Inverse Laplace Transform of Function of Root by Reciprocal

Theorem

Let $f: \R \to \R$ be a real function.

Then:

$\invlaptrans {\dfrac {\map F {\sqrt s} } s} = \dfrac 1 {\sqrt {\pi t} } \ds \int _0^\infty e^{-{u^2} / 4 t} \map f u \rd u$

where $\invlaptrans {\map F {\sqrt s} }$ denotes the inverse Laplace transform of $\map F {\sqrt s}$.

Proof

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Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Appendix $\text A$: Table of General Properties of Laplace Transforms: $16.$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of General Properties of Laplace Transforms: $32.18$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 33$: Laplace Transforms: Table of General Properties of Laplace Transforms: $33.18$