Example. Find the image of the function

$$
f(t)=\left[\begin{array}{ll}
1, & 0 \leqslant t \leqslant 1 \\
0, & t > 1
\end{array}\right.
$$
Solution. Let's represent $f(t)$ as

$$
f(t)=\eta(t)-\eta(t-1)
$$

1. We have

$$
\eta(t) \longleftrightarrow \frac{1}{p}
$$

By the delay theorem for $\tau=1$

$$
\eta(t-1) \longleftrightarrow \frac{e^{-p}}{p}
$$

2. Using the linearity property, we find the desired image

$$
F(p)=\frac{1}{p}-\frac{e^{-p}}{p}
$$

Answer. $F(p)=\frac{1-e^{-p}}{p}$.

Remark. The function $\eta(t)$ is usually omitted. In cases where the delay theorem is applicable, this can lead to errors.

Conditions of the PROBLEM. Find the image of the function $f(t)$.

1. $f(t)=\left[\begin{array}{cl}(t-1)^{2}, & t>1, \\ 0, & t2, \\ 0, & t2 .\end{array}\right.$
3. $f(t)=\left[\begin{array}{ll}t, & 02 \\ 0, & t2 .\end{array}\right.$
5. $f(t)=\left[\begin{array}{cc}e^{-(t-1)}, & t>1, \\ 0, & t1, \\ 0, & t0, \\ 0, & t0, \\ 0, & t<0 .\end{array}\right.$

Answers. 1. $F(p)=e^{-p} / p^{3}$. 2. $F(p)=p e^{-2 p} / p^{2}+1$. 3. $F(p)=$ $=\left(1-2 e^{-p}+e^{-2 p}\right) / p . \quad$ 4. $F(p)=\left(1-e^{-2 p}\right) / p^{2}$. 5. $F(p)=$ $=\left(1-2 e^{-p}+e^{-2 p}\right) / p^{2}$. 6. $F(p)=e^{-p} /(p+1)$. 7. $F(p)=e^{-p} /\left(p^{2}+p\right)$. 8. $F(p)=\left(1-e^{-p}\right) /\left(p+p e^{-p}\right)$. 9. $F(p)=\left(1+e^{-\pi p}\right) /\left[\left(p^{2}+1\right)\left(1-e^{-\pi p}\right)\right]$. 10. $F(p)=\left(p+2 e^{-\pi p / 2}-p e^{-\pi p}\right) /\left[\left(p^{2}+1\right)\left(1-e^{-\pi p}\right)\right]$.

## 2.6. Image of a function of the form $t^{n} f(t)$

STATEMENT OF THE PROBLEM. Find the image of the function $t^{n} f(t)$ $(n \in \mathbb{N})$.

## PLAN OF SOLUTION.

If the function $f(t)$ is an original and $F(p)$ is its image, then by the theorem on differentiation of the image we have $\forall n \in \mathbb{N}$

$$
t^{n} f(t) \longleftrightarrow(-1)^{n} F^{(n)}(p)
$$

1. Find the image $F(p)$ of the function $f(t)$.
2. Compute the derivatives $F^{(n)}(p)$.
3. Find the image of the function $t^{n} f(t)$ using formula (1).