Enter values and click Calculate
Calculator Use
The Laplace Transform Calculator is a specialized tool designed to convert functions from the time domain, usually denoted as f(t), into the complex frequency domain, denoted as F(s). This mathematical transformation is a cornerstone of engineering, physics, and mathematics, particularly in the study of differential equations and control systems.
By selecting the appropriate function type and entering the necessary parameters such as constants, exponents, or frequencies, you can instantly find the resulting s-domain expression without performing complex integration by hand.
- Function Type
- Choose the mathematical form of your time-domain function (e.g., exponential, trigonometric, or power functions).
- Parameters (a, n, c)
- Input the coefficients specific to your equation. 'a' typically represents a rate or frequency, 'n' represents a power of t, and 'c' is a scale constant.
- Show Steps
- Enable this option to view the underlying transform formula used to derive the result.
How It Works
The Laplace transform is defined by an improper integral. For a function f(t) defined for all real numbers t ≥ 0, the Laplace transform is the function F(s) defined by:
F(s) = ∫ [0 to ∞] e^(-st) f(t) dt
Our laplace transform calculator uses established tables and identities to provide results for the most common functions used in academic and professional settings:
- Linearity: The transform of a sum is the sum of the transforms.
- Frequency Shifting: Multiplying by e^(at) in the time domain results in a shift in the s-domain.
- Time Scaling: Adjusting the 't' variable impacts the magnitude and frequency of the s-domain result.
- Power Functions: Specifically using the Gamma function properties (simplified to factorials for integers).
Laplace Transform Table & Formulas
Below are the standard formulas this laplace transform calculator employs for various operations:
| Time Domain f(t) | s-Domain F(s) |
|---|---|
| Constant (1) | 1 / s |
| t^n | n! / s^(n+1) |
| e^(at) | 1 / (s – a) |
| sin(at) | a / (s² + a²) |
| cos(at) | s / (s² + a²) |
Calculation Example
Example: Find the Laplace transform of the function f(t) = t² * e^(3t).
Step-by-step solution:
- Identify the basic components: This is a "Damped Power" function.
- Set the parameters: n = 2 (the power of t) and a = 3 (the exponent coefficient).
- Recall the formula: L{t^n * e^(at)} = n! / (s – a)^(n+1).
- Substitute the values: 2! / (s – 3)^(2+1).
- Calculate the factorial: 2! = 2 * 1 = 2.
- Final Result: F(s) = 2 / (s – 3)³
Common Questions
Why do we use Laplace transforms?
Laplace transforms are primarily used to simplify differential equations. By converting a calculus problem (differentiation/integration) into an algebraic problem, they make it significantly easier to solve systems, especially those involved in circuit analysis and mechanical vibrations.
What is the "s" in the result?
In the Laplace domain, s is a complex frequency variable, defined as s = σ + jω. It represents the "spatial frequency" of the signal. While the time domain deals with "when" something happens, the s-domain deals with "how fast" and with what damping the system behaves.
Can this calculator do Inverse Laplace transforms?
This specific laplace transform calculator focuses on the forward transform (Time to s-domain). For the inverse, you would typically use partial fraction decomposition on the F(s) expression to match it back to the entries in the Laplace table.
Does the power 'n' have to be an integer?
In introductory physics and engineering, 'n' is almost always a non-negative integer. If 'n' were a fraction, the transform would involve the Gamma function Γ(n+1). For simplicity, this calculator currently supports integer powers.