Laplace–Carson transform

Let ${\displaystyle V(j,t)}$ be a function and ${\displaystyle p}$ a complex variable. The Laplace–Carson transform is defined as:[1]

${\displaystyle V^{\ast }(j,p)=p\int _{0}^{\infty }V(j,t)e^{-pt}\,dt}$

The inverse Laplace–Carson transform is:

${\displaystyle V(j,t)={\frac {1}{2\pi i}}\int _{a_{0}-i\infty }^{a_{0}+i\infty }e^{tp}{\frac {V^{\ast }(j,p)}{p}}\,dp}$

where ${\displaystyle a_{0}}$ is a real-valued constant, ${\displaystyle i\infty }$ refers to the imaginary axis, which indicates the integral is carried out along a straight line parallel to the imaginary axis lying to the right of all the singularities of the following expression:

${\displaystyle e^{tp}{\frac {V(j,t)}{p}}}$

• Laplace transform