Example. Find the sine Fourier transform of the function $f(x)$ defined in the interval $(0, \infty)$:

$$
f(x)=\left[\begin{array}{ll}
0, & 0<x<1 \\
1, & 1<x<2 \\
0, & 2<x
\end{array}\right.
$$
Solution.

1. Condition (1) is satisfied, since

$$
\int_{0}^{\infty}|f(x)| d x=\int_{1}^{2} d x=11.
$$
2. $f(x)=x$ when $0<x<1$.
3. $f(x)=e^{-x}$.
4. $f(x)=x e^{-x}$.
5. $f(x)=e^{-x} \cos x$.
6. $f(x)=x e^{-x} \cos x$.
7. $f(x)=\frac{x}{1+x^{2}}$.
8. $f(x)=\frac{x \cos x}{1+x^{2}}$.
9. $f(x)=x e^{-x^{2}}$.
10. $f(x)=0$ when $x>1$.

Answers.

1. $F(p)=\sqrt{\frac{2}{\pi}} \frac{1-\cos p}{p}$.
2. $F(p)=\sqrt{\frac{2}{\pi}} \frac{\sin p-p \cos p}{p^{2}}$.
3. $F(p)=\sqrt{\frac{2}{\pi}} \frac{p}{p^{2}+1}$.
4. $F(p)=\sqrt{\frac{2}{\pi}} \frac{2 p}{\left(p^{2}+1\right)^{2}}$.
5. $F(p)=\frac{1}{\sqrt{2 \pi}}\left[\frac{p+1}{(p+1)^{2}+1}+\frac{p-1}{(p-1)^{2}+1}\right]$.
6. $F(p)=\sqrt{\frac{2}{\pi}}\left\{\frac{p+1}{\left[(p+1)^{2}+1\right]^{2}}+\frac{p-1}{\left[(p-1)^{2}+1\right]^{2}}\right\}$.
7. $F(p)=\sqrt{\frac{\pi}{2}} \operatorname{sign}(p) e^{-|p|}$.
8. $F(p)=\frac{1}{2} \sqrt{\frac{\pi}{2}}\left[\operatorname{sign}(p+1) e^{-|p+1|}+\operatorname{sign}(p-1) e^{-|p-1|}\right]$.
9. $F(p)=\frac{1}{2 \sqrt{2}} p e^{-p^{2} / 4}$.
10. $F(p)=\sqrt{\frac{\pi}{2}} J_{0}(p)$.

## 4.2. Cosine Fourier Transform

Problem Statement. Find the cosine Fourier transform of the function $f(x)$ defined in the interval $(0, \infty)$.

Solution Plan. If the function $f(x)$ is defined in the interval $(0, \infty)$ and

$$
\int_{0}^{\infty}|f(x)| d x<\infty \text { or } \int_{0}^{\infty}|f(x)|^{2} d x<\infty
$$

then its cosine Fourier transform is the function

$$
F(p)=\sqrt{\frac{2}{\pi}} \int_{0}^{\infty} f(x) \cos (p x) d x, \quad p \in(-\infty, \infty)
$$

1. Check that condition (1) is satisfied.
2. Compute the integral on the right-hand side of (2) and write down the answer.