On Some Relations between the Laplace and Mellin Transforms

Several earlier papers have applied some identities relating the Laplace and Mellin transforms; this note develops certain such identities, achieving greater generality and rigor. Specifically each of these transforms is expressed as a contour integral involving the other, and an expansion of the Laplace transform is derived in terms of the functions (s·d/ds)
ⁿ
(1+s)
⁻¹
with coefficients defined by the Mellin transform.