$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\const}{\operatorname{const}}$ $\newcommand{\Hess}{\operatorname{Hess}}$ $\newcommand{\supp}{\operatorname{supp}}$ $\newcommand{\arg}{\operatorname{arg}}$
We consider \begin{equation} I(k)= \int_L e^{k \phi(z)}f(z)\,dz \label{eq-2.5.1} \end{equation} where now $L\subset \Omega$ is a contour from $z_0$ to $z_1$, $\Omega \subset \bC$ is a simple-connected domain and $\phi$, $f$ are holomorphic in $\Omega$. Recall from complex variables that in this framework $I(k)$ does not depend on the choice of $L$.
We are interested in the asymptotics of $I(k)$ as $k\to +\infty$.
Obviously we need to select a contour $L$ in such way that $\max_{z\in L} \Re\phi(z)$ was a small as possible. Consider a landscape of $\Re \phi(z)$. From Complex Variables we know that both $\Re \phi(z)$ and $\Im \phi(z)$ are harmonic functions. We also know that harmonic functions do not have local maxima or minima in the inner points, only saddle points.
Lemma 1.
Proof. From Complex Variables.
If $\phi' (z_0)=\ldots =\phi^{(m-1)}(z_0)=0$ and $\phi^{(m)}(z_0)\ne 0$ we call $(m-1)$ multiplicity of the saddle. If $m=2$ then the saddle is simple.
Lemma 2. Consider lines along which $\Re \phi$ changes the fastest. Those are tangent to $\nabla \Re \phi$. Along these lines $\Im \phi=\const$.
Lemma 3. Let $\phi' (z_0)=\ldots =\phi^{(m-1)}(z_0)=0$ and $A:=\phi^{(m)}(z_0)\ne 0$. Then in the vicinity of $z_0$
Proof. Sufficient to consider a toy-model $\phi(z)=Az^m$ with $z_0=0$.
$m=2$ | $m=3$ |
Gray lines are $\{z:\,\Re \phi(z) =\Re \phi(z_0)\}$, blue lines are of the steepest descent and red lines of the steepest ascent |
Lemma 4. One can select contour from $z_0$ to $z_1$ in such a way that it can be broken into several contours $L_1,\ldots ,L_K$ such that
Proof. This lemma looks intuitively obvious (but the rigorous proof is a bit tedious).
Obviously we need to calculate only contributions of the contours of type (a). We consider just one contour $L$ of type (a) from $z^*$ to some point (does not matter).
Theorem 1. Let $L$ be a contour from $z^*$ to some point (does not matter) and $\Re\phi(z)<\Re\phi(z^*)$ in each point of this contour $z\ne z^*$. Let $\phi' (z^*)=\ldots =\phi^{(m-1)}(z^*)=0$ and $A:=\phi^{(m)}(z^*)\ne 0$, $m\ge 2$. Let $L$ be a contour of the steepest descent.
Then \begin{equation} I(k)\sim e^{ik\phi(z^*)}\sum_{n\ge 0}\kappa_n k^{-(n+1)/m} \label{eq-2.5.5} \end{equation} where \begin{equation} \kappa_0= \Gamma ((m+1)/m) |f^{(m)}(z^*)/m!|^{-1/m}e^{i\theta}f(z^*) \label{eq-2.5.6} \end{equation} where $e^{i\theta}$ is a direction of $L$ in $z^*$.
Remark 1. If $m=1$ (\ref{eq-2.5.5}) holds with \begin{equation} \kappa_0= -(\phi'(z^*))^{-1}f(z^*). \label{eq-2.5.7} \end{equation}
$\Leftarrow$ $\Uparrow$ $\Rightarrow$
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