# Carson transform

From Encyclopedia of Mathematics

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) |

**How to Cite This Entry:**

Carson transform.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Carson_transform&oldid=32847

This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article