Example 4. Find the function $f(x)$, the complex Fourier transform of which is

$$
F(p)=-i p \ln \frac{1+p^{2}}{p^{2}}
$$
## Solution.

1. We check that condition (4) is satisfied:

$$
\begin{aligned}
\int_{-\infty}^{+\infty}|F(p)|^{2} d p=\int_{-\infty}^{+\infty} p^{2} \ln ^{2} & \frac{p^{2}+1}{p^{2}} d p=2 \int_{0}^{\infty} p^{2} \ln ^{2} \frac{p^{2}+1}{p^{2}} d p= \\
& =2 \int_{0}^{1} p^{2} \ln ^{2} \frac{p^{2}+1}{p^{2}} d p+2 \int_{1}^{\infty} p^{2} \ln ^{2} \frac{p^{2}+1}{p^{2}} d p
\end{aligned}
$$

Here, the first integral converges because the integrand is bounded $\left(\lim _{p \rightarrow 0} p^{2} \ln \left[\left(p^{2}+1\right) / p^{2}\right]=0\right)$. The second integral converges because

$$
p^{2} \ln ^{2} \frac{p^{2}+1}{p^{2}}=p^{2} \ln ^{2}\left(1+\frac{1}{p^{2}}\right) \sim \frac{1}{p^{2}}
$$

2. By formula (5) we have

$$
f(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} F(p) e^{-i p x} d p=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty}(-i p) \ln \frac{p^{2}+1}{p^{2}} e^{-i p x} d p
$$

To simplify the calculations, note that we can first omit the factor $i p$, find

$$
f_{1}(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} \ln \frac{p^{2}+1}{p^{2}} e^{-i p x} d p
$$

and then obtain $f(x)$ using the formula

$$
f(x)=\frac{d}{d x} f_{1}(x)
$$

The integral in (6) is computed by integration by parts followed by the application of Cauchy's residue theorem. We get

$$
f_{1}(x)=\sqrt{2 \pi} \frac{1-e^{-|x|}}{|x|}
$$

Using formula (7), we get

$$
f(x)=\sqrt{2 \pi} \frac{e^{-|x|}(|x|+1)-1}{x^{2}} \operatorname{sign}(x)
$$

Answer. $f(x)=\sqrt{2 \pi} \frac{e^{-|x|}(|x|+1)-1}{x^{2}} \operatorname{sign}(x)$.

Problem conditions. Find the function $f(x)$, the Fourier transform of which is $F(p)$. In problems 1-3, $F(p)$ is the sine Fourier transform. In problems 4-6, $F(p)$ is the cosine Fourier transform. In problems 7-10, $F(p)$ is the complex Fourier transform.

1. $F(p)=p$ for $|p|1 . \quad 2 . \quad F(p)=p e^{-|p|}$.
2. $F(p)=\frac{1}{2}\left[(p+1) e^{-|p+1|}+(p-1) e^{-|p-1|}\right]$.
3. $F(p)=p$ for $|p|1$.
4. $F(p)=e^{-|p|}$.
5. $F(p)=\frac{1}{2}\left[e^{-|p+1|}+e^{-|p-1|}\right]$.
6. $F(p)=p$ for $01$.
7. $F(p)=\frac{1}{p^{2}+2 p+2}$.
8. $F(p)=-i p \operatorname{arctg} p^{2}$.
9. $F(p)=\frac{-i p e^{3 i p}}{4 p^{2}+8 p+5}$.

Answers.

1. $f(x)=\sqrt{\frac{2}{\pi}} \frac{\sin x-x \cos x}{x^{2}}$.
2. $f(x)=\sqrt{\frac{2}{\pi}} \frac{2 x}{\left(x^{2}+1\right)^{2}}$.
3. $f(x)=\sqrt{\frac{2}{\pi}} \frac{2 x \cos x}{\left(x^{2}+1\right)^{2}}$.
4. $f(x)=\sqrt{\frac{2}{\pi}} \frac{x \sin x-1+\cos x}{x^{2}}$.
5. $f(x)=\sqrt{\frac{2}{\pi}} \frac{1}{x^{2}+1}$.
6. $f(x)=\sqrt{\frac{2}{\pi}} \frac{\cos x}{x^{2}+1}$.
7. $f(x)=\frac{1}{\sqrt{2 \pi} i} \frac{(1+i x) e^{-i x}-1}{x^{2}}$. 8. $f(x)=\sqrt{\frac{\pi}{2}} e^{i x} e^{-|x|}$.
8. $f(x)=\sqrt{2 \pi} e^{-|x| / \sqrt{2}} \frac{x \cos (x / \sqrt{2}+\pi / 4)-\sin (x / \sqrt{2})}{x^{2}}$.
9. $f(x)=\frac{1}{2} \sqrt{\frac{\pi}{2}}\left(i-\frac{1}{2} \operatorname{sign}(x-3)\right) e^{i(x-3)} e^{-|x-3| / 2}$.
$\Gamma$ l a 5

## EQUATIONS OF MATHEMATICAL PHYSICS

When studying the topic of EQUATIONS OF MATHEMATICAL PHYSICS, you will learn to solve basic problems for equations of electrostatics, electrodynamics, heat conduction, and continuum mechanics. These equations are called Laplace's equation, Poisson's equation, the wave equation, or the equation of oscillations, and the heat conduction equation. All these equations are partial differential equations. You will learn to find general solutions and other partial differential equations that have not yet found practical applications.

The selection of problems for this chapter and the choice of methods for their solution are carried out in such a way that any typical problem for equations of mathematical physics can either be easily reduced to those included in this chapter or can be solved by one of the presented methods. Exceptions are the Cauchy problems for the wave equation and the heat conduction equation on the plane and in space. It is advisable to solve them using Fourier or Laplace integral transforms.

When solving problems containing partial differential equations, it is usually necessary to make complex variable substitutions, compute integrals, find sums of series, and plot these sums. These actions distract from the essence