\[ \mathcal{L}\{f(t)\} = F(s), \quad F(s) = \int_0^{\infty} e^{-st} f(t)\, dt \]
| Function \( f(t) \) | Laplace Transform \( F(s) \) |
|---|---|
| \( 1 \) | \( \frac{1}{s}, \quad \text{Re}(s) > 0 \) |
| \( t \) | \( \frac{1}{s^2} \) |
| \( t^n \) | \( \frac{n!}{s^{n+1}} \) |
| \( e^{at} \) | \( \frac{1}{s - a} \) |
| \( \sin(at) \) | \( \frac{a}{s^2 + a^2} \) |
| \( \cos(at) \) | \( \frac{s}{s^2 + a^2} \) |
| \( \sinh(at) \) | \( \frac{a}{s^2 - a^2} \) |
| \( \cosh(at) \) | \( \frac{s}{s^2 - a^2} \) |
| \( u_c(t) \) | \( \frac{e^{-cs}}{s} \) |
| \( \delta(t - a) \) | \( e^{-as} \) |
Time Shifting:
\[ \mathcal{L}\{f(t - a)u(t - a)\} = e^{-as}F(s) \]
Frequency Shifting:
\[ \mathcal{L}\{e^{at}f(t)\} = F(s - a) \]
First Derivative:
\[ \mathcal{L}\{f'(t)\} = sF(s) - f(0) \]
Second Derivative:
\[ \mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f'(0) \]
n-th Derivative:
\[ \mathcal{L}\{f^{(n)}(t)\} = s^nF(s) - s^{n-1}f(0) - \dots - f^{(n-1)}(0) \]
Integration:
\[ \mathcal{L}\left\{\int_0^t f(\tau)\,d\tau\right\} = \frac{F(s)}{s} \]
Linearity:
\[ \mathcal{L}\{a f(t) + b g(t)\} = a F(s) + b G(s) \]
Convolution:
\[ \mathcal{L}\{f(t) * g(t)\} = F(s)G(s) \]
Initial Value Theorem:
\[ \lim_{t \to 0^+} f(t) = \lim_{s \to \infty} sF(s) \]
Final Value Theorem:
\[ \lim_{t \to \infty} f(t) = \lim_{s \to 0} sF(s), \quad \text{if limit exists} \]