Find Laplace transform of 1/t | Laplace of 1/t

The Laplace transform of 1/t does not exist. In this post, using the gamma function we will prove that the Laplace of 1 over t does not exist.

Laplace of 1/t

Question: Show that the Laplace transform of 1/t does not exist.

Answer:

By definition of Laplace transforms, the Laplace of 1/t will be equal to

L{1/t} = $\int_0^\infty \dfrac{1}{t}$ e^-stdt …(I)

Put u=st.

So du=s dt ⇒ dt = du/s.

Also, t=u/s.

t	u
0	0
∞	∞

So from (I) we have that

L{1/t} = $\int_0^\infty \dfrac{s}{u} e^{-u} \dfrac{du}{s}$

= $\int_0^\infty u^{-1} e^{-u} du$

= $\int_0^\infty u^{0-1} e^{-u} du$

= Γ(0) as the Gamma function is defined by Γ(z) = $\int_0^\infty$ t^z-1 e^-t dz. So we have obtained that

L{1/t} = Γ(0).

Final Conclusion:

We know Γ(n) = (n-1)!, so Γ(0) is not defined.

As L{1/t}=Γ(0), we conclude that the Laplace transform of 1/t is not defined, hence the Laplace of 1/t does not exist.

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