$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$
$\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\Ai}{\mathrm{Ai}}$
Definition 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.
Problem 2. As $\bR\ni z\to +\infty$ calculate
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 \begin{equation} \iint_D e^{k\sin(x)\sin(y)}\,dxdy\tag{4} \end{equation} where
Problem 5. Calculate the first term in the asymptotics as $k\to +\infty$ of \begin{equation} \iiint_D e^{k\sin(x)\sin(y)\cos(z)}\,dxdydz\tag{5} \end{equation} where
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 \begin{equation} \iint_D e^{k\sin(x)\sin(y)})\,dxdy\tag{7} \end{equation} where
Problem 8. Calculate the first term in the asymptotics as $k\to +\infty$ of \begin{equation} \iiint_D e^{i k\sin(x)\sin(y)\sin(z)}\,dxdydz\tag{8} \end{equation} where
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.
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 $x\to +\infty$.