Laplace Transform Calculator
Find Laplace transforms for common time-domain functions used in differential equations, circuits, control systems, and engineering math.
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What this calculator teaches
The Laplace transform converts a function of time into a function of s. This often turns differential equations into algebraic equations.
It is especially important in circuits, mechanical vibrations, controls, and systems engineering.
Formula & Symbols
| Concept | Formula | Meaning |
|---|---|---|
| Definition | L{f(t)} = ∫₀∞ e^{-st}f(t)dt | Transforms time-domain function into s-domain. |
| Power | L{t^n}=n!/s^{n+1} | For nonnegative integer n. |
| Exponential shift | L{e^{at}f(t)}=F(s-a) | Shift s by a. |
Step-by-Step Examples
- Use L{tⁿ}=n!/sⁿ⁺¹.
- For n=2, n! = 2.
- Denominator is s³.
- Use L{sin(bt)}=b/(s²+b²).
- Substitute b=4.
Where students use this
Common Mistakes to Avoid
The original function uses t. The transform answer uses s.
Multiplying by e^{at} changes F(s) to F(s-a).
The simple factorial formula is for nonnegative integer powers.
Frequently Asked Questions
Related Math Calculators
Formula Explorer connections
Interpretation: This relationship models the evolution, stability, transform or response of an engineering and differential-equation system. Assumption: Initial conditions, boundary conditions, units, model order and numerical step size must match the physical system; approximations require convergence checks.