Does the laplace transform of 1 / t exist? If yes, how do we calculate it? Putting it in ∫ 0 ∞ ( e − s t / t ) d t won't solve. Is there any other way?

Question

Does the laplace transform of   1      /    t exist? If yes, how do we calculate it? Putting it in      ∫    0    ∞    (      e          −      s      t            /    t  )  d  t won&#039;t solve. Is there any other way?

Milton Hampton · Accepted Answer

No, it doesn&#039;t exist. In general the Laplace transform of       t    n   is             Γ      (      n      +      1      )              s              n        +        1            , and   Γ  (  n  ) isn&#039;t defined on 0,−1,−2,−3... This integral is the definition of the Laplace transform, so the transform doesn&#039;t exist if the integral doesn&#039;t. While there are other integral transforms that could transform       1    t   in a useful way, anything other than what you gave wouldn&#039;t be considered a Laplace transform anymore.

Jaylyn Horne · Accepted Answer

You can actually simplify it further by substituting   s  t  =  x, so you&#039;ll get  L      (          1      t        )    =      ∫    0    ∞                      e                    −        x              x           d    xwhich is a divergent integral. In other words, the transform doesn&#039;t converge for any value of s.

Does the laplace transform of 1/t exist? If yes, how do we calculate it?

Answered question

Answer & Explanation

New Questions in Differential Equations