List of formulas related to Laplace’s local and integral theorems:
Local Laplace theorem
The local Laplace theorem states that for a function ( f(x) ) that satisfies the regularity conditions, one can write:
[
f(x) \approx f(a) + f'(a)(x – a) + \frac{f”(a)}{2!}(x – a)^2 + \frac{f”’ (a)}{3!}(x – a)^3 + \ldots
]
where (a) is the point in the neighborhood of which the decomposition is performed.
Laplace’s integral theorem
Laplace’s integral theorem, also known as the limit passage theorem, can be written as:
[
\int_{-\infty}^{\infty} e^{-tx} f(x) \, dx = \mathcal{L}{f(t)}
]
where ( \mathcal{L} ) is the Laplace transform of the function ( f(t) ).