Editing Laplace transform (section)

=== Inverse Laplace transform ===
{{Main article|Inverse Laplace transform}}
Two integrable functions have the same Laplace transform only if they differ on a set of [[Lebesgue measure]] zero. This means that, on the range of the transform, there is an inverse transform. In fact, besides integrable functions, the Laplace transform is a [[one-to-one function|one-to-one mapping]] from one function space into another in many other function spaces as well, although there is usually no easy characterization of the range.

Typical function spaces in which this is true include the spaces of bounded continuous functions, the space {{math|[[Lp space|''L''<sup>∞</sup>(0, ∞)]]}}, or more generally [[tempered distributions]] on {{open-open|0, ∞}}. The Laplace transform is also defined and injective for suitable spaces of tempered distributions.

In these cases, the image of the Laplace transform lives in a space of [[analytic function]]s in the [[#Region of convergence|region of convergence]].  The [[inverse Laplace transform]] is given by the following complex integral, which is known by various names (the '''Bromwich integral''', the '''Fourier–Mellin integral''', and '''Mellin's inverse formula'''):
{{Equation box 1
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|equation = <math>f(t) = \mathcal{L}^{-1}\{F\}(t) = \frac{1}{2 \pi i} \lim_{T\to\infty} \int_{\gamma - i T}^{\gamma + i T} e^{st} F(s)\, ds,</math>
|ref = Eq. 3
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where {{mvar|γ}} is a real number so that the contour path of integration is in the region of convergence of {{math|''F''(''s'')}}. In most applications, the contour can be closed, allowing the use of the [[residue theorem]]. An alternative formula for the inverse Laplace transform is given by [[Post's inversion formula]]. The limit here is interpreted in the [[Weak topology#Weak-* topology|weak-* topology]].

In practice, it is typically more convenient to decompose a Laplace transform into known transforms of functions obtained from a table and construct the inverse by inspection.