2.P. Problems to Chapter 2

$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$

$\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\Ai}{\mathrm{Ai}}$

Problems to Chapter 2

Definition 1. $\Gamma$-function is defined as $$\Gamma (z)= \int_0^\infty e^{-t}t^{z-1}\,dz \label{eq-2.P.1}$$ as $\Re z>0$; it satisfies $$\Gamma(z)=(z-1)\Gamma(z-1) \label{eq-2.P.2}$$ 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.

1. 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.
2. As $m =2,4,\ldots$ calculate \begin{equation*} \int_{-\infty}^\infty e^{-y^m/m} y^n\,dy. \end{equation*}

Problem 2. As $\bR\ni z\to +\infty$ calculate

1. First term in the asymptotics of $\Gamma(z+1)$ and thus justify Stirling formula
2. Calculate the second term in this approximation.
3. Justify these asymptotics by plugging $y=zt$.

Problem 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. Calculate the first term in the asymptotics as $k\to +\infty$ of $$\iint_D e^{k\sin(x)\sin(y)}\,dxdy\tag{4}$$ where

1. $D=\{(x,y): 0 < x < \pi, 0 < y < \pi\}$,
2. $D=\{(x,y): x^2+y^2 < 5\}$.

Problem 5. Calculate the first term in the asymptotics as $k\to +\infty$ of $$\iint_D e^{k\sin(x)\sin(y)\cos(z)}\,dxdydz\tag{5}$$ where

1. $D=\{(x,y): 0 < x < \pi, 0 < y < \pi, -\pi/2 < z < \pi/2 \}$,
2. $D=\{(x,y): x^2+y^2 +z^2< 5\}$.

Problem 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. Calculate the first term in the asymptotics as $k\to +\infty$ of $$\iint_D e^{k\sin(x)\sin(y)})\,dxdy\tag{7}$$ where

1. $D=\{(x,y): -\pi/3 < x < 4\pi/3, -\pi/3 < y < 4\pi/3\}$,
2. $D=\{(x,y): x^2+y^2 < 10\}$.

Problem 8. Calculate the first term in the asymptotics as $k\to +\infty$ of $$\iint_D e^{i k\sin(x)\sin(y)\sin(z)}\,dxdydz\tag{8}$$ where

1. $D=\{(x,y): -\pi/3 < x < 4\pi/3, -\pi/3 < y < 4\pi/3, -\pi/3 < z < 4\pi/3 \}$,
2. $D=\{(x,y): x^2+y^2 +z^2< 10\}$.

Definition 2.

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.

1. Using stationary phase method calculate the first term in $\Ai(x)$ asymptotics as $x\to -\infty$.
2. Approximately find zeroes of $\Ai(x)$ as $x<0$, $|x|\gg 1$ and estimate an error.
3. Prove that $\Ai(x)=O(x^{-\infty})$ as $x\to +\infty$ (in fact it decays exponentially, but we will prove this later).

Problem 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 $x\to +\infty$.