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) ).

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