Laplace Transform Calculator

Enter Time-Domain Function (t):

Result:

Laplace Transform Theory

The Laplace Transform is defined as:

\[ \mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt \]

Common Transforms

Time Domain (f(t))	Frequency Domain (F(s))
1	\(\frac{1}{s}\)
\(e^{at}\)	\(\frac{1}{s-a}\)
\(t^n\)	\(\frac{n!}{s^{n+1}}\)
\(\sin(\omega t)\)	\(\frac{\omega}{s^2 + \omega^2}\)

Properties

• Linearity: \(\mathcal{L}\{af(t) + bg(t)\} = aF(s) + bG(s)\)
• Differentiation: \(\mathcal{L}\{f'(t)\} = sF(s) - f(0)\)
• Integration: \(\mathcal{L}\left\{\int_0^t f(\tau)d\tau\right\} = \frac{F(s)}{s}\)

Note: This calculator supports basic transforms. Complex functions may show integral form.