$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\Ai}{\mathrm{Ai}}$ ##[Problems to Chapter 2](id:sect-2.P) > 1. [Problem 1](#problem-2.P.1) > 2. [Problem 2](#problem-2.P.2) > 3. [Problem 3](#problem-2.P.3) > 4. [Problem 4](#problem-2.P.4) > 5. [Problem 5](#problem-2.P.5) > 6. [Problem 6](#problem-2.P.6) > 7. [Problem 7](#problem-2.P.7) > 8. [Problem 8](#problem-2.P.8) > 9. [Problem 9](#problem-2.P.9) > 10. [Problem 9](#problem-2.P.10) **[Definition 1.](id:definition-1.P.1)** $\Gamma$-function is defined as \begin{equation} \Gamma (z)= \int\_0^\infty e^{-t}t^{z-1}\,dz \label{eq-2.P.1} \end{equation} as $\Re z>0$; it satisfies \begin{equation} \Gamma(z)=(z-1)\Gamma(z-1) \label{eq-2.P.2} \end{equation} and therefore could be extended to $\bC$ as a meromorphic function with poles at $0,-1,-2,\ldots$. Also as $z=1,2,\ldots$ $\Gamma(z)=(z-1)!$ **[Problem 1.](id:problem-2.P.1)** a. Plugging $t=y^m/m$ calculate \begin{equation\*} \int\_0^\infty e^{-y^m/m} y^n\,dy \end{equation\*} in terms of $\Gamma$-function. b. As $m =2,4,\ldots$ calculate \begin{equation\*} \int\_{-\infty}^\infty e^{-y^m/m} y^n\,dy. \end{equation\*} **[Problem 2.](id:problem-2.P.2)** As $\bR\ni z\to +\infty$ calculate a. First term in the asymptotics of $\Gamma(z+1)$ and thus justify [Stirling formula](https://en.wikipedia.org/wiki/Stirling%27s\_approximation) b. Calculate the second term in this approximation. c. Justify these asymptotics by plugging $y=zt$. **[Problem 3.](id:problem-2.P.3)** Calculate first two terms in the asymptotics as $k\to +\infty$ of \begin{gather} \int\_0^{\pi/3} e^{k\sin(x)}\,dx,\tag{3a}\\\\ \int\_0^{\pi/2} e^{k\sin(x)}\,dx,\tag{3b}\\\\ \int\_0^{\pi} e^{k\sin(x)}\,dx,\tag{3c}\\\\ \int\_0^{2\pi} e^{k\sin(x)}\,dx,\tag{3d}\\\\ \int\_0^{3\pi} e^{k\sin(x)}\,dx.\tag{3e} \end{gather} **[Problem 4.](id:problem-2.P.4)** Calculate the first term in the asymptotics as $k\to +\infty$ of \begin{equation} \iint\_D e^{k\sin(x)\sin(y)}\,dxdy\tag{4} \end{equation} where a. $D=\\{(x,y): 0 < x < \pi, 0 < y < \pi\\}$, b. $D=\\{(x,y): x^2+y^2 < 5\\}$. **[Problem 5.](id:problem-2.P.5)** Calculate the first term in the asymptotics as $k\to +\infty$ of \begin{equation} \iint\_D e^{k\sin(x)\sin(y)\cos(z)}\,dxdydz\tag{5} \end{equation} where a. $D=\\{(x,y): 0 < x < \pi, 0 < y < \pi, -\pi/2 < z < \pi/2 \\}$, b. $D=\\{(x,y): x^2+y^2 +z^2< 5\\}$. **[Problem 6.](id:problem-2.P.6)** Calculate first two terms in the asymptotics as $k\to +\infty$ of \begin{gather} \int\_0^{\pi/3} e^{ik\sin(x)}\,dx,\tag{6a}\\\\ \int\_0^{\pi/2} e^{ik\sin(x)}\,dx,\tag{6b}\\\\ \int\_0^{\pi} e^{ik\sin(x)}\,dx,\tag{6c}\\\\ \int\_0^{2\pi} e^{ik\sin(x)}\,dx,\tag{6d}\\\\ \int\_0^{3\pi} e^{ik\sin(x)}\,dx.\tag{6e} \end{gather} **[Problem 7.](id:problem-2.P.7)** Calculate the first term in the asymptotics as $k\to +\infty$ of \begin{equation} \iint\_D e^{k\sin(x)\sin(y)})\,dxdy\tag{7} \end{equation} where a. $D=\\{(x,y): -\pi/3 < x < 4\pi/3, -\pi/3 < y < 4\pi/3\\}$, b. $D=\\{(x,y): x^2+y^2 < 10\\}$. **[Problem 8.](id:problem-2.P.8)** Calculate the first term in the asymptotics as $k\to +\infty$ of \begin{equation} \iint\_D e^{i k\sin(x)\sin(y)\sin(z)}\,dxdydz\tag{8} \end{equation} where a. $D=\\{(x,y): -\pi/3 < x < 4\pi/3, -\pi/3 < y < 4\pi/3, -\pi/3 < z < 4\pi/3 \\}$, b. $D=\\{(x,y): x^2+y^2 +z^2< 10\\}$. **[Definition 2.](id:def-2.P.2)** [Airy function](https://en.wikipedia.org/wiki/Airy_function) could be defined as \begin{equation\*} \Ai(x):=\frac{1}{2\pi}\int_{-\infty}^\infty e^{i(t^3x+tx)}\,dt. \end{equation\*} **[Problem 9.](id:problem-2.P.9)** a. Using stationary phase method calculate the first term in $\Ai(x)$ asymptotics as $x\to -\infty$. b. Approximately find zeroes of $\Ai(x)$ as $x<0$, $|x|\gg 1$ and estimate an error. c. Prove that $\Ai(x)=O(x^{-\infty})$ as $x\to +\infty$ (in fact it decays exponentially, but we will prove this later). **[Problem 10.](id:problem-2.P.10)** For Airy function using deformation of the contour $(-\infty,\infty)$ and the method of the steepest descent calculate main term in the asymptotics as $x\to +\infty$. ________ [$\Leftarrow$](./L2.5.html) [$\Uparrow$](../contents.html) [$\Rightarrow$](../Chapter3/L3.1.html)