Namespaces
Variants
Actions

Difference between revisions of "Carson transform"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
The result of transformation of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020500/c0205001.png" /> defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020500/c0205002.png" /> and vanishing when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020500/c0205003.png" />, into the function
+
{{TEX|done}}
 +
The result of transformation of a function $f(t)$ defined for $-\infty<t<\infty$ and vanishing when $t<0$, into the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020500/c0205004.png" /></td> </tr></table>
+
$$F(s)=s\int\limits_0^\infty f(t)e^{-st}dt,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020500/c0205005.png" /> is a complex variable. The inversion formula is
+
where $s$ is a complex variable. The inversion formula is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020500/c0205006.png" /></td> </tr></table>
+
$$\frac{1}{2\pi i}\int\limits_{\sigma_1-i\infty}^{\sigma_1+i\infty}\frac1sF(s)e^{st}ds.$$
  
The difference between the Carson transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020500/c0205007.png" /> and its [[Laplace transform|Laplace transform]] is the presence of the factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020500/c0205008.png" />.
+
The difference between the Carson transform of $f(t)$ and its [[Laplace transform|Laplace transform]] is the presence of the factor $s$.
  
  

Latest revision as of 06:56, 12 August 2014

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://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