chapter  3
39 Pages

## The laplace transform

ByBaidyanath Patra

In article 1.3 of chapter 1 it is noted that if ∫ - ∞ + ∞ | f ( t ) | d t $\int_{ - \infty }^{ + \infty } |f(t)|dt$ is not convergent, Fourier transform F(ξ) of the function f(t) need not exist for all real ξ. For example, when f ( t ) = sin ω t , $f(t) = {\text{sin }}\omega t,$ ω =  real, F(ξ) does not exist. But such situations do arise occasionally in practice. To handle this situation, we consider a new function f 1(t) connected to f(t) defined by