Inverse Laplace Transform

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Inverse Laplace Transform
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1 Inverse Laplace transform

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):


LaTeX: f(t) = \mathcal{L}^{-1} \{F(s)\} = \frac{1}{2 \pi i} \int_{ \gamma - i \cdot \infty}^{ \gamma + i \cdot \infty} e^{st} F(s)\,ds,


where

LaTeX: \gamma is a real number so that the contour path of integration is in the region of convergence of LaTeX: F\left(s\right) normally requiring LaTeX: \gamma > Re \left\{s_p\right\} for every singularity LaTeX: s_p of LaTeX: F\left(s\right) and LaTeX: i^2 = -1.

If all singularities are in the left half-plane, that is LaTeX: Re\left\{s_p\right\} < 0 for every LaTeX: s_p, then LaTeX: \gamma can be set to zero and the above inverse integral formula becomes identical to the inverse Fourier transform.

2 See Also