(s-a) | s > a |
| Sine Function | sin(at) | a/(s^2+a^2) | s > 0 |
| Cosine Function | cos(at) | s/(s^2+a^2) | s > 0 |
| Hyperbolic Sine Function | sinh(at) | a/(s^2-a^2) | s > |a| |
| Hyperbolic Cosine Function | cosh(at) | s/(s^2-a^2) | s > |a| |
| Exponentially Decaying Sine Function | e^(-at)sin(bt) | b/((s+a)^2+b^2) | s > -a |
| Exponentially Decaying Cosine Function | e^(-at)cos(bt) | (s+a)/((s+a)^2+b^2) | s > -a |
| Step Function | u(t-a) | e^(-as)/s | s > 0, a > 0 |
| Impulse Function | δ(t-a) | e^(-as) | a ≥ 0 |
| Ramp Function | t | 1/s^2 | s > 0 |
| Quadratic Function | t^2 | 2/s^3 | s > 0 |
| t times Exponential Function | t*e^(at) | 1/(s-a)^2 | s > a |
| t times Sine Function | t*sin(at) | 2as/((s^2+a^2)^2) | s > 0 |
| t times Cosine Function | t*cos(at) | s^2-a^2/((s^2+a^2)^2) | s > 0 |
Laplace Transform Theory
What is the Laplace Transform?
The Laplace transform is an integral transform that converts a function of time f(t) to a function of complex frequency s, F(s). The Laplace transform has widespread applications in solving differential equations, analyzing control systems, and signal processing.
Definition of the Laplace Transform
The Laplace transform of a function f(t) is defined as:
Inverse Laplace Transform
The inverse Laplace transform converts a function F(s) back to the time domain function f(t):
Application Areas
Circuit Analysis
In circuit analysis, Laplace transform is used to convert differential equations in the time domain to algebraic equations in the s-domain, simplifying the analysis process.
Control Systems
In control systems, Laplace transform is used to analyze system stability, response characteristics, and transfer functions.
Signal Processing
In signal processing, Laplace transform is used to analyze the frequency domain characteristics of continuous-time signals.
Differential Equation Solving
Laplace transform is a powerful tool for solving linear constant-coefficient differential equations, especially for non-homogeneous equations and initial value problems.
Mechanical Vibrations
In mechanical vibration analysis, Laplace transform is used to solve for the response of vibrating systems.
Heat Conduction
In heat conduction problems, Laplace transform is used to solve the heat conduction equation.