6.4 Inversion
Theorem 6.4.1 (Lerch). If \(f\) and \(g\) are piecewise continuous, of exponential order, and \(F(s)=G(s)\) for all \(s\) in some half-plane, then \(f=g\) at every point of continuity. The inverse transform is therefore essentially unique.
In practice inversion is done by recognising entries of Table 6.1, after partial fractions and the shifting rules have been used to reduce \(F(s)\) to recognisable pieces. For completeness, the general formula is the following.
Theorem 6.4.2 (Bromwich integral). \[f(t) = \frac {1}{2\pi i}\int ^{\gamma +i\infty }_{\gamma -i\infty }F(s)\,e^{st}\,ds,\] the contour being any vertical line with \(\gamma \) to the right of every singularity of \(F\).
Remark 6.4.3. The Bromwich integral is the Laplace analogue of the Fourier inversion formula of Section 5.9, and the resemblance is not accidental: substituting \(s=\gamma +i\omega \) turns it into a Fourier inversion of \(e^{-\gamma t}f(t)\). This is the precise sense in which the Laplace transform is the Fourier transform of a function damped by \(e^{-\gamma t}\) and cut off below \(t=0\), and it is why the two transforms share their operational properties. Evaluating the contour integral requires the residue calculus and is left to a course in complex analysis.
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