An integral formula for the inverse Laplace transform, called the Bromwich integral, the Fourier–Mellin integral, and Mellin's inverse formula, is given by the line integral:
where the integration is done along the vertical line in the complex plane such that is greater than the real part of all singularities of F(s). This ensures that the contour path is in the region of convergence. If all singularities are in the left half-plane, or F(s) is a smooth function on - ∞ < Re(s) < ∞ (i.e. no singularities), then can be set to zero and the above inverse integral formula above becomes identical to the inverse Fourier transform.
In practice, computing the complex integral can be done by using the Cauchy residue theorem.
It is named after Hjalmar Mellin, Joseph Fourier and Thomas John I'Anson Bromwich.
If F(s)is the Laplace transform of the function f(t),then f(t) is called the inverse Laplace transform of F(s).
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