数理解析研究所講究録
931 巻 1995 年 70-99

70

On the connection formulafor the first Painlev\’e equation–from the viewpoint of the exact $WKB$ analysis –.
Yoshitsugu TAKEI

Research Institute for Mathematical Sciences
Kyoto University
Kyoto, 606-01, JAPAN

竹痔義次)(京油田裡厨Introduction.

1

In the workshop (held at the Research Institute for Mathematical Sciences,
Kyoto University during the period of April 26–28, 1995) I gave a series oflectures on the exact WKB analysis of Painlev\’e transcendents with a large
parameter, the subject of my recent research jointly done with T. Kawai(RIMS, Kyoto Univ.) and T. Aoki (Kinki Univ.). Starting with the existenceof the J-th Painlev\’e equation $(P_{J})(J=$of formal power series solutions, VI) and the definition of turning points and Stokes curves for them,I,
I explained the following topics mainly in my lectures:$\lambda_{J}$.. .

1. local equivalence between$\lambda_{\mathrm{I}}$and$\lambda_{J}(J=\mathrm{I}\mathrm{I}, \ldots, \mathrm{V}\mathrm{I})$2. construction of instanton-type formal solutions

3. description of the connection formula for,
type solutions$\lambda_{\mathrm{I}}$$\lambda_{J,\alpha,\beta}$,of$(P_{J})$,in terms of the instanton-$\lambda_{\mathrm{I},\alpha,\beta}$

4. local equivalence between$\lambda_{\mathrm{I},\alpha,\beta}$

and$\lambda_{\mathrm{I}\mathrm{I},\alpha,\beta}$.

and, in particular, their local equivaThe formal power series solutions
lence are discussed in [KT2] in detail, while the readers should be referredto [AKT3] concerning the topics 2 and 4 in the above list. In this report we$\lambda_{J}$71

concentrate our consideration on the topic 3, i.e., on the connection formulafor the first Painlev\’e equation.
There have been several works about the connection formula for the first
Painlev\’e equation. For example, some Russian mathematicians investigate
Painlev\’e equations by using their relationship with the isomonodromic deformations of linear equations associated with them. (See e.g. [FIK], andreferences cited there.) In particular, Kapaev has performed asymptotic
analysis of the associated linear equation to obtain the leading coefficient of. On the otherthe connection formula for the first Painlev\’e equation
and Kruskal
Joshi
analysis,
multiple-scale
hand, employing the so-called
study the first (and the second) Painlev\’e equation without resorting to the..
.
isomonodromic method
Now the purpose of our research is to discuss the Painlev\’e equation
in a unified manner through the exact WKB analysis (cf. [Tke]). As,anda matter of fact, as is shown by the local equivalence between
equation
canonicalasaregardedbePainlev\’ecanequation
the first
from the viewpoint of the exact WKB
near a simple turning point ofanalysis (just like the Airy equation in the case of linear ordinary differential
equations of second order). This, being one of the supporting evidences for, implies that explicit
the possibility of such a unified treatment ofshould be a crucially important step for the determination
analysis ofnear a simple turning point. To determineof the connection formula for, we shall compute thethe explicit form of the connection formula forby using the
Stokes multipliers of the linear equation associated with
only theemploynot
weshall
specific,
more
to
be
analysis;exact WKB
to overcome the difficulty ofbut also Kapaev’s idea
Voros theory
the double turning point whose appearance is a characteristic phenomenon). At the present stage the results we havein our theory (cf. [KT1,obtained so far are not satisfactory enough (in the sense that the explicit
formulas are obtained only for the leading term, not for arbitrarily higher
orders), although we hope our method will give us a complete answer in thenear future.
By the way, we also note the following fact here; the number of theis exactly equal to the number of the local
formal power series solutions
charts, which are constructed explicitly by Takano et al., of ‘the space of’(due to Okamoto, cf. [Tka]). As I mentioned at theinitial values fordream (just a dream in the true sense) is that thislectures,ourend of thecoincidence is not accidental at all but may be a key property in considering. We hopeofthe analytic meaning of instanton-type solutions$([\mathrm{K}])$$([\mathrm{J}\mathrm{K}1, \mathrm{J}\mathrm{K}2])$$(P_{J})$$\lambda_{\mathrm{I}}$$(P_{\mathrm{I}})$$(P_{J})$$(P_{J})$$(P_{\mathrm{I}})$$(P_{J})$$(P_{\mathrm{I}})$$P_{\mathrm{I}}$$(1\mathrm{K}])$$([\mathrm{V}])$$\mathrm{K}\mathrm{T}2]$$\lambda_{J}$$(P_{J})$$\lambda_{J,\alpha,\beta}$$(P_{J})$$\lambda_{J}$72also that this dream will be realized in some future.
In ending this introduction, the author would like to thank Professor
T. Kawai and Professor T. Aoki for their many valuable suggestions and thestimulating discussions with them. A part of this work was prepared during
the author’s stay at the Isaac Newton Institute for Mathematical Sciences
supported by the joint research project
Analysis and Geometry”
between the Newton Institute and RIMS, which is sponsored by the Royal
Society and the Japan Society for the Promotion of Science.
($‘ \mathrm{I}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}$2Basic part of the exact WKB analysis for–A brief review.

$(P_{\mathrm{I}})$.

First of all, to fix our notations, let us review the basic part of the exact
WKB analysis for the first Painlev\’e equation.
The concrete form of the equation is given by the following:(1)$\frac{d^{2}\lambda}{dt^{2}}=\eta^{22}(6\lambda+t)$,

which is equivalent to the following Hamiltonian system:(2)$\frac{d\lambda}{dt}$$=$$\frac{d\nu}{dt}$$=$$\eta\frac{\partial I\mathrm{f}_{\mathrm{I}}}{\partial\nu}$

,

$\{$

$- \eta\frac{\partial IC_{\mathrm{I}}}{\partial\lambda}$

,

where
. Here and in what follows denotes a large
parameter. We also denote the polynomial
by. As is wellknown, the equation (1) (or rather the system (2)) represents the condition
for isomonodromic deformations of the following Schr\"odinger equation in
the sense of [JMU] (see [O] and [KT2] also):$IC_{\mathrm{I}}=\nu^{2}/2-(2\lambda^{3}+t\lambda)$$\eta$$6\lambda^{2}+t$(3)$(- \frac{\partial^{2}}{\partial x^{2}}+\eta^{2}Q_{\mathrm{I}}(X, t, \eta).)\psi(x, \iota, \eta)=0$

where the potential(4)$F_{\mathrm{I}}(\lambda, t)$$Q_{\mathrm{I}}$,is given by$Q_{\mathrm{I}}=4x^{3}+2t_{X}+ \nu-2(4\lambda 32+t\lambda)-\eta^{-}\frac{\nu}{x-\lambda}1+\eta^{-}2_{\frac{3}{4(x-\lambda)^{2}}}$.

73

Observing this concrete form of the equation, we can easily find thatpower
the system (2) (hence the equation (1) also) has the:, with respect toseries solution, which is denoted by$\mathrm{f}\mathrm{o}\mathrm{l}1_{0}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{f}-1\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}1$$(\lambda_{\mathrm{I}}, \nu_{\mathrm{I}})$$\lambda_{1}$(5)$\eta$$=$$\lambda_{0}(\iota)+\eta-1\lambda_{1}(t)+\eta-2\lambda_{2}(t)+\cdots$$=$$\nu_{0}(t)+\eta-1\nu_{1}(t)+\eta-2\nu_{2}(t)+\cdots$$\{$$\nu_{1}$,.

should be a solution of
An important point is that the requirement
to satisfy the algebraic equations(2) forces its leading term$(\lambda_{1}, \nu_{\mathrm{I}})$$(\lambda 0, \nu_{0})$

(6)

and

$F_{\mathrm{I}}(\lambda \mathrm{o}(t),t)=0$$\nu_{0}(t)=0$

to be determined recursively fromand also the other terms. In particular, for the odd order terms all vanish identically andevery even order term can be expressed in the following way:$(\lambda_{j}, \nu_{j})(j\geq 1)$$(\lambda_{0}, \nu_{0})$$\lambda_{\mathrm{I}}$

(7)

where

(8)

$\lambda_{2j}=c_{j}(-\frac{t}{6})^{(5j)/2}1-$$\{c_{j}\}$$(j=0,1,2, \cdots)$ ,is a series of constants defined by1,$c_{0}$$=$$c_{j}$$=$$c_{1}=- \frac{1}{12^{3}}$,$c_{2}=- \frac{49}{2\cdot 12^{6}}$

,

$\{$

$(j\geq 3)$$\frac{25}{12^{3}}(j-1)2c_{j}-1-\frac{1}{2}k,l+l_{-,\geq}-j\sum_{k}2c_{k}c_{l}$.

vanish for ) As is always
the case with singular perturbations, the recursive formula (8) makes
diverge. To give an analytic meaning to , we consider. its Borel sum,which is the central object of our discussion in this report.
In [KT2] we have defined the turning points and the Stokes curves for. In the case of the equatin (1), i.e., theformal power series solutions of$t=0$
Painlev\’eis the unique turning point (which
equation, the origin
firstis simple) and the Stokes curves become the following:(Similarly all of the odd order terms$\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}_{\mathrm{C}}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{y}$$\nu_{\mathrm{I}}.$$\lambda_{\mathrm{I}}$$\lambda_{\mathrm{I}}$$(P_{J})$(9)$\{t\in C;{\rm Im}\int^{t}0\sqrt{\frac{\partial F_{\mathrm{I}}}{\partial\lambda}(\lambda_{0}(t),\iota)}dt=0\}$,

namely $\{t\in C;{\rm Im}(-t)5/4\}=0$ , five straight lines emanating from the origin.
Note that the turning points and the Stokes curves for thus defined have a$\lambda_{\mathrm{I}}$74

close relationship with those of the associated Schr\"odinger equation (3) $-(4)$(see Proposition 1 in \S 4.1 below). This relationship (and, in the case of thefirst Painlev\’e equation, the concrete analysis of the Borel transform of
to be done in the subsequent section as well) explains the reason why the
Borel sum of is expected to have a discontinuity on the Stokes curves, as
we shall see later.
Our goal in this report is to present, as explicitly as possible, the conon anection formula describing the discontinuity of the Borel sum of
Stokes curve. Recalling the definition of the Borel resummation, we mayeasily guess that the connection formula should be described in terms ofthe ‘instanton-type’ solutions introduced in [KT1] first. As a matter ofoffact, in [AKT3] we have constructed the following formal solutions
the equation (1) with two free parameters by employing the multiple-scale
analysis:$\lambda_{\mathrm{I}}$$\lambda_{\mathrm{I}}$$\lambda_{\mathrm{I}}$$\lambda_{\mathrm{I},\alpha,\beta}$

(10)

,

$\lambda_{\mathrm{I},\alpha,\beta}=\lambda 0(t)+\eta\lambda_{1}-1/2-+\eta 1/2(t, \eta)\lambda 1(t, \eta)+\cdots$

is the same as that ofwhere the leading term$(s=-t/6)$ , the sub-leading term
is given by$\lambda_{0}(t)$$\lambda_{\mathrm{I}}$, i.e.,$\lambda_{0}(t)=s^{1/2}$$\lambda_{1/2}(t, \eta)$

(11)

with

$\lambda_{1/2}(t, \eta)=(12\lambda 0)^{-1}/4[\alpha \mathit{8}^{5\alpha\beta}/2e^{\mathcal{T}}+\beta S^{-5\alpha\beta/\tau}2e^{-}]$$\alpha$andbeing arbitrary complex numbers and$\beta$(12)$\tau=\eta\int_{0}^{t}\sqrt{\frac{\partial F_{\mathrm{I}}}{\partial\lambda}(\lambda_{0}(t),t)}dt=-\frac{48\sqrt{3}}{5}(-\frac{t}{6})^{5}/4\eta$,and the higher order terms $\lambda_{j/2}(t, \eta)(j\geq 2)$ , which are determined recur, have the following form:
andsively from$\lambda_{0}(t)$(13)$\lambda_{1/2}(t, \eta)$$\lambda_{j/2}(t, \eta)=\sum_{=k0}^{j}\lambda_{j}^{(j}/2)e-2k(\iota)(j-2k)_{\mathcal{T}}$.

Although it is possible to introduce more arbitrary constants
Remark.
in determining
into the formal solution
for every positive integer , we assume throught this report that all of them are zero
exceptcontained in the expression (11) of. In fact, for our
purposes it is sufficient to consider such a simple case due to the distinctive
homogeneity (with respect to and ) that the solution
as well as the$\lambda 2((2\iota_{-}1)/t, \eta)$$\lambda_{\mathrm{I},\alpha,\beta}$$l$$(\alpha, \beta)$$\lambda_{1/2}(t, \eta)$$t$$\lambda_{\mathrm{I}}$$\eta$

75

equation (1) enjoys. For the details concerning these solutions see [AKT3,\S 1], where the other Painlev\’e equations are also discussed.
Furthermore, if we assume one of the free parameters, say , to be zero,)then we find that the ‘negative instanton-terms’ (i.e., the terms
all vanish and the formal
with $j-2k<0$ in the expression (13) of
namelyway,
inanother
expressedbecansolution$\beta$$\lambda(t)e^{(j-}2k$$\tau$$\lambda_{j/2}(t, \eta))$$\lambda_{\mathrm{I},\alpha,0}$

(14)

.

$=\lambda(0)+\eta^{-1}\lambda/2(1)e+\mathcal{T}\eta\lambda 2)e+-1(2\mathcal{T}\ldots$$\lambda_{\mathrm{I},\alpha_{:}}0$

where eachparticular,
the form,)$(t)kj$is a formal power series of$\lambda^{(j)}=\sum_{k\geq 0}\eta^{-k}\lambda_{j}^{(+}/2$$\lambda^{(0)}$(15)

coincides with the formal power series solution$\lambda^{(1)}=\alpha(12\sqrt{s})^{-1/4}[1-\frac{15}{4}(12\sqrt{s})^{-5/1]}2\eta+-\ldots$$\eta^{-1}$. In$\lambda^{(1)}$$\lambda_{\mathrm{I}},$has,

are determined recursively. (A similar expression
and the other(or), which) This particular solution
is possible also for, should be an importantis nothing but the solution discussed iningredient of the connection formula for . In the sequel we try to determine
the explicit form (or, equivalently, the explicit value of the constant (orof the solution that appears in the connection formula for .
$\lambda^{(j)}(j\geq 2)$$\lambda_{\mathrm{I},\alpha,0}$$\lambda_{\mathrm{I},0,\beta}.$$\lambda_{\mathrm{I},0,\beta}$$[\mathrm{K}\mathrm{T}1, \S 3]$$\lambda_{\mathrm{I}}$$\alpha$$\lambda_{\mathrm{I}}$$\beta))$3

Analytical structure of the Borel transform andthe connection formula for .
$\lambda_{\mathrm{I}}$

Before discussing the Stokes multipliers of the linear equation (3) $-(4)$ , weinvestigate some analytical properties of the Borel transform of without
applying the isomonodromic method. The content of this section is essen.
tially the same as that ofof . Since the Borel sum of
the
Borel
transform
denoteLet
is defined as a Laplace integral$\lambda_{\mathrm{I}}$$[\mathrm{K}\mathrm{T}1, \S 3]$$\lambda_{\mathrm{I},B}(t, y)$$\lambda_{\mathrm{I}}$$\lambda_{\mathrm{I}}$,$\eta\int_{0}^{\infty}e^{-y\eta}\lambda_{\mathrm{I}},B(t, y)dy$

it is reasonable to guess that the Borel sum of may have a discontinuity, whichat a point where a singular point (as a function of y) of$\lambda_{\mathrm{I}}$$t_{*}$$\lambda_{\mathrm{I},B}(t_{*}, y)$76, lies on the positive real axis and that it should satisfy
we denote by
the following connection formula there:$\phi(t_{*})$(16)$\lambda_{\mathrm{I}}arrow\lambda_{\mathrm{I}}+e^{-\phi(t)\eta}\tilde{\lambda}+\cdot$.,.

corresponds to (the Borel sum of) the singular part ofHere
at the singular point $y=\phi(t)$ . (In (16) we use the same symbol
todesignate its Borel sum also.) In this manner the analytical structure of the(i.e., the location of its singular points and its singular
Borel transformpart at every singular point) completely determines the explicit form of theconnection formula for .
On the other hand, we find that the right-hand side of (16) has thesame form as that of one parameter instanton-type solutions defined by(14). As a matter of fact, it must be one of the instanton-type solutions
due to the “uniqueness” of this kind of solutions (, Theorem 3.1]) andto the homogeneity of . This implies, under the assumption of the Borel
summability of , that the explicit description of instanton-type solutions
explained in \S 2 should be translated into the corresponding informationon the analytical structure ofand as a consequence the connection
formula for should be determined completely except one constant whichis contained in the expression (14) $-(15)$ of one parameter instanton-type
solutions.
From now on let us investigate the analytical structure of
by using
its relationship with the explicit description of instanton-type solutions andtry to fix the undetermined constant mentioned above. By the definition
the explicit form ofis the following:$\tilde{\lambda}$$\lambda_{\mathrm{I},B}(t, y)$$\lambda_{\mathrm{I}}$$\lambda_{\mathrm{I},B}$$\lambda_{\mathrm{I}}$$[\mathrm{K}\mathrm{T}1$$\lambda_{\mathrm{I}}$$\lambda_{\mathrm{I}}$$\lambda_{\mathrm{I},B}$$\lambda_{\mathrm{I}}$$\alpha$$\lambda_{\mathrm{I},B}$$\alpha$$\lambda_{\mathrm{I},B}$(17)$\lambda_{\mathrm{I},B}(t, y)=\sum_{j=0}\frac{c_{j}}{(2j)!}\infty(-\frac{t}{6}\mathrm{I}^{(1-\mathrm{s}}j)/2y^{2j}$,where the coefficient has already been defined by (8). Taking the homogeneity of andinto account, we rewrite (17) as follows:$c_{j}$$\lambda_{\mathrm{I}}$(18)$\lambda_{\mathrm{I},B}$$\lambda_{\mathrm{I},B}(t, y)=(-\frac{t}{6})^{1}/\sum_{j=0}^{2\infty}d_{j}zj$

,

where
(19)

$z= \frac{25}{4\cdot 12^{3}}(-\frac{t}{6})^{-5}/2y^{2}$,$d_{j}=( \frac{4\cdot 12^{3}}{25})^{j}\frac{c_{j}}{(2j)!}$.

77

It follows from the recursive formula (8) that the series
example, the following estimate:(20)$\{d_{j}\}$satisfies, for$(j\geq 1)$$C4^{j} \frac{(j-1)!^{2}}{(2j)!}\leq|d_{j}|\leq 4^{j+1}\frac{j!^{2}}{(2j+2)!}$is a positive constant. By using this estimate (20) we can readily
whereconverges for small and that its radius ofverify that the sum
convergence is equal to 1. Furthermore, since is negative for every $j\geq 1$ ,. This immediately implies
the point $z=1$ is a singular point of$t\neq0$
defines an analytic
the Borel transformthat for each fixed$C$$\sum d_{j}z^{j}$$z$$d_{j}$$\sum d_{j}z^{j}$$\lambda_{\mathrm{I},B}(t, y)$function near $y=0$ and has a singularity at $y= \pm\frac{48\sqrt{3}}{5}(-t/6)5/4$ which
hereinafter. The location of this singularity exactly
is denoted bysign)with the one predicted by the explicit form of
the(except
coincides
instanton-type solutions (14) (cf. (12)). As the singular part of)ofcorresponds to the coefficientat this singular pointin the expression (14), we can further expect that the function
should have the following local expression near the singular point$z=1$ :$\sum d_{j}z^{j}=(1-Z)^{1}/2f(Z)+\mathit{9}(z)$(21)$\pm\phi_{\mathrm{I}}(t)$$\lambda_{\mathrm{I},B}(t, y)$$\eta^{-1/2}\lambda^{(}1$$y=\phi_{\mathrm{I}}(t)$$e^{\tau}=e^{-\phi_{1}(t)}\eta$$\sum d_{j}z^{j}$where and are holomorphic functions in a neighborhood of $z=1$ . Asa matter of fact, $(1 -z)^{1/2}f(Z)$ (or, more precisely, the discontinuity of$(-t/6)^{1/2}(1-z)^{1/2}f(z))$ corresponds to the Borel transform of.
) In particular, it follows(The term $(1-z)1/2$ essentially comes fromfrom this correspondence that the undetermined constant in the connecis connected with the value of $f(z)$ at $z=1$ by thetion formula forrelation $\alpha=i\sqrt{5\pi}/12\mu(\mu=f(z)|_{z=}1)$ . (Note that, strictly speaking, thisrelation is correct except the sign. The sign may depend not only on the
choice of various branches but also on which Stokes curves we are consideringat.)Thus our remaining task is to fix the value $\mu=f(1)$ . Let us recallin (14) represents the locationof “instantons”
that the exponent. On the other hand, it follows from (12) thatof singular points of. This strongly
is only an integral multiple ofthe possible value ofsuggests that the point $z=1$ should be the unique singular point ofon its circle of convergence. Let us now assume this fact for the moment.
Then it enables us to apply the Darboux method to the analytic function. Namely, let $\sum f_{k}(z-1)k$ denote the Taylor expansion of $f(z)$ at
$f$

$g$

$\eta^{-1/2}\lambda^{(1)}$$\eta^{-1/2}.$$\alpha$$\lambda_{\mathrm{I}}$$j\tau/\eta$$e^{j\tau}$$\lambda_{\mathrm{I},B}(t, y)$$j\tau/\eta$$\phi_{\mathrm{I}}(t)$$\sum d_{j}z^{j}$$\sum d_{j}z^{j}$78$z=1$ (hence $f_{0}=\mu$ ) and consider the function$F_{N}(_{Z})= \sum_{j=0}d_{j}z-\infty j(1-Z)1/2k\sum_{=0}^{N-1}fk(Z-1)^{k}$for a positive integer $N$ . It follows from the above assumption and theexpression (21) that the holomorphic function $F_{N}(z)$ in $\{z\in C;|z|<1\}$defines an $N$ -times continuously differentiable function on the unit circle$\{z\in C;|z|=1\}$ . Hence the coefficient of the Taylor expansion of $F_{N}(z)$ at$z=0$ , which is explicitly given by$d_{j}- \sum_{k=0}^{N-1}(-1)j+kf_{k^{\frac{\Gamma(k+3/2)}{j!\Gamma(k+3/2-j)}}}$,

becomes $O(j^{-M})$ for an arbitrarily given positive number $M$ providing $N$
is sufficiently large, as it is at the same time the Fourier coefficient of the-periodic
function. Since $(j!\Gamma(k+3/2-j))^{-1}=O(j^{-k-}3/2)$
holds for every non-negative integer , this immediately implies$C^{N}$$2\pi$$F_{N}(e^{i\theta})$

$k$

.

(22)

$\mu=f_{0}=\lim_{jarrow\infty}(-1)^{j}d_{j}\frac{j!\Gamma(3/2-j)}{\Gamma(3/2)}$

Combining (22) and the recursive formula for
we thus obtain the following expression of :$\{d_{j}\}$(cf. (8) and (19)) together,

$\mu$

(23)

with

(24)

$\mu=\lim_{jarrow\infty}\mu j$$\{\mu_{j}\}$being defined by the following:$\mu_{0}$$=$1,$\mu_{j}$$–$$\frac{(2j-2)^{2}}{(2j-1)(2j-3)}\mu_{j-1}$$\mu_{1}=\frac{4}{25}$$\mu_{2}=\frac{392}{1875}$$\{$$+ \frac{1}{2}\sum\frac{(2k-1)!!(2k-3)!!(2l-1)!!(2l-3)!!}{(2j-1)!!(2j-3)!!}\mu_{k}\mu_{l}$$k+l^{--j}k,l\geq 2$$(j\geq 3)$(where $(2n-1)!!=(2n-1)\cdot(2n-3)\cdots\cdot 1$ ). A numerical computation of(23) $-(24)$ shows, for example,(25)$0.24656174\leq\mu\leq 0.24656184$.

79(Note that for the numerical computation of it is better to use some error
estimate together since the convergence of (23) $-(24)$ is considerably slow.)
The conclusion of this section can be summarized as follows: The study
based on the relationship between the analytical structure of the Borel
and the explicit description of instanton-type solutions tells
transform
us that under several assumptions such as the Borel summability ofon each Stokes curve
etc. we have the following connection formula for$\{t\in C;{\rm Im}(-t)^{5/4}=0\}$ :$\mu$$\lambda_{\mathrm{I},B}$$\lambda_{\mathrm{I}}$$\lambda_{\mathrm{I}}$(26)$\lambda_{\mathrm{I}}$$arrow$$=$$\lambda_{\mathrm{I},\alpha,0}$$\lambda_{\mathrm{I}}+\eta\lambda-1/2(1)-e\phi 1(\iota)\eta+\eta\lambda-1(2)e-2\mathrm{I}(\iota)\eta+\emptyset\ldots$

,

where
(27)

$\phi_{\mathrm{I}}(t)$(28)$\lambda^{(1)}$

$=$

$=$

$\frac{48\sqrt{3}}{5}s^{5/4}$$(s=- \frac{t}{6})$,$i \sqrt{\frac{5\pi}{12}}\mu(12\sqrt{s})^{-1/4}[1-\frac{15}{4}(12\sqrt{s})-5/2-1]\eta+\cdots$,

and the other $\lambda^{(j)}(j\geq 2)$ are determined recursively. (On some Stokes) The
being replaced bycurves the formula (26) holds with
(24).
bydefinedtheseries
of
limit
the
is
constant$\lambda_{\mathrm{I},0,\beta}.$$\lambda_{\mathrm{I},\alpha,0}$$\{\mu_{j}\}$$\mu$

Isomonodromic deformations of the associatedlinear equation and the connection formula for4$\lambda_{\mathrm{I}}$.

In the preceding section we have presented a characterization of the undetermined constant and determined the explicit form of the connection formulain a quite satisfactory manner. However we have to admit that theforexact value of the constant is still not fixed; we can compute it only numerically. In this section we consider our problem by using the relationship ofthe equation (1) with isomonodromic deformations of the associated linear
equation (3) $-(4)$ and try again to fix the constant explicitly.
$\lambda_{\mathrm{I}}$4.1

Preliminaries.

First of all, let us substitute(29)$\lambda=\lambda_{0}(t)+\eta^{-}\Lambda 1/2$,$\nu=\eta^{-1/2}N$

80

into the linear equation (3) $-(4)$ . After the substitution the
becomes the following:$(\lambda_{0}(t)=\sqrt{-t}/6)$

potential(30)$Q_{\mathrm{I}}$$=$$Q_{\mathrm{I}}$$4x^{3}+2tx-(4\lambda_{0}^{3}+2t\lambda_{0})+\eta^{-1}(N^{22}-12\lambda 0\Lambda)$$- \eta^{-3/2}(4\Lambda^{3}+\frac{N}{x-\lambda_{0}}\mathrm{I}+\eta^{-2_{\frac{-N\Lambda+3/4}{(x-\lambda_{0})^{2}}+}}\cdots$.

Now our strategy is as follows: We first compute a pair of two independent Stokes multipliers of the linear equation (3) $-(30)$ by employing theexact WKB analysis due to Voros [V] (see [DDPI] and [AKT2] also). Since
a double turning point naturally appears in our situation, we also makeuse of a local reduction theorem of [AKT3], based on Kapaev’s idea,
near the double turning point. As a consequence we will obtain an explicit
description of the Stokes multipliers, which actually are infinite series ofwith their coefficients being functions of , A and $N$ . Then we solve
equations” (i.e., equations that require these Stokes multipliers to be constant) for A and $N$ , regarding and as parameters, to seek
solutions of the first Painlev\’e equation (1). If we perform this procedurein two adjacent Stokes regions (i.e., sectorial regions surrounded by Stokes
curves for ) respectively, the connection formula for will be determined
explicitly.
Before starting the computation of Stokes multipliers, let us make some
important remarks concerning the WKB-theoretic structure of the equation(3) $-(30)$ here.
First, since the top termofis factorized as$([\mathrm{K}])$$\eta^{-1/2}$($t$$‘ \mathrm{i}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{C}$$t$$\eta$$\lambda_{\mathrm{I}}$$\lambda_{\mathrm{I}}$$\mathrm{W}$$Q_{0}$$Q_{\mathrm{I}}$$Q_{0}=4x^{3}+2tx-(4\lambda_{0}^{3}+2t\lambda_{0})=4(x-\sqrt{s})2(X+2\sqrt{s})$,where denotes $-t/6$ as usual, the equation (3) $-(30)$ has a double turning
point at $x=\sqrt{s}$ . This is really an unpleasant feature; the existence ofa double turning point prevents us from employing the Voros theory tocompute the Stokes mul.t ipliers. We will discuss how to deal with this double
turning point in the next subsection.
Second, in connection with this double turning point, the Stokes geometry (i.e., the turning points and the Stokes curves) of the linear equation(3) $-(30)$ is controlled by that of the Painlev\’e equation (1) in the following$s$manner:

Proposition 1 (i) At the turning point for($i.e.$ , at $t=0$ ) the double
turning point $x=\sqrt{s}$ and the other simple turning point $x=-2\sqrt{s}$ of the$\lambda_{\mathrm{I}}$81

collapse to one point.
linear equation (3)(9) forthe two turning points $\sqrt{s}and-2\sqrt{s}$curve(ii) On each Stokes
are connected by a Stokes curve of the equation (3) $-(so)$ .
$-(\mathit{3}\mathit{0})$$\lambda_{\mathrm{I}}$For example, on $\{t\in C;\arg t=3\pi/5\}$ the following degeneracy of Stokes
curves of the linear equation (3) $-(30)$ occurs:$\lrcorner C_{x}$Figure 1: $\arg t=3\pi/5$

This implies that the configuration of Stokes curves of (3) $-(30)$ is completely
different according as $\arg t$ is smaller or greater than $3\pi/5$ , as the following
two pictures show:

Figure 2: $\arg t<3\pi/5$

Figure 3: $\arg t>3\pi/5$

This difference of the configuration of Stokes curves of the linear equation
causes the Borel sum of to have a discontinuity on its Stokes curves (inother words, the formal expression of the Borel sum of to become different
in an adjacent Stokes region), as we shall see in \S 4.4.
$\lambda_{\mathrm{I}}$$\lambda_{\mathrm{I}}$

Remark. These two WKB-theoretically important properties, that is, theexistence of a double turning point and the relationship between the two
Stokes geometries, are commonly observed for every Painlev\’e equationand its associated linear equation. For the details see [KT2].
$(P_{J})$

82

4.2

Canonical form near the double turning point.

In order to compute the Stokes multipliers of the equation (3) $-(30)$ we need
the connection formula for WKB solutions of (3) $-(30)$ on each Stokes curve
emanating from the double turning point $x=\sqrt{s}$ . One problem is that the
point $x=\sqrt{s}$ is not only a double turning point but also a singular pointof (higher order terms of) the potential (30). To overcome this difficulty wemake use of a local transformation theorem (Proposition 2 below) near theand proved indouble turning point, which is based on Kapaev’s idea
) in [AKT3]. In this subsection we study
all orders (with respect tothe canonical equation which appears in this local transformation theorem.
See also [DDP2] for a general treatment of double turning points.
Now let us first state the local transformation theorem.
$(1^{\mathrm{K}}])$$\eta^{-1/2}$

Proposition 2 There exist a neighborhood
(31)

$U$

of

$x=\sqrt{s}$$\zeta(x, t, \eta)=j=\sum^{\infty}0\zeta_{j}/2(_{X}, t)\eta-j/2$

whose coefficients
formal series$\zeta_{j/2}(x, t)$, a formal series,

are holomorphic in $x\in U$ (and(32)

$E(t, \eta)$$=$$\sum_{j=0}^{\infty}E/2(ji)\eta^{-j}/2$(33)$\rho(t, \eta)$$=$$\sum_{j=0}^{\infty}\rho j/2(t)\eta^{-j}/2$$al_{\mathit{8}}o$in ), and,,so that the following four conditions are satisfied:
(34)
(35)
(36)
(37)

$\frac{\partial\zeta_{0}}{\partial x}$

never vanishes,$\zeta_{0}(\sqrt{s}, t)=0$$\zeta_{1/2}$,

identically vanishes,$Q_{\mathrm{I}}(x, t, \eta)=(\frac{\partial\zeta}{\partial x})^{2}[4\zeta(x, t, \eta)2+\eta^{-1}E(t, \eta)$$+ \frac{\eta^{-3/2}\rho(t,\eta)}{\zeta(x,t,\eta)..-\zeta(\sqrt{s}+\eta^{-1/2}\Lambda,t,\eta)}$$+ \frac{3\eta^{-2}}{4(\zeta(_{X},t,\eta)-\zeta(\sqrt{s}+\eta-1/2\Lambda,\iota,\eta))2}]$$- \frac{1}{2}\eta^{-2}\{\zeta(x, t, \eta);x\}$

,

$t$

83

where

$\{\zeta;x\}$denotes the Schwarzian derivative:$\frac{\partial^{3}\zeta/\partial x^{3}}{\partial\zeta/\partial x}-\frac{3}{2}(\frac{\partial^{2}\zeta/\partial x^{2}}{\partial\zeta/\partial x})^{2}$

For the proof of Proposition 2 we refer the reader to [AKT3]. In fact, it canbe verified in the same way as [AKT3, Theorem 3.1], although the situation
is a little different in that here we assume only the relation (29) while in[AKT3] an instanton-type solution of the system (2) has already been substituted into the potential . Proposition 2 claims that in a neighborhoodof the double turning point $x=\sqrt{s}$ the linear equation (3) $-(30)$ should betransformed to the following canonical equation by the transformation (31):$Q_{\mathrm{I}}$

,

(38)

$(- \frac{\partial^{2}}{\partial\zeta^{2}}+\eta^{2}Q\mathrm{c}\mathrm{a}\mathrm{n}(\zeta, t, \eta))\varphi(\zeta, t, \eta)=0$

where
(39)

.

$Q_{\mathrm{C}} \mathrm{a}\mathrm{n}=4\zeta 2E\eta^{-1}(+t, \eta)+\frac{\eta^{-3/2}\rho(t,\eta)}{\zeta-\eta^{-1/2}\sigma(t,\eta)}+\frac{3\eta^{-2}}{4(\zeta-\eta^{-1/2}\sigma(t,\eta))2}$(which is of order
byHere we have denoted$E(t,\eta),$,
power
andseriesformalin by (35) . Note that these$N$
which depend also on A and , are not independent; they satisfy$\eta\zeta 1/2(\sqrt{s}+\eta^{-}1/2\Lambda, t, \eta)$$0$$\sigma(t, \eta)$$\rho(t, \eta)$$)$$\eta$(40)$\sigma(t, \eta)$$E(t, \eta)=\rho(t, \eta)^{2}-4\sigma(t, \eta)^{2}$(cf. [AKT3, Remark 3.3]). Furthermore they are determined uniquely in theconstruction of the transformation. For example, the explicit form of their
top order terms is given by
(41)
(42)

(43)

$E_{0}$$=$$(3\sqrt{s})^{-}1/2(N^{2}-12\sqrt{s}\Lambda^{2})$$\rho 0$$=$$-(3\sqrt{s})^{-1/4}N$$\sigma_{0}$$=$$(3\sqrt{s})^{1/4}\Lambda$

,

,

.

In particular, the formal power series $E(t, \eta)$ is related to the original equation (3) $-(30)$ (and hence can be computed easily) by the following relation:(44)$E(t, \eta)=4{\rm Res}_{x=\sqrt{s}\circ}S\mathrm{d}\mathrm{d}$denotes the generalized odd part (in the(cf. [AKT3, (3.33)]), wheresense of [AKT3, Definition 2.1] of the solution of the Riccati equation$S_{\mathrm{o}\mathrm{d}\mathrm{d}}$

$)$

$S$

84

associated with the original equation (3) $-(30)$ , that is,withbeing the two formal power series solutions of the Riccati equation.
In the rest of this subsectin we discuss the connection formula for WKB
solutions of the canonical equation (38) $-(39)$ . Let us denote the solutionof the associated Riccati equation by $T= \sum_{j\geq-2}\eta^{-j/2}\tau(j/2\zeta)$ and its generalized odd part byfor example, $\tau_{-1}=2\zeta$ andidenticallyvanishes. By the induction it can be readily verified that, for every $j\geq 0$ ,contains terms with negative powers in only and$S_{\mathrm{o}\mathrm{d}\mathrm{d}}=(S^{+}-S-)/2$$S^{\pm}$$T_{\mathrm{o}\mathrm{d}\mathrm{d};}$$\tau_{-1/2}(\zeta)$$T_{j/2}(\zeta)$$\zeta$(45)${\rm Res}_{\zeta=}0T_{\mathrm{o}} \mathrm{d}\mathrm{d}={\rm Res}_{\zeta=0}(T_{\mathrm{o}\mathrm{d}\mathrm{d}}-\eta\tau_{-}1)=\frac{E(t,\eta)}{4}$holds. Thus the following normalized WKB solutions of the equation (38)(39), which we use as a fundamental system of solutions hereinafter, become
well-defined:(46)$\varphi\pm$$=$$\frac{\sqrt{2\eta}}{\sqrt{T_{\mathrm{o}\mathrm{d}\mathrm{d}}}}\zeta^{\pm E/4}$$\cross\exp\pm\{\eta\int_{0}^{\zeta}\tau-1d\zeta+\int_{\infty}^{\zeta}(\tau_{\mathrm{o}\mathrm{d}\mathrm{d}}-\eta\tau_{-}1-\frac{E}{4\zeta})d\zeta\}$that is,(47)$\varphi_{\pm}--\exp(\pm\eta\zeta 2)\zeta-1/2\pm E/4(1+O(\eta^{1}\zeta/2)-1)$,.

Note that the asymptotic behavior (47) uniquely determines the normalization of WKB solutions (in the formal sense). In what follows we consider
these WKB solutions
the Stokes regions (Region I
Region III)
indicated in Figure 4.
$\varphi\pm \mathrm{i}\mathrm{n}$$\sim$

Figure 4(For our present purposes, i.e., to determine the connection formula for ,it is sufficient to consider the problem in these three regions.) We place a$\lambda_{\mathrm{I}}$85etc. so thatcut on the negative imaginary axis and define a branch of$0<\arg\zeta<3\pi/2$ may be satisfied.
Now our task is to determine the connection formula forIII. To do so, we transform thetwo adjacent regions of Region$-(39)$(38)
into the Weber equation. As a matter of fact,canonical equation
if we first define and by$\zeta^{\pm E/4}$$\varphi\pm \mathrm{b}\mathrm{e}\mathrm{t}\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{n}$$\mathrm{I}\sim \mathrm{R}\mathrm{e}\mathrm{g}\mathrm{i}_{\mathrm{o}\mathrm{n}}$$\tilde{v}$$\tilde{u}$$\tilde{v}$

(48)

,

$=$

$(\zeta-\eta^{-1}\sigma/2)^{1}/2\varphi$$=$$( \zeta-\eta^{-1/-1}2\sigma)\{\frac{d}{d\zeta}\tilde{v}+\eta^{1/2}\rho\tilde{v}\}$$\{$$\tilde{u}$$=$,and if we, after that, apply the linear transformation(49)

we find(50),$\frac{d}{d\zeta}=\eta$ .

In particular,$u$thus defined satisfies

,

(51)

$\frac{d^{2}u}{d\zeta^{2}}=\eta^{2}(4\zeta 2\eta+-1(E+2))u$and the equation (51) can be transformed by the scaling
independent variable into the Weber equation:(52)$\frac{d^{2}u}{dz^{2}}+(\kappa+\frac{1}{2}-\frac{z^{2}}{4}\mathrm{I}u=0$,

where $\kappa=-E/4-1$ . In other words, if for a solution
equation (38) $-(39)$ we define $u(z)$ by(53)$\varphi(\zeta)$–of the$z=2\sqrt{\eta}\zeta$$\varphi$of the canonical$u(z)^{\mathrm{d}\mathrm{f}}=^{\mathrm{e}} \frac{\sqrt{2}\eta^{1/4}}{\sqrt{z-2\sigma}}[2\frac{d}{dz}(\varphi(\frac{z}{2\sqrt{\eta}}))$$+(z+ \rho-2\sigma+\frac{1}{z-2\sigma})\varphi(\frac{z}{2\sqrt{\eta}})]$,

86

then the resulting function $u(z)$ satisfies (52), and conversely(54)$u(z)$$\mapsto$$\varphi(\zeta)^{\mathrm{d}}=^{\mathrm{e}\mathrm{f}}\frac{1}{4\eta^{1/2}\sqrt{\zeta-\eta^{-1/2}\sigma}}$$\cross[\frac{1}{\rho-2\sigma}(-2\frac{du}{dz}+zu)+u]|_{z=2\sqrt{\eta}\zeta}$

transforms a solution of the Weber equation (52) to a solution of (38) $-(39)$ .
Let us denote the former correspondence byand the latter by. Making use of this equivalence between the canonical equationand the Weber equation, we compute the connection formula forin thefollowing way:
It is well known that the Weber equation (52) has the following formal:
solution denoted by$u=\mathcal{G}\varphi$$\varphi=\mathcal{G}^{-1}u$$\varphi_{\pm}$$\overline{D_{\kappa}}(Z)$(55)$\overline{D_{\kappa}}(z)=e-z^{2}/4\kappa\sum^{\infty}Zn=0(-1)n_{\frac{\kappa(\kappa-1)\cdots(\kappa-2n+1)}{n!2^{n_{Z}2n}}}$.

andSimilarly
also satisfy the equation. Now we pickup two of them appropriately and regard them as a fundamental system ofsolutions of (52) respectively in the Region
in the complex-plane as is indicated below:$\overline{D\kappa}(-Z)$$D_{\overline{-\kappa}-1}(\pm iz)$$\mathrm{I}’\sim \mathrm{R}\mathrm{e}\mathrm{g}\mathrm{i}_{0}\mathrm{n}\mathrm{I}\mathrm{I}\mathrm{I}’$$z$

Figure 587Here we choose a branch ofeach region:

Region$\{$RegionRegion$\mathrm{I}’$etc. so that the following may be satisfied in$0<\arg z<\pi/2,$ $-\pi/2<\arg(-iz)<0$ ,:$-\pi/2<\arg(-Z)<0,0<\arg(-iz)<\pi/2$ ,:$\mathrm{I}\mathrm{I}’$$z^{\kappa}$$\mathrm{I}\mathrm{I}\mathrm{I}’$: $0<\arg(-Z)<\pi/2,$$-\pi/2<\arg(i_{Z})<0$.

These formal solutions are important in the sense that in every region they
the equation (38) $-(39)$ by the transcorrespond to the WKB solutions
a straightforward computation
example,bove.
Fordefinedformationshows that in Region$\varphi\pm \mathrm{o}\mathrm{f}$$\dot{\mathrm{a}}$$\mathcal{G}$$\mathrm{I}\ \mathrm{I}’$(56)$\mathcal{G}^{-1}\overline{D_{\kappa}}(Z)=\frac{1}{\rho-2\sigma}(2\sqrt{\eta})^{\kappa}e^{-}\eta\zeta^{2}\zeta\kappa+1/2(1+o(\sqrt{\eta}\zeta)-1)$.

This immediately implies(57).

$\mathcal{G}^{-1}\overline{D_{\kappa}}(z)=\frac{1}{\rho-2\sigma}(2\sqrt{\eta})^{\kappa}\varphi-=\frac{1}{\rho-2\sigma}(2\sqrt{\eta})^{-}E/4-1\varphi_{-}$The other correspondences can be computed.in a similar manner (though
it is necessary to pay some attention to the choice of branches). We write
down the complete list of them below:
(58)

(59)

(60)

Region I&I’ :,$\mathcal{G}^{-1}\overline{D_{\kappa}}(z)$$=$$\frac{1}{\rho-2\sigma}(2\sqrt{\eta})-E/4-1\varphi_{-}$$\mathcal{G}^{-1}D_{-\hslash-}1(-iz)$$=$$\frac{1}{2}e^{-i\pi E/8}(2\sqrt{\eta})^{E/}4-1\varphi+$$\mathcal{G}^{-1}\overline{D_{\kappa}}(-z)$$=$$- \frac{1}{\rho-2\sigma}e^{i\pi E//-}4(2\sqrt{\eta})-E41\varphi_{-}$$\mathcal{G}^{-1}D_{\overline{-\kappa}-}1(-iz)$$=$$\frac{1}{2}e^{-i\pi E/8}(2\sqrt{\eta})^{E/}4-1\varphi+$$=$$- \frac{1}{\rho-2\sigma}e^{i\pi E//-}4(2\sqrt{\eta})-E41\varphi_{-}$$=$$\frac{1}{2}e^{-3i\pi E/8}(2\sqrt{\eta})^{E/}4-1\varphi+\cdot$,

Region II&II’ :,,Region III&III’ :.

$\mathcal{G}^{-1}\overline{D_{\kappa}}(-z)$.
$\mathcal{G}^{-1}D_{\overline{-\kappa}-1}(iZ)$,88

On the other hand, according to the well-known asymptotic formula(cf. [BMP, Chapter 8.4]) we canfor the classical Weber function(resp.
regard the Weber function) as a Borel(resp.
sum of the formal) in each region.
Furthermore, the well-known formulas (cf. [BMP, Chapter 8.2])$D_{\kappa}(z)$$D_{\kappa}(z)$$\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\overline{D_{\kappa}}(Z)$$D_{\kappa}(-z)\underline{D}_{-\hslash-1},(\pm iz)$$\overline{D_{\kappa}}(-Z),$$D_{-}\kappa-1(\pm iz)$(61)

$D_{\kappa}(z)$$=$$e^{i\pi\kappa}D_{\kappa}(-z)+ \frac{\sqrt{2\pi}}{\Gamma(-\kappa)}e^{i\pi(\kappa+}1)/2D-\kappa-1(-iz)$(62)$D_{-\kappa-1}(-iz)$$=$$e^{-i\pi(\kappa+1)}D- \kappa-1(iZ)+\frac{\sqrt{2\pi}}{\Gamma(\kappa+1)}e^{-i\pi\kappa/}D_{\kappa}2(-z)$

for the Weber function
then play a role of the connection formula for
the formal
etc. between theregions in question.
(Note that on the positive imaginary axis $D_{-\kappa-1}(-iz)$ is a subdominanton the negative real axis.)
solution and so is
Hence, combining the correspondences (58) $\sim(60)$ and the classical formulas (61) $-(62)$ together, we obtain the connection formula for the WKB
the canonical equation (38) $-(39)$ . The result is the following:
solutions$D_{\kappa}(z)$$\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\mathrm{s}\overline{D_{\kappa}}(z)$$\mathrm{t}\mathrm{w}\underline{\mathrm{o}\mathrm{a}\mathrm{d}}\mathrm{j}\mathrm{a}\mathbb{C}\mathrm{e}\mathrm{n}\mathrm{t}$$\overline{D_{\kappa}}(-Z)$$\varphi\pm \mathrm{o}\mathrm{f}$(resp.
Proposition 3 Let) denote the Borel sum oftheRegion I (resp. II, III) (cf. Figure 4). Then we have the following relation:$\varphi_{\pm}^{\mathrm{I}}$$\varphi_{+}^{\mathrm{I}}$

(63)

$=$

$\varphi_{\pm}^{\mathrm{I}\mathrm{I}},$$\varphi_{\pm}^{\mathrm{I}\mathrm{I}\mathrm{I}}$$\varphi\pm in$$\varphi_{+}^{\mathrm{I}1}$$\{$$\varphi_{-}^{\mathrm{I}}$,$=$$\varphi_{-+\frac{\rho-2\sigma}{2}\frac{\sqrt{2\pi}}{\Gamma(E/4+1)}}^{\mathrm{I}\mathrm{I}\pi E}e-i/4(2\sqrt{\eta})E/2\varphi_{+}^{\mathrm{I}\mathrm{I}}$$=$$\varphi_{+}-\mathrm{I}\mathrm{I}\mathrm{I}\frac{2}{\rho-2\sigma}\frac{\sqrt{2\pi}}{\Gamma(-E/4)}e)/2(i\pi(E+12\sqrt{\eta})-E/2\varphi_{-}\mathrm{I}\mathrm{I}\mathrm{I}$and$\varphi_{+}^{\mathrm{I}\mathrm{I}}$(64)$\varphi_{-}^{\mathrm{I}\mathrm{I}}$

4.3

$=$

$\varphi_{-}^{\mathrm{I}\mathrm{I}\mathrm{I}}$.

Computation of the Stokes multipliers.

Making use of the connection formula of the canonical equation near the
double turning point, i.e., Proposition 3 verified in the preceding subsection,and the Voros theory
as well, we compute the Stokes multipliers of thelinear equation associated with the first Painlev\’e equation in this subsection.
For the sake of specification, in other words, in order to fix the configurationof Stokes curves and the branch of various multi-valued functions, we restrict$([\mathrm{V}])$89

our considerations here to a neighborhood of the Stokes curve $\{t\in C;\arg t=$$3\pi/5\}$ for. The other cases can be discussed in the same way.
given
Now the equation we want to discuss is (3) with the potentialby (30). To compute its Stokes multipliers we need a well-normalized formal
solution, which in our situation can be expressed in terms of WKB solutions
as follows:$\lambda_{\mathrm{I}}$$Q_{\mathrm{I}}$(65)$\psi_{\pm}=\frac{\sqrt{2\eta}}{\sqrt{S_{\mathrm{o}\mathrm{d}\mathrm{d}}}}\exp\pm\{\eta\int_{-2\sqrt{s}}^{x}S_{-1}dX+\int_{\infty}^{x}(s_{\mathrm{o}\mathrm{d}\mathrm{d}\eta}-S-1)dx\}$,

where $S= \sum_{j>_{-}2}\eta^{-}s_{j/}2(j/2x)$ denotes the solution of the Riccati equation
designates its generalized odd part (cf.
andassociated[AKT3, Definition 3.1] . For example, the top term $S_{-1}(x)$ is given by$\mathrm{w}\mathrm{i}\mathrm{t}\overline{\mathrm{h}}(3)-(30)$$S_{\mathrm{o}\mathrm{d}\mathrm{d}}$$)$$S_{-1}(x)=2(x-\sqrt{s})^{\sqrt{x+2\sqrt{s}}}$$(s=-t/6)$ whose branch is determined so that $\arg S_{-}1(X)=\pi/5$ may
be satisfied on the segment connecting $x=\sqrt{s}$ and $x=-2\sqrt{s}$ (which is
nothing but a Stokes curve of the equation (3) $-(30))$ when $\arg t=3\pi/5$
identically vanishes(i.e., $\arg s=8\pi/5$ and $\arg\sqrt{s}=4\pi/5$ ). Note that
) holds for every $j\geq 0$ . Hence the solutions
and $S_{j/2}=O(x^{-3}/2)$ (as(65) are well-defined in the formal sense and, since$S_{-1/2}$$xarrow\infty$(66)$\int^{x}s_{-1(}X)dX$$=$

2 $\int^{x}(x-\sqrt{s})(x+2\sqrt{s})1/2dx$$=$$\frac{2}{5}(x-3\sqrt{s})(X+2\sqrt{s})3/2$

holds (modulo constants depending on ), they have the following asymptotic
behavior as $xarrow\infty$ :
$t$

(67)

$\psi_{\pm}=x^{-3/4}\exp\{\pm\eta(\frac{2}{5}X5/2+\frac{t}{2}x^{1/2}\mathrm{I}\}(1+O(x^{-1/}2))$.

Our next problem is to determine the connection formula for the noron a Stokes curve emanating from the double
malized WKB solutions
turning point. We will solve this by using the local transformation between
the equation (3) $-(30)$ in question and the canonical equation (38) $-(39)$ pro, III) denote the Borel sum(
vided by Proposition 2. First letof the WKB solutions (65) in each region indicated in Figure 6.
$\psi_{\pm}$$\psi_{\pm}^{J}$$J=\mathrm{I},$$\mathrm{I}\mathrm{I}$90$\lrcorner C_{x}$Figure 6: (A local picture near $x=\sqrt{s}.$ )Since the leading term (i.e., the geometric part)
formation (31) is defined by(68)

of the local trans-$\zeta_{0}(X, \iota)$$\zeta_{0}(X, \iota)=(2\int_{\sqrt{s}}^{x}(x-\sqrt{s})(X+2\sqrt{s})^{1/2}dX)^{1}/2$,$\llcorner C_{\zeta}$$\lrcorner C_{x}$$\zeta=\zeta_{0}(_{X}, t)$

Figure 7: Transformation$\zeta=\zeta_{0}(x, t)$.

(, III) should correspondthe Borel resummed WKB solution
respectively under the transformation (31). As a matter of fact, it
toandthe following relation in the formal
follows from (37) that$\psi_{\pm}^{J}$$J=\mathrm{I},$$\mathrm{I}\mathrm{I}$$\varphi_{\pm}^{J}$$\psi_{\pm}$$\varphi\pm \mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{S}\mathrm{f}\mathrm{y}$sense:
(69)

where

,

$\psi_{\pm}(x, t, \eta)=C_{\pm}(t, \eta)(\frac{\partial\zeta}{\partial x})^{-1/2}\varphi_{\pm}(\zeta(x, t, \eta), t, \eta)$$C_{\pm}(t, \eta)$are “constants” (i.e., not depending on ) of the following
$x$

91

form:

.

(70)

$C_{\pm}(t, \eta)=\exp(\eta c^{(\pm)}-1(t))\sum_{0}^{\infty}\eta c_{-j}j=)-j/2(\pm)/2(\iota$

The relation (69) together with the geometric picture of the transformation
andguarantees the correspondence between the Borel sums
, III) (cf. [AKTI]). Furthermore, combining the relation (69)(
, we obtain the following
with the connection formula for:connection formula for$\psi_{\pm}^{J}$$\zeta=\zeta_{0}(x, t)$$\varphi_{\pm}^{J}$$J=\mathrm{I},$$\mathrm{I}\mathrm{I}$$\varphi\pm(\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{p}0\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}_{0}\mathrm{n}3)$$\psi_{\pm}$

Proposition 4 The following formulas hold for the:I, II, III) of the equation (3)$\mathit{8}olutions\{\psi_{\pm}^{J}\}(J=$$-(\mathit{3}\mathit{0})$$\psi_{+}^{\mathrm{I}}$

(71)

$=$

$\psi_{+}^{\mathrm{I}\mathrm{I}}$$\psi_{+}^{\mathrm{I}\mathrm{I}}$

and

$\{$
$\psi_{-}^{\mathrm{I}}$$=$$\psi_{-}^{\mathrm{I}\mathrm{I}}$$\psi_{-}^{\mathrm{I}\mathrm{I}}+a_{12}\psi^{\mathrm{I}\mathrm{I}}+$$=$$\psi_{+}^{\mathrm{I}\mathrm{I}\mathrm{I}}+a23\psi^{\mathrm{I}\mathrm{I}\mathrm{I}}-$

$\{$

$=$

$\psi_{-}^{\mathrm{I}\mathrm{I}\mathrm{I}}$

,

where
(72)
(73)

,

$a_{12}$$=$$\frac{C_{-}}{c_{+}}\frac{\rho-2\sigma}{2}\frac{\sqrt{2\pi}}{\Gamma(E/4+1)}e^{-i}(\pi E/42\sqrt{\eta})E/2$$a_{23}$$=$$- \frac{c_{+}}{C_{-}}\frac{2}{\rho-2\sigma}\frac{\sqrt{2\pi}}{\Gamma(-E/4)}e^{i}(\pi(E+1)/22\sqrt{\eta})-E/2$$=$$\frac{c_{+}}{C_{-}}\frac{\rho+2\sigma}{2}\frac{\sqrt{2\pi}}{\Gamma(-E/4+1)}e^{i\pi(E1)}(+/22\sqrt{\eta})-E/2$.

we have used the formula $\Gamma(1+\alpha)=$(To obtain the last expression ofand the relation (40).) As is obvious from Proposition 4,
are uniquelyplay an important role in our problem. These constants
computedprinciple,
and,
(31)
in
can
be
determined by the transformation
recursively just like and , although the actual computation is not easy
(especially for higher order terms). Here we only perform the calculation of, which is sufficient for the present purposes.
and$a_{23}$$C_{\pm}\mathrm{s}\mathrm{h}_{0}\mathrm{u}\mathrm{l}\mathrm{d}$$\alpha\Gamma(\alpha)$$c_{\pm}$$\rho$$\sigma$$c_{0}^{(\pm)}(t)$$c_{-1}^{(\pm)}(t)$Lemma 1(74) $c_{\pm}= \exp(\pm\eta\int_{-2\sqrt{s}}\sqrt{s}1s-d_{X})(4(3\sqrt{s})5/4)\mp E\mathrm{o}/4(1+o(\eta-1/2))$$– \exp(\mp\frac{24\sqrt{3}}{5}s^{5}/4\dot{\eta})(4(3\sqrt{s})5/4)\mp E_{0}/4(1+o(\eta)-1/2)$.

92

denote the part of order (with respect to
Proof. Let. By a straightforward computation we find$0$

$S_{\mathrm{o}\mathrm{d}\mathrm{d},0}$$\eta^{-1/2}$) of$S_{\mathrm{o}\mathrm{d}\mathrm{d}}$$S_{\mathrm{o}\mathrm{d}\mathrm{d},0=} \frac{N^{2}-12\lambda_{0}\Lambda^{2}}{4(_{X-}\sqrt{s})\sqrt{x+2\sqrt{s}}}=\frac{(3\sqrt{s})^{1/2}E0}{4(x-\sqrt{s})\sqrt{x+2\sqrt{s}}}$.

(Here we have used (41).) Since$(3 \sqrt{s})^{1/2}\int x\frac{dx}{(x-\sqrt{s})\sqrt{x+2\sqrt{s}}}$$=$$\log\frac{\sqrt{x+2\sqrt{s}}-\sqrt{3\sqrt{s}}}{\sqrt{x+2\sqrt{s}}+\sqrt{3\sqrt{s}}}$$=$$\log\frac{x-\sqrt{s}}{(\sqrt{x+2\sqrt{s}}+\sqrt{3\sqrt{s}})^{2}}$,we obtain(75)$\psi_{\pm}=((x-\sqrt{s})^{\sqrt{x+2\sqrt{s}}})-1/2\exp(\pm\eta\int_{-2\sqrt{s}}xS_{-}1dX\mathrm{I}$$\cross(x-OS\pm E\mathrm{o}/4(\sqrt{x+2\sqrt{s}}+\sqrt{3\sqrt{s}})^{\mp}E_{0}/2-(1+O(\eta 1/2))$.

On the other hand, it follows from (47) and (68) that(76)$( \frac{\partial\zeta}{\partial x})^{-}1/2\varphi_{\pm}(\zeta(X, b, \eta), t, \eta)$$=( \frac{\partial\zeta_{0}}{\partial x})^{-}\exp\pm(\eta\zeta^{2}0+2\zeta_{0}\zeta 1)1/2\zeta_{0^{-1}}/2\pm E\mathrm{o}/4(1+O(\eta-1/2))$$=((x- \sqrt{s})^{\sqrt{x+2\sqrt{s}}})-1/2\mathrm{x}\mathrm{e}\mathrm{p}(\pm\eta\int_{\sqrt{s}}xs_{-}1dX\mathrm{I}$$\cross\exp(\pm 2\zeta 0\zeta 1)(\frac{\zeta_{0}}{x-\sqrt{s}})^{\pm E\mathrm{o}/}4)(X-\sqrt{s})^{\pm E\mathrm{o}/}4(1+o(\eta^{-}1/2)$.

Let us compare (75) and (76). We have already known that $(\sqrt{x+2\sqrt{s}}+$anddiffer only by a multiplicative constant. (This is an immediate consequence of (69). Using the differential equation for , we can confirm this fact also by a straightforward computation.) Hence, if we evaluate them at $x=\sqrt{s}$ (and use),$\sqrt{3\sqrt{s}})^{\mp B\mathrm{o}/2}$$\exp(\pm 2\zeta_{0}\zeta 1)(\zeta 0/(x-\sqrt{s}))^{\pm}E\mathrm{o}/4$$\zeta_{1}$$\zeta_{0}’(\sqrt{s})=(3\sqrt{s})^{1/2}$

93

we can easily obtain (74). The last expression of (74) also follows from (66).

Q.E.D.

Now the Stokes multipliers of the equation (3) $-(30)$ can be readily computed. First let us consider the case where $\arg t$ is smaller than $3\pi/5$ .
$\lrcorner C_{x}$The configuration of Stokes curves (Figure 8) and Proposition 4 imply that
corresponding to the transfer from Region A
the connection matrix fortriangular matrix:
following
thebeshouldRegionto$\psi_{\pm}$$\mathrm{B}$(77)

a true solution of the equation (3) $-(30)$Since the Borel sum ofwith the asymptotic expansion (67) in an appropriate sectorial region (withthe center at infinity), the matrix (77) is really one of the Stokes matrices
gives a Stokes multiplier. We denote thisand its off-diagonal component
Stokes multiplier by $M_{AB}$ in what follows; namely$\psi_{\pm}\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}_{\mathrm{S}}$$a_{12}$(78)$M_{AB}=a_{12}$(if $\arg t<3\pi/5$ ).

the connection matrix corresponding to the transfer
Similarly,
from Region to Region , which is also a Stokes matrix and whose offdiagonal element gives another Stokes multiplier (independent of the previous one) denoted by $M_{BC}$ , we obtain the following:$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}\dot{\mathrm{r}}\mathrm{i}\mathrm{n}\mathrm{g}$$\mathrm{B}$(79)$\mathrm{C}$$M_{BC}--a_{23}$(if $\arg t<3\pi/5$ ).

On the other hand, when $\arg t$ is greater than $3\pi/5$ , the configuration of
Stokes curves changes as was explained in \S 4.1 (see Figure 9).

94$\lrcorner C_{x}$Figure 9: $\arg t>3\pi/5$
Observing this new configuration, we find that the same result holds for the
Stokes multiplier $M_{BC}$ , i.e.,(80)$M_{BC}=a_{23}$(if $\arg t>3\pi/5$ ).

However, for the Stokes multiplier $M_{AB}$ the result differs from the previous
one due to the existence of a Stokes curve emanating from the simple turning
point $x=-2\sqrt{s}$ . In fact, the connection matrix from Region A to Regionis the product of two triangular matrices as follows:$\mathrm{B}$

$(01$

(81)

$i \exp 2\int_{-}\infty 12\sqrt{s}(S1\mathrm{o}\mathrm{d}\mathrm{d}-\eta S-)dx)$,

where the second matrix is a contribution from the simple turning point.
(Its off-diagonal component is the so-called Voros coefficient. Cf. [V].) Note
that
(82)
2
$\int_{-2\sqrt{s}}^{\infty}(s_{\mathrm{o}\mathrm{d}\mathrm{d}\eta}-s_{-1})d_{X}=\oint_{\mathrm{a}\Gamma\circ \mathrm{u}\mathrm{n}}\mathrm{d}x=\sqrt{s}(S_{\mathrm{o}\mathrm{d}\mathrm{d}}-\eta S_{-1})d_{X}$is single-valued on the Riemann sphere with a cut
and , and the point $x=\sqrt{s}$ is the unique singular pointon this sphere with the cut). Taking account of the relationof(44), we thus obtain the following:holds (as$s_{\mathrm{o}\mathrm{d}\mathrm{d}}-\eta S-1$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}-2\sqrt{s}$$\infty$$s_{\mathrm{o}\mathrm{d}\mathrm{d}}-\eta S-1$

(83)

$M_{AB}=a_{1}2+ie^{i\pi E/2}$(if $\arg t>3\pi/5$ ).

Here let us introduce the following new symbol(84)$\beta,$$\gamma$and $F$ :$\beta=\frac{i}{2}(\rho+2\sigma)=\frac{i}{2}(-\frac{N}{(3\sqrt{s})^{1/4}}+2(3\sqrt{s})1/4\Lambda)+O(\eta^{-1/2})$

,

95

,

(85)

$\gamma=\frac{i}{2}(\rho-2\sigma)=\frac{i}{2}(-\frac{N}{(3\sqrt{s})^{1/4}}-2(3\sqrt{s})^{1/}4\Lambda \mathrm{I}+O(\eta^{-1/2})$(86)$F= \log\{\frac{c_{+}}{C_{-}}e^{i\pi E/4-E/2}(2\sqrt{\eta})\}$$=- \frac{48\sqrt{3}}{5}s^{5/4}\eta-\frac{E_{0}}{4}\log(263^{5}/2s\eta)5/4+\frac{i\pi E_{0}}{4}+o(\eta^{-1/2})$.

(We have used (42), (43) and Lemma 1 to obtain the explicit form of theleading term of each symbol.) Using these symbols, we can summarize theresults obtained in this subsection as follows: The Stokes multipliers $M_{AB}$and $M_{BC}$ (corresponding to the transfer from Region A to Region andfrom Region to Region respectively) of the linear equation (3) $-(30)$ has
the following expression:$\mathrm{B}$$\mathrm{C}$$\mathrm{B}$$(\arg t<3\pi/5)$(87)$M_{AB}$$=$$-i\gamma e^{(-F})_{\frac{\sqrt{2\pi}}{\Gamma(E/4+1)}}$$M_{BC}$$=$$\beta e^{(i\pi E/+}\frac{\sqrt{2\pi}}{\Gamma(-E/4+1)}4F)$$M_{AB}$$=$$ie^{i\pi E}-/2i \gamma e^{(}-p)\frac{\sqrt{2\pi}}{\Gamma(E/4+1)}$$M_{BC}$$=$$\beta e^{(i\pi E/+}\frac{\sqrt{2\pi}}{\Gamma(-E/4+1)}4F)$

$\{$

,

$(\arg t>3\pi/5)$

(88)

4.4

$\{$

Connection formula for$\lambda_{\mathrm{I}}$.

.

By the end of the preceding section we have succeeded in computing (the first
few terms of) a pair of Stokes multipliers of the linear equation associatedwith the first Painlev\’e equation. Making use of this expression of the Stokes
in themultipliers, we explicitly determine the connection formula forfollowing way:equation”):
Let us consider the following equation (“$\lambda_{\mathrm{I}}$$\mathrm{i}_{\mathrm{S}\mathrm{o}\mathrm{m}\mathrm{o}}\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{i}_{\mathrm{C}}$(89)$M_{AB}=m_{1}$and$M_{BC}=m_{2}$,are given complex constants. Note that $\{M_{AB}, M_{B}c\}$ is
and
where
a pair of independent Stokes multipliers of the linear equation (3) $-(30)$ and$m_{1}$$m_{2}$

96

every Stokes multiplier of (3) $-(30)$ can be expressed as a rational functionof $M_{AB}$ and $M_{BC}$ (cf. [K]). Since the Hamiltonian system (2) represents
the condition for isomonodromic deformations of (3) $-(30)$ , we can obtain asolution of the system (2) (and, hence, of the first Painlev\’e equation (1)) bysolving the equation (89) for A and $N$ with regarding and as parameters
for any pairof complex numbers. Keeping this fact in mind, let us
first solve the equation (89) by using the explicit formula (87) of $M_{AB}$ and$M_{BC}$ to obtain a solution of the first Painlev\’e equation when $\arg t<3\pi/5$ .
Then we consider the same equation (89) with the other expression (88)of $M_{AB}$ and $M_{BC}$ . Although the expressions of the Stokes multipliers aredifferent for (87) and (88), the analytic continuation of the solution thus
obtained in the region $\{t\in C;\arg t<3\pi/5\}$ should satisfy the equation(89) with the same constants
andin the region $\{t\in C;\arg t>3\pi/5\}$
also. Therefore, if we solve the equation (89) with the same constantsby using (88), we can obtain an explicit representation of the analytic
andcontinuation of the original solution in the region $\{\arg t<3\pi/5\}$ to the new
region $\{\arg t>3\pi/5\}$ . In this way the connection formula across the Stokes
curve $\{t\in C;\arg t=3\pi/5\}$ should be determined explicitly.
As an example, let us investigate the case $m_{1}=m_{2}=0$ in greater detail.
When $\arg t<3\pi/5$ , the equation $M_{AB}=M_{BC}=0$ immediately implies$t$$\eta$$(m_{1}, m_{2})$$m_{1}$$m_{2}$$m_{1}$$m_{2}$(90)$\beta=\gamma=0$.

Assuming A and $N$ have the expansion of the form
(91)

A

$=$

$\Lambda_{0}+\eta^{-}\Lambda 1/2+1/2\ldots$

$N$

$=$

$N_{0+}\eta^{-1/2}N1/2+\cdots$

$\{$

,

we can easily solve the equation (90). The result is $\Lambda_{0}=N_{0}=0$ (cf. (84)and (85) and the corresponding solution of the first Painlev\’e equation (1)becomes the formal power series solution . On the other hand, when$\arg t>3\pi/5$ , we still have $\beta=0$ and hence$)$$\lambda_{\mathrm{I}}$(92)$E=p^{2}-4\sigma^{2}=-4\beta\gamma=0$

follows from (40), (84) and (85). However, $\gamma=0$ does not hold. Instead wehave to solve the following system of equations:(93)$\beta=0$

and$\sqrt{2\pi}e^{(-F)}\gamma=1$,

97

or, more explicitly,(94)$\beta=0$

and$\gamma=\frac{1}{\sqrt{2\pi}}\exp(-\frac{48\sqrt{3}}{5}s^{5/}\eta 4)(1+O(\eta^{-}1/2))$.

If we assume the expansion (91) again and further regard $\exp(-48\sqrt{3}s^{5/4}\eta/5)$
as a quantity of order in (just like in the construction of the multiple-scale
solution (10) discussed in [AKT3] , the equation (94) implies$0$$\eta$$)$(95)$\Lambda_{0}=i\frac{1}{2\sqrt{\pi}}(12\sqrt{s})-1/4\exp(-\frac{48\sqrt{3}}{5}s^{5/}\eta 4)$,that is,(96)$\lambda=\sqrt{s}+\eta^{-}i\frac{1}{2\sqrt{\pi}}1/2(12\sqrt{s})-1/4\exp(-\frac{48\sqrt{3}}{5}s\eta 5/4)+\cdots$

This expression (96) of should be the analytic continuation ofregion $\{\arg t>3\pi/5\}$ .
Thus we have explicitly determined the connection formula for
the Stokes curve $\{t\in c_{;\arg}t=3\pi/5\}$ as follows:.

$\lambda$(97)$\lambda_{\mathrm{I}}$$arrow$$\lambda_{\mathrm{I}}$$\lambda_{\mathrm{I}}$in theacross$\lambda_{\mathrm{I},i/2\sqrt{\pi},0}$$\sqrt{s}+\eta-1/2i\frac{1}{2\sqrt{\pi}}(12\sqrt{s})-1/4\mathrm{p}\mathrm{e}\mathrm{x}(-\frac{48\sqrt{3}}{5}s^{s/4}\eta)+\cdots$$=$.

In particular, by the comparison of (97) and (26) $\sim(28)$ we have
(98)$\mu=\sqrt{\frac{3}{5}}\frac{1}{\pi}=0.24656177\cdots$,

which coincides with the previous numerical result (25) very accurately.
This is our determination of the connection formula forby using theisomonodromic method. In my opinion the isomonodromic method is more
powerful than the argument performed in the previous section (\S 3). (Forexample, it enables us to determine the undetermined constant (or, more
specifically,in the formula (28) in a completely explicit manner.) We
hope that this method will determine the connection formula not only forthe formal power series solution
but also for any multiple-scale solution
with two free parameters. (Here a free parameter means an arbitrary$\lambda_{\mathrm{I}}$$\alpha$

$\mu$

$)$

$\lambda_{\mathrm{I}}$$\lambda_{\mathrm{I},\alpha,\beta}$

98

function or formal series ofthat is free from . See Remark in \S 2 also.)
However, to realize this expectation we have to compute the connection
formula for WKB solutions of the associated linear equation near the double
turning point $x=\sqrt{s}$ , in particular the constantsin (69) $-(70)$ , up to
an arbitrarily higher order, and at thestage this is still an open
problem.
$\eta^{-1/2}$$t$$C_{\pm}$$\mathrm{p}\mathrm{r}.\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}$References[AKTI] T. Aoki, T. Kawai and Y. Takei: The Bender-Wu analysis and the
Voros theory. Special Functions, Springer-Verlag, 1991, pp. 1-29.
[AKT2] –: Algebraic analysis of singular perturbations –on exact WKB analysis –. S\^ugaku, 45 (1993), 299-315. (In Japanese.
English translation is to appear in S\^ugaku Expositions.)[AKT3] –: WKB analysis of Painlev\’e transcendents with a large
parameter. II. –Multiple-scale analysis of Painlev\’e transcend.ents.
Preprint (RIMS-1038).

[BMP]

A. Erd\’elyi et al.: Higher Transcendental Functions. Bateman
Manuscript Project (Vol. I, II, III). California Institute of Tech-Hill, 1953.
nology,$\mathrm{M}\mathrm{c}\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{W}$[DDPI] E. Delabaere, H. Dillinger and F. Pham: R\’esurgence de Voros et
p\’eriodes des courbes hyperelliptiques. Ann. Inst. Fourier (Grenoble), 43 (1993), 163-199.
[DDP2] –: Exact semi-classical expansions for one dimensional
quantum oscillators. Pr\’epublications No. 398, Univ. Nice-SophiaAntipolis, 1994.
[FIK]

A.S. Fokas, A.R. Its and A.A. Kapaev: On the asymptotic analysisof the Painlev\’e equations via the isomonodromic method. Nonlinearity, 7 (1994), 1291-1325.
;

[JK1]

N. Joshi and M.D. Kruskal: An asymptotic approach to the connection problem for the first and the second Painlev\’e equations.
Phys. Lett., 130 (1988), 129-137.

[JK2]–: Connection results for the first Painlev\’e equation.
Painlev\’e RanScendents, Plenum, 1992, pp. 61-79.

99

[JMU]

M. Jimbo, T. Miwa and K. Ueno: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. Physica D, 2 (1981), 306-352.

[K]

A.A. Kapaev: Asymptotics of solutions of the Painlev\’e equation of
the first kind. Differential Equations, 24 (1989), 1107-1115.
lated from the Russian original (1988).)$(^{r}\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{S}-$[KT1]

T. Kawai and Y. Takei: WKB analysis and deformation of
Schr\"odinger equations. RIMS K\^oky\^uroku, No. 854, 1993, pp. 22-42.

[KT2]–: WKB analysis of Painlev\’e transcendents with a large
parameter. I. To appear in Adv. in Math.

[O]K. Okamoto: Isomonodromic deformation and Painlev\’e equations,and the Garnier systems. J. Fac. Sci. Univ. Tokyo, Sect. IA, 33(1986), 575-618.

[Tka]

K. Takano: His lecture at the workshop and his report in this vol-

ume.
[Tke]

Y. Takei: On a WKB-theoretic approach to the Painlev\’e transcendents. To appear in the Conf. Proc. “XIth International Congressof Mathematical Physics (Paris, 1994)”.

[V]

A. Voros: The return of the quartic oscillator. The complex WKB
method. Ann. Inst. H. Poincar\’e, 39 (1983), 211-338.