Laplace transform

Definition

The (unilateral) Laplace transform $\mathcal{L}$ is defined by $$\mathcal{L}[f(t)](s) \equiv \int_0^\infty f(t) e^{-st} dt$$ The inverse of the Laplace transform, i.e. the Bromwich integral, is given by $$F(t) = \frac{1}{2\pi i} \int_{\gamma - i \infty}^{\gamma + i \infty} e^{st} f(s) ds$$

Table

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References

1. https://mathworld.wolfram.com/LaplaceTransform.html
2. https://mathworld.wolfram.com/BromwichIntegral.html