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}.
Table of Contents
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 |
FAQs
Answer: The Laplace transform of cos(at)/t does not exist.
Answer: The Laplace transform of cos(at) is a/(s2+a2).
