$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ ###[Problems to Sections 5.1, 5.2](id:sect-3.2.P) > 1. [Problem 1](#problem-5.2.P.1) > 2. [Problem 2](#problem-5.2.P.2) > 3. [Problem 3](#problem-5.2.P.3) > 4. [Problem 4](#problem-5.2.P.4) > 5. [Problem 5](#problem-5.2.P.5) > 6. [Problem 6](#problem-5.2.P.6) > 7. [Problem 7](#problem-5.2.P.7) Some of the problems could be solved based on the other problems and properties of Fourier transform (see [Section 5.2](./S5.2.html)) and such solutions are much shorter than from the scratch; seeing and exploiting connections is a plus. **[Problem 1.](id:problem-5.2.P.1)** Let $F$ be an operator of Fourier transform: $f(x)\to \hat{f}(k)$. Prove that a. $F^\*F=FF^\*=I$; b. $(F^2 f)(x)=f(-x)$ and therefore $F^2f=f$ for even function $f$ and $F^2=-f$ for odd function $f$; c. $F^4=I$; d. If $f$ is a real-valued function then $\hat{f}$ is real-valued if and only if $f$ is even and $i\hat{f}$ is real-valued if and only if $f$ is odd. **[Problem 2.](id:problem-5.2.P.2)** Let $\alpha\>0$. Find Fourier transforms of a. $e^{-\alpha|x|}$; b. $e^{-\alpha|x|}\cos(\beta x)$, $e^{-\alpha|x|}\sin (\beta x)$ with $\beta\>0$; c. $x e^{-\alpha|x|}$ with $\beta\>0$; d. $xe^{-\alpha|x|}\cos(\beta x)$, $x e^{-\alpha|x|}\sin (\beta x)$ with $\beta\>0$. **[Problem 3.](id:problem-5.2.P.3)** Let $\alpha\>0$. Find Fourier transforms of a. $(x^2+\alpha^2)^{-1}$; b. $x(x^2+\alpha^2)^{-1}$; c. $(x^2+\alpha^2)^{-1}\cos (\beta x)$, $(x^2+\alpha^2)^{-1}\sin(\beta x)$; d. $x(x^2+\alpha^2)^{-1}\cos (\beta x)$, $x(x^2+\alpha^2)^{-1}\sin(\beta x)$. **[Problem 4.](id:problem-5.2.P.4)** Let $\alpha\>0$. Based on Fourier transform of $e^{-\alpha x^2/2}$ find Fourier transforms of a. $e^{-\alpha x^2/2}\cos (\beta x)$, $e^{-\alpha x^2/2}\sin (\beta x)$; b. $ x e^{-\alpha x^2/2}\cos (\beta x)$, $x e^{-\alpha x^2/2}\sin (\beta x)$. **[Problem 5.](id:problem-5.2.P.5)** Find Fourier transforms of a. $f(x)=\left\\{\begin{aligned} & 1&& |x|\le a,\\\\ & 0 && |x|\> a;\end{aligned}\right.$ b. $f(x)=\left\\{\begin{aligned} & x && |x|\le a,\\\\ & 0 && |x|\> a;\end{aligned}\right.$ c. Using (a) calculate $\int\_{-\infty}^\infty \frac{\sin (x)}{x}\,dx$. **[Problem 6.](id:problem-5.2.P.6)** a. Prove the same properties as in [Problem 1](#problem-5.2.P.1) for multidimensional Fourier tramsform (see [Subection 5.2.A](./S5.2.A.html). b. Prove that $f$ if multidimensional function $f$ has a rotational symmetry (that means $f(Q\mathbf{x})= f(\mathbf{x})$ for all orthogonal transform $Q$) then $\hat{f}$ also has a rotational symmetry (and conversely). *Note.* Equivalently $f$ has a rotational symmetry if $f(\mathbf{x})$depend only on $|\mathbf{x}|$. **[Problem 7.](id:problem-5.2.P.7)** Find multidimenxional Fourier transforms of a. $f(x)=\left\\{\begin{aligned} & 1&& |\mathbf{x}|\le a,\\\\ & 0 && |\mathbf{x}|\> a;\end{aligned}\right.$ b. $f(x)=\left\\{\begin{aligned} &a-|\mathbf{x}| &&|\mathbf{x}|\le a,\\\\ &0 &&|\mathbf{x}|\> a, \end{aligned}\right.$; c. $f(x)=\left\\{\begin{aligned} &a^2-|\mathbf{x}|^2 &&|\mathbf{x}|\le a,\\\\ &0 &&|\mathbf{x}|\> a, \end{aligned}\right.$; d. $f(x)=e^{-\alpha |\mathbf{x}|}$. *Hint.* Using [Problem 6(b)](id:problem-5.2.P.6) observe that we need to find only $\hat{f}(0,\ldots,0, k)$ and use appropriate coordinate system (polar as $n=2$, or spherical as $n=3$ and so one). *Note.* This problem could be solved as $n=2$, $n=3$ or $n\ge 2$ (any). ___________________ [$\Uparrow$](../contents.html)  [$\uparrow$](./S5.2.html)  [$\Rightarrow$](./S5.3.html)