Laplace Transform of cos(t)/t

The Laplace transform of cos(t)/t does not exist as we get a divergent integral from the definition of the Laplace transform. In this article, we will discuss about L{cos(t)/t}.

What is the Laplace Transform of cos(t)/t?

Answer: The Laplace transform of cos(t)/t does not exist.

Proof:

We know that the Laplace transform of a function f(t) divided by t, denoted by L{f(t)/t}, is given by the following division by t Laplace transform formula:

$L\{\frac{f(t)}{t} \} =\int_s^\infty F(s) ds$, where $L\{f(t)\}=F(s)$ …(I)

Step 1: Put f(t) = cos(t) in the above formula.

∴ F(s) = L{f(t)} = L{cos(t)} = s/(s²+1)

Step 2: Now, applying the formula (I), the Laplace transform of cos(t)/t is equal to

L{cos(t)/t} = $\int_s^\infty \dfrac{s}{s^2+1} ds$

Step 3: Put s²+1 = z.

∴ 2s ds=dz

t	z
s	s2+1
∞	∞

Step 4: Then L{cos(t)/t} = $\dfrac{1}{2}\int_s^\infty \dfrac{2s}{s^2+1} ds$

= $\dfrac{1}{2}\int_{s^2+1}^\infty \dfrac{dz}{z}$

= $\dfrac{1}{2}$ $\Big[ \log(z)\Big]_{s^2+1}^\infty$ as the integration of dx/x is log x.

= $\dfrac{1}{2}$ $\Big[ \log \infty – \log(s^2+1)\Big]$

→ ∞.

Thus, the Laplace transform of cos(t)/t is a divergent integral and so it does not exist. This is proved by the definition of Laplace transforms.

Find the Laplace transform of cos(t)/t. Summary: L{cos(t)/t} does NOT exist.

Also Read:

Laplace transform of t:	1/s2
Laplace transform of sin t:	1/(s2+1)
Laplace transform of sin(t)/t:	tan-1(1/s)
Laplace transform of cos t:	s/(s2+1)
Laplace transform of e-t:	1/(s+1)
Laplace transform of 1:	1/s

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