Setting

We study integrals of the form

${\displaystyle I(\alpha ,N):=\int _{C}e^{-Nf(\alpha ,t)}g(\alpha ,t)dt,}$

where ${\displaystyle C}$ is a contour and

• ${\displaystyle f,g}$ are two analytic functions in the complex variable ${\displaystyle t}$ and continuous in ${\displaystyle \alpha }$.
• ${\displaystyle N}$ is a large number.

Suppose we have two saddle points ${\displaystyle t_{+},t_{-}}$ of ${\displaystyle f(\alpha ,t)}$ with multiplicity ${\displaystyle 1}$ that depend on a parameter ${\displaystyle \alpha }$. If now an ${\displaystyle \alpha _{0}}$ exists, such that both saddle points coalescent to a new saddle point ${\displaystyle t_{0}}$ with multiplicity ${\displaystyle 2}$, then the steepest descent method no longer gives uniform asymptotic expansions.

Procedure

Suppose there are two simple saddle points ${\displaystyle t_{-}:=t_{-}(\alpha )}$ and ${\displaystyle t_{+}:=t_{+}(\alpha )}$ of ${\displaystyle f}$ and suppose that they coalescent in the point ${\displaystyle t_{0}:=t_{0}(\alpha _{0})}$.

We start with the cubic transformation ${\displaystyle t\mapsto w}$ of ${\displaystyle f(\alpha ,t)}$, this means we introduce a new complex variable ${\displaystyle w}$ and write

${\displaystyle f(\alpha ,t)={\tfrac {1}{3}}w^{3}-\eta (\alpha )w+A(\alpha ),}$

where the coefficients ${\displaystyle \eta$ :=\eta (\alpha )} and ${\displaystyle A:=A(\alpha )}$ will be determined later.

We have

${\displaystyle {\frac {dt}{dw}}={\frac {w^{2}-\eta }{f_{t}(\alpha ,t)}},}$

so the cubic transformation will be analytic and injective only if ${\displaystyle dt/dw}$ and ${\displaystyle dw/dt}$ are neither ${\displaystyle 0}$ nor ${\displaystyle \infty }$. Therefore ${\displaystyle t=t_{-}}$ and ${\displaystyle t=t_{+}}$ must correspond to the zeros of ${\displaystyle w^{2}-\eta }$, i.e. with ${\displaystyle w_{+}:=\eta ^{1/2}}$ and ${\displaystyle w_{-}:=-\eta ^{1/2}}$. This gives the following system of equations

${\displaystyle {\begin{cases}f(\alpha ,t_{-})=-{\frac {2}{3}}\eta ^{3/2}+A,\\f(\alpha ,t_{+})={\frac {2}{3}}\eta ^{3/2}+A,\end{cases}}}$

we have to solve to determine ${\displaystyle \eta }$ and ${\displaystyle A}$. A theorem by Chester–Friedman–Ursell (see below) says now, that the cubic transform is analytic and injective in a local neighbourhood around the critical point ${\displaystyle (\alpha _{0},t_{0})}$.

After the transformation the integral becomes

${\displaystyle I(\alpha ,N)=e^{-NA}\int _{L}\exp \left(-N\left({\tfrac {1}{3}}w^{3}-\eta w\right)\right)h(\alpha ,w)dw,}$

where ${\displaystyle L}$ is the new contour for ${\displaystyle w}$ and

${\displaystyle h(\alpha ,w):=g(\alpha ,t){\frac {dt}{dw}}=g(\alpha ,t){\frac {w^{2}-\eta }{f_{t}(\alpha ,t)}}.}$

The function ${\displaystyle h(\alpha ,w)}$ is analytic at ${\displaystyle w_{+}(\alpha ),w_{-}(\alpha )}$ for ${\displaystyle \alpha \neq \alpha _{0}}$ and also at the coalescing point ${\displaystyle w_{0}}$ for ${\displaystyle \alpha _{0}}$. Here ends the method and one can see the integral representation of the complex Airy function.

Chester–Friedman–Ursell note to write ${\displaystyle h(\alpha ,w)}$ not as a single power series but instead as

${\displaystyle h(\alpha ,w)=\sum \limits _{m}q_{m}(\alpha )(w^{2}-\eta )^{m}+\sum \limits _{m}p_{m}(\alpha )w(w^{2}-\eta )^{m}}$

to really get asymptotic expansions.

Theorem by Chester–Friedman–Ursell

Let ${\displaystyle t_{+}:=t_{+}(\alpha ),t_{-}:=t_{-}(\alpha )}$ and ${\displaystyle t_{0}:=t_{0}(\alpha _{0})}$ be as above. The cubic transformation

${\displaystyle f(t,\alpha )={\tfrac {1}{3}}w^{3}-\eta (\alpha )w+A(\alpha )}$

with the above derived values for ${\displaystyle \eta (\alpha )}$ and ${\displaystyle A(\alpha )}$, such that ${\displaystyle t=t_{\pm }}$ corresponds to ${\displaystyle u=\pm \eta ^{1/2}}$, has only one branch point ${\displaystyle w=w(\alpha ,t)}$, so that for all ${\displaystyle \alpha }$ in a local neighborhood of ${\displaystyle \alpha _{0}}$ the transformation is analytic and injective.