# Discuss is there a solution or not (why) of Laplace transformation of L(frac{1}{t})

Discuss is there a solution or not (why) of Laplace transformation of
$L\left(\frac{1}{t}\right)$
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Tuthornt
Step 1
The Laplace transform of the function f(t) is defined by the integral:
$L\left(f,s\right)={\int }_{0}^{\mathrm{\infty }}{e}^{-st}f\left(t\right)dt$
for those s where the integral converges
Step 2
First find the Laplace transform for the value of $f\left(t\right)=\frac{1}{t}$
$L\left(f,s\right)={\int }_{0}^{\mathrm{\infty }}{e}^{-st}f\left(t\right)dt$
$f\left(t\right)=\frac{1}{t}$
$L\left(\frac{1}{t},s\right)={\int }_{0}^{\mathrm{\infty }}{e}^{-st}\frac{1}{t}dt$

${\int }_{0}^{\mathrm{\infty }}{e}^{-x}\cdot \frac{s}{x}\cdot \frac{1}{s}dx={\int }_{0}^{\mathrm{\infty }}{e}^{-x}\cdot \frac{\left(dx\right)}{x}$
${\int }_{0}^{\mathrm{\infty }}{e}^{\left(}-x\right)\cdot \frac{\left(dx\right)}{x}\ge {\int }_{0}^{1}{e}^{-x}\cdot \frac{\left(dx\right)}{x}>{e}^{-1}{\int }_{0}^{1}\frac{\left(dx\right)}{x}$

Step 3
Hence, $L\left(\frac{1}{t}\right)$ does not exist.