Example. Find the image of the function

$$
f(t)=\cos \omega t \quad(\omega>0)
$$
Solution. We have

$$
\cos t \longleftrightarrow \frac{p}{p^{2}+1}
$$

Then, by the similarity theorem $\forall \omega>0$

$$
\cos \omega t \longleftrightarrow \frac{1}{a} \frac{p / \omega}{(p / \omega)^{2}+1}=\frac{p}{p^{2}+\omega^{2}}
$$

Answer. $\quad \cos \omega t \longleftrightarrow \frac{p}{p^{2}+\omega^{2}}$.

Remark. By calculating the image from the definition or using Euler's formula and the linearity property, it is easy to verify that the obtained formula is valid $\forall \omega \in \mathbb{C}$.

Conditions of the Problems. Find the image of the function $f(t) \eta(t)$.

1. $f(t)=\sin 2 t . \quad 2 . f(t)=\operatorname{ch} 2 t . \quad 3 . f(t)=\operatorname{sh} 2 t . \quad 4 . f(t)=$ $=\sin (t / 2) . \quad 5 . f(t)=\cos (t / 2) . \quad 6 . f(t)=\sin ^{2} t . \quad 7 . f(t)=\cos ^{2} t$. 8. $f(t)=\sin 3 t \cdot \cos 2 t . \quad$ 9. $f(t)=\sin (\omega t+\varphi) . \quad$ 10. $f(t)=\cos (\omega t+\varphi)$.

Answers. 1. $F(p)=2 /\left(p^{2}+4\right) . \quad 2 . \quad F(p)=p /\left(p^{2}-4\right) . \quad 3 . F(p)=$ $=2 /\left(p^{2}-4\right)$. 4. $F(p)=2 /\left(4 p^{2}+1\right)$. 5. $F(p)=4 p /\left(4 p^{2}+1\right)$. 6. $F(p)=2 /\left(p^{3}+4 p\right)$. 7. $F(p)=\left(p^{2}+2\right) /\left(p^{3}+4 p\right)$. 8. $F(p)=$ $=\left(3 p^{2}+15\right) /\left(p^{4}+26 p^{2}+25\right)$. 9. $F(p)=(\omega \cos \varphi+p \sin \varphi) /\left(p^{2}+\omega^{2}\right)$. 10. $F(p)=(p \cos \varphi-\omega \sin \varphi) /\left(p^{2}+\omega^{2}\right)$.

## 2.4. Image of a Function of the Form $e^{-a t} f(t)$

Statement of the Problem. Find the image of the function $e^{-a t} f(t)$ $(a \in \mathbb{C})$, if $f(t) \longleftrightarrow F(p)$

## Plan of Solution.

Damping (or mixing) theorem.

If the function $f(t)$ is an original and $F(p)$ is its image, then $\forall a \in \mathbb{C}$ the function $e^{-a t} f(t)$ is also an original and its image is given by the formula

$$
e^{-a t} f(t) \longleftrightarrow F(p+a)
$$