Carson transform

The result of transformation of a function $f(t)$ defined for $-\infty<t<\infty$ and vanishing when $t<0$, into the function

$$F(s)=s\int\limits_0^\infty f(t)e^{-st}dt,$$

where $s$ is a complex variable. The inversion formula is

$$\frac{1}{2\pi i}\int\limits_{\sigma_1-i\infty}^{\sigma_1+i\infty}\frac1sF(s)e^{st}ds.$$

The difference between the Carson transform of $f(t)$ and its Laplace transform is the presence of the factor $s$.

Comments

Two well-known references for the Laplace transformation are [a1], which stresses the theory, and [a2], which stresses applications.

References

[a1]	D.V. Widder, "The Laplace transform" , Princeton Univ. Press (1972)
[a2]	G. Doetsch, "Introduction to the theory and application of the Laplace transformation" , Springer (1974) (Translated from German)