first shifting theorem
Often the function you want to transform is some basic function dressed up by multiplying it with a decaying exponential — think of a vibration that dies out, e^{-t} sin(omega t). The first shifting theorem says you do not need a new table entry for the dressed-up version: you take the transform of the plain function and simply slide it along the s-axis.
Precisely: if L{f(t)} = F(s), then L{e^{a t} f(t)} = F(s - a). Multiplying f(t) by e^{a t} in the time domain corresponds to replacing s by s - a in the transform — a horizontal shift of the s-variable. This is why it is also called s-shifting or frequency shifting. For example, since L{cos(omega t)} = s/(s^2 + omega^2), the theorem gives L{e^{a t} cos(omega t)} = (s - a)/((s - a)^2 + omega^2) instantly, by writing s - a wherever s appeared.
In applied work this theorem is the reason damped oscillations are so easy to handle. The decay rate a moves the poles of the transform off the imaginary axis, and reading the rule backward is the standard inverse-transform move: when partial fractions leave you with a term like 1/((s - a)^2 + omega^2), you recognize the shift and recover e^{a t} sin(omega t)/omega. It pairs with completing the square, which is how you spot the hidden s - a in a quadratic denominator.
To invert 1/(s^2 + 4 s + 13), complete the square: s^2 + 4 s + 13 = (s + 2)^2 + 9. The shift a = -2 and omega = 3 give the inverse e^{-2 t} sin(3 t)/3.
Completing the square exposes the s + 2, which the first shifting theorem reads as an e^{-2 t} envelope on a sine.
Do not confuse this with the second shifting theorem: the first shifts the transform variable s (an exponential factor in time), while the second shifts the time variable t (a delayed switch-on). They look superficially similar but apply in opposite directions.