Laplace transform
Notation
$$ F(s) = \mathcal{L}\{ f(t) \} $$
Inverse transform
$$ f(t) = \mathcal{L}^{-1}\{ F(s) \} $$
Example: Find the Laplace transform of \( f(t) = a \) (constant)
Start from the definition
$$ F(s) = \int_{0}^{\infty} f(t) e^{-st} dt $$
Substitute \( f(t)=a \)
$$ F(s) = \int_{0}^{\infty} a\, e^{-st} dt $$
Factor out the constant
$$ F(s) = a \int_{0}^{\infty} e^{-st} dt $$
Integrate with respect to \(t\)
$$ a \left[ \frac{e^{-st}}{-s} \right]_{t=0}^{\infty} $$
Evaluate the limits (for \(\operatorname{Re}(s) > 0\))
$$ a\, \frac{\lim\limits_{t\to\infty} e^{-st} - e^{0}}{-s}
= a\, \frac{0 - 1}{-s} = \frac{a}{s} $$
Note. The limit \(\lim_{t\to\infty} e^{-st}=0\) holds for \(\operatorname{Re}(s) > 0\).