Two error bounds of the elliptic asymptotics for the fifth Painlevé transcendents

The fifth Painlevé equation

$$y^{\prime\prime}=\Bigl{(}\frac{1}{2y}+\frac{1}{y-1}\Bigr{)}(y^{\prime})^{2}-\frac{y^{\prime}}{x}+\frac{(y-1)^{2}}{x^{2}}\Bigl{(}a_{\theta}y-\frac{b_{\theta}}{y}\Bigr{)}+c_{\theta}\frac{y}{x}-\frac{y(y+1)}{2(y-1)},$$

(PV)

in which $8a_{\theta}=(\theta_{0}-\theta_{1}+\theta_{\infty})^{2},$ $8b_{\theta}=(\theta_{0}-\theta_{1}-\theta_{\infty})^{2}$, $c_{\theta}=1-\theta_{0}-\theta_{1}$ with $\theta_{0},\theta_{1},\theta_{\infty}\in\mathbb{C}$, governs the isomonodromy deformation of a linear system of the form

	$\displaystyle\frac{d\Xi}{d\lambda}=$	$\displaystyle\Bigl{(}\frac{x}{2}\sigma_{3}+\frac{\mathcal{A}_{0}}{\lambda}+\frac{\mathcal{A}_{1}}{\lambda-1}\Bigr{)}\Xi,$
		$\displaystyle\sigma_{3}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix},\quad\mathcal{A}_{0}=\begin{pmatrix}\mathfrak{z}+\theta_{0}/2&-u(\mathfrak{z}+\theta_{0})\\ \mathfrak{z}/u&-\mathfrak{z}-\theta_{0}/2\end{pmatrix},$
		$\displaystyle\mathcal{A}_{1}=\begin{pmatrix}-\mathfrak{z}-(\theta_{0}+\theta_{\infty})/2&uy(\mathfrak{z}+(\theta_{0}-\theta_{1}+\theta_{\infty})/2)\\ -(uy)^{-1}(\mathfrak{z}+(\theta_{0}+\theta_{1}+\theta_{\infty})/2)&\mathfrak{z}+(\theta_{0}+\theta_{\infty})/2\end{pmatrix}$

(cf. [1], [10, (1.1)], [11, (3.1)]) with the monodromy data $(M^{0},M^{1})=((m^{0}_{ij}),(m^{1}_{ij}))\in SL_{2}(\mathbb{C})^{2}$ defined along loops surrounding $\lambda=0$ and $1$, respectively. Then a general solution $y(x)$ of (P_V) is parametrised by $(M^{0},M^{1})$ [1, Section 2]. As in [10, 11, Theorem 2.1], for each $\phi$ such that $0<|\phi|<\pi/2,$ $y(x)$ admits an expression of the form

$$\frac{y(x)+1}{y(x)-1}=A_{\phi}^{1/2}\mathrm{sn}((x-x_{0})/2+\Delta(x);A^{1/2}_{\phi})$$

(1.1)

with $\Delta(x)=O(x^{-2/9+\varepsilon})$ for any $\varepsilon$ satisfying $0<\varepsilon<2/9$ as $x=e^{i\phi}t\to\infty$ through the cheese-like strip $S(\phi,t_{\infty},\kappa_{0},\delta_{0}),$ where $v=\mathrm{sn}(z;k)$ is the Jacobi elliptic function such that $v_{z}^{2}=(1-v^{2})(1-k^{2}v^{2}),$ and the symbols $A_{\phi}$, $x_{0}$ and $S(\phi,t_{\infty},\kappa_{0},\delta_{0})$ are as in (1) and (3) below. Since $A_{\phi}$ does not depend on the solution $y(x)$, the leading term of the expression above contains the integration constant $x_{0}$ depending on $(M^{0},M^{1})$ and the other integration constant appears in the error term $\Delta(x).$ Moreover $\Delta(x)$ may be treated in studying, say, the $\tau$-function [8, p. 121], degeneration into trigonometric asymptotics [8, Section 4]. For these facts detailed study on $\Delta(x)$ is desirable. Under the supposition $\Delta(x)=O(x^{-1})$, an asymptotic form of $\Delta(x)$ containing the other integration constant is discussed in [10, Theorem 2.3 and Corollary 2.4]. For the $\tau$-function associated with (P_I) Iwaki [5], by the method of topological recursion, obtained a conjectural full-order expansion yielding the elliptic expression of solutions.

In this paper we unconditionally present an explicit expression of $\Delta(x)$, which leads to the estimate $\Delta(x)=O(x^{-1})$ as was conjectured. The correction function $B_{\phi}(t)$ [11, (5.5)] for the Lagrangian of $y(x)$ contains information about asymptotics (see also [8, Section 3]). Analogous explicit formula is obtained for the error term of the asymptotic expression of $B_{\phi}(t)$.

Our results are stated in Theorems 2.1, 2.2 and 2.3. In Section 3, from a system of equations equivalent to (P_V) we derive integral equations containing the error term $h(x)=\Delta(x)/2.$ The final section is devoted to the proofs of main theorems by using these equations, in which our argument is quite different from those in [4], [2, Chapter 8], [6] and [9] applied to (P_II) and (P_I).

Throughout this paper we use the following symbols.

(1) For each $\phi\in\mathbb{R}$, $A_{\phi}\in\mathbb{C}$ is a unique solution of the Boutroux equations

$$\mathrm{Re\,}e^{i\phi}\int_{\mathbf{a}}\sqrt{\frac{A_{\phi}-z^{2}}{1-z^{2}}}dz=\mathrm{Re\,}e^{i\phi}\int_{\mathbf{b}}\sqrt{\frac{A_{\phi}-z^{2}}{1-z^{2}}}dz=0$$

[10, Section 7]. Here $\mathbf{a}$ and $\mathbf{b}$ are basic cycles as in Figure 1.1 on the elliptic curve $\Pi^{*}=\Pi^{*}_{+}\cup\Pi^{*}_{-}$ given by $w(A_{\phi},z)=\sqrt{(1-z^{2})(A_{\phi}-z^{2})}$ such that $\Pi^{*}_{+}$ and $\Pi^{*}_{-}$ are glued along the cuts $[-1,-A^{1/2}_{\phi}]\cup[A^{1/2}_{\phi},1]$ with $0\leq\mathrm{Re\,}A_{\phi}^{1/2}\leq 1;$ and the branches of the square roots

$$\sqrt{\frac{A_{\phi}-z^{2}}{1-z^{2}}}=\frac{\sqrt{A_{\phi}-z^{2}}}{\sqrt{1-z^{2}}},\quad\sqrt{(A_{\phi}-z^{2})(1-z^{2})}=\sqrt{A_{\phi}-z^{2}}\,\sqrt{1-z^{2}}$$

are determined by $z^{-1}\sqrt{A_{\phi}-z^{2}}\to i$ and $z^{-1}\sqrt{1-z^{2}}\to i$ as $z\to\infty$ on the upper sheet $\Pi^{*}_{+}$.

(2) The periods of $\Pi^{*}$ along $\mathbf{a}$ and $\mathbf{b}$ are

$$\Omega_{\mathbf{a}}=\int_{\mathbf{a}}\frac{dz}{w(A_{\phi},z)},\quad\Omega_{\mathbf{b}}=\int_{\mathbf{b}}\frac{dz}{w(A_{\phi},z)},$$

and write

$$\mathcal{E}_{\mathbf{a}}=\int_{\mathbf{a}}\sqrt{\frac{A_{\phi}-z^{2}}{1-z^{2}}}\,dz,\quad\mathcal{E}_{\mathbf{b}}=\int_{\mathbf{b}}\sqrt{\frac{A_{\phi}-z^{2}}{1-z^{2}}}\,dz.$$

(3) Set

$$x_{0}\equiv\frac{-1}{\pi i}(\Omega_{\mathbf{b}}\log(m^{0}_{21}m^{1}_{12})+\Omega_{\mathbf{a}}\log\mathfrak{m}_{\phi})-(\tfrac{1}{2}\Omega_{\mathbf{a}}+\Omega_{\mathbf{b}})(\theta_{\infty}+1)\quad\mod 2\Omega_{\mathbf{a}}\mathbb{Z}+2\Omega_{\mathbf{b}}\mathbb{Z},$$

in which $\mathfrak{m}_{\phi}=m^{0}_{11}$ if $-\pi/2<\phi<0$, and $=e^{-\pi i\theta_{\infty}}(m^{1}_{11})^{-1}$ if $0<\phi<\pi/2.$ For given positive numbers $\kappa_{0},$ $\delta_{0}$ and $t_{\infty}$,

$$S(\phi,t_{\infty},\kappa_{0},\delta_{0})=\{x=e^{i\phi}t\,|\,\mathrm{Re\,}t>t_{\infty},\,\,\,|\mathrm{Im\,}t|<\kappa_{0}\}\setminus\bigcup_{\sigma\in\mathcal{P}_{0}}\{|x-\sigma|<\delta_{0}\}$$

with $\mathcal{P}_{0}=\{\sigma\,|\,\mathrm{sn}((\sigma-x_{0})/2;A^{1/2}_{\phi})=\infty\}=\{x_{0}+\Omega_{\mathbf{a}}\mathbb{Z}+\Omega_{\mathbf{b}}(2\mathbb{Z}+1)\},$ and

$$\check{S}(\phi,t_{\infty},\kappa_{0},\delta_{0})={S}(\phi,t_{\infty},\kappa_{0},\delta_{0})\setminus\bigcup_{\sigma\in\mathcal{Q}}\{|x-\sigma|<\delta_{0}\}$$

with $\mathcal{Q}=\{{\sigma}\,|\,\mathrm{sn}(({\sigma}-x_{0})/2;A^{1/2}_{\phi})=\pm A^{-1/2}_{\phi},\pm 1\}.$ For $\sigma=e^{i\phi}t_{\sigma}\in\mathcal{Q}$ let $l(\sigma)$ be the line defined by $x=e^{i\phi}(\mathrm{Re\,}t_{\sigma}+i\eta)$ with $\eta\geq\mathrm{Im\,}t_{\sigma}$ if $\mathrm{Im\,}t_{\sigma}\geq 0$ (respectively, $\eta\leq\mathrm{Im\,}t_{\sigma}$ if $\mathrm{Im\,}t_{\sigma}<0$). Let $\check{S}_{\mathrm{cut}}(\phi,t_{\infty},\kappa_{0},\delta_{0})$ denote $\check{S}(\phi,t_{\infty},\kappa_{0},\delta_{0})$ with cuts along the lines $l(\sigma)$ for all $\sigma\in\mathcal{Q},$ where some local segments contained in $l(\sigma)$ may be replaced with arcs, if necessary, not to touch another circle $|x-\sigma^{\prime}|=\delta_{0}$ with $\sigma^{\prime}\in\mathcal{P}_{0}\cup\mathcal{Q},$ $\sigma^{\prime}\not=\sigma.$

(4) For $\mathrm{Im\,}\tau>0,$

$$\vartheta(z,\tau)=\sum_{n\in\mathbb{Z}}e^{\pi i\tau n^{2}+2\pi izn}$$

with $\vartheta^{\prime}(z,\tau)=(d/dz)\vartheta(z,\tau)$ is the $\vartheta$-function [3], [12]. Note that $\vartheta(z\pm 1,\tau)=\vartheta(z,\tau),$ $\vartheta(z\pm\tau,\tau)=e^{-\pi i(\tau\pm 2z)}\vartheta(z,\tau),$

(5) We write $f\ll g$ or $g\gg f$ if $f=O(g).$

Our results are stated as follows.

Set

	$\displaystyle\psi_{0}(x)$	$\displaystyle=A_{\phi}^{1/2}\mathrm{sn}((x-x_{0})/2;A_{\phi}^{1/2}),$
	$\displaystyle b_{0}(x)$	$\displaystyle=\beta_{0}-\frac{2\mathcal{E}_{\mathbf{a}}}{\Omega_{\mathbf{a}}}x-\frac{8}{\Omega_{\mathbf{a}}}\frac{\vartheta^{\prime}}{\vartheta}\Bigl{(}\frac{1}{2\Omega_{\mathbf{a}}}(x-x_{0}),\tau_{0}\Bigr{)},\quad\tau_{0}=\frac{\Omega_{\mathbf{b}}}{\Omega_{\mathbf{a}}},$
	$\displaystyle\mathfrak{b}(x)$	$\displaystyle=\frac{\mathcal{E}_{\mathbf{a}}}{4}(x-x_{0})+\frac{\vartheta^{\prime}}{\vartheta}\Bigl{(}\frac{1}{2\Omega_{\mathbf{a}}}(x-x_{0}),\tau_{0}\Bigr{)}=-\frac{\Omega_{\mathbf{a}}}{8}(b_{0}(x)-b_{0}(x_{0}))$

with

$$\beta_{0}=-\frac{8}{\Omega_{\mathbf{a}}}(\log(m^{0}_{21}m^{1}_{12})+\pi i(\theta_{\infty}+1)),\quad b_{0}(x_{0})=\beta_{0}-\frac{2\mathcal{E}_{\mathbf{a}}}{\Omega_{\mathbf{a}}}x_{0},$$

where $\psi_{0}(x),$ $b_{0}(x)$ and $\mathfrak{b}(x)$ are bounded in $S(\phi,t_{\infty},\kappa_{0},\delta_{0})$.

Recall the correction function $B_{\phi}(t)$ such that $a_{\phi}=A_{\phi}+t^{-1}B_{\phi}(t),$ where

	$\displaystyle a_{\phi}=$	$\displaystyle 1-\frac{4(e^{-2i\phi}(y^{*})^{2}-y^{2})}{y(y-1)^{2}}+4e^{-i\phi}(\theta_{0}+\theta_{1})\frac{y+1}{y-1}t^{-1}$
		$\displaystyle+e^{-2i\phi}\frac{(y-1)}{y}((\theta_{0}-\theta_{1}+\theta_{\infty})^{2}y-(\theta_{0}-\theta_{1}-\theta_{\infty})^{2})t^{-2}$

with $x=e^{i\phi}t$ [11, (3.5)]. In particular, $a_{\phi,\mathrm{Lag}}:=a_{\phi}|_{y^{*}=dy/dt}$ is the Lagrangian of $y=y(e^{i\phi}t).$ In this case, let $b(x)$ be such that

$$a_{\phi,\mathrm{Lag}}=A_{\phi}+\frac{b(x)}{x}=A_{\phi}+\frac{e^{-i\phi}b(e^{i\phi}t)}{t}.$$

If $y^{*}=dy/dt$, then $b(x)=e^{i\phi}B_{\phi}(t).$ (In [10], $a_{\phi}$ and $B_{\phi}(t)$ are defined under the condition $y^{*}=dy/dt.$)

To prove our theorems, we recall the following facts [10, Section 6].

(1) For the solution $y(x)$ of (P_V), $(\psi(x),b(x))$ with $\psi(x)=(y(x)+1)(y(x)-1)^{-1}$ solves a system of equations

	$\displaystyle 4(\psi^{\prime})^{2}=$	$\displaystyle(1-\psi^{2})(A_{\phi}-\psi^{2})-(1-\psi^{2})(4(\theta_{0}+\theta_{1})\psi-b)x^{-1}$
		$\displaystyle+4(2(\theta_{0}-\theta_{1})\theta_{\infty}\psi+(\theta_{0}-\theta_{1})^{2}+\theta_{\infty}^{2})x^{-2},$		(3.1)
	$\displaystyle b^{\prime}=$	$\displaystyle-2(A_{\phi}-\psi^{2})+4\psi^{\prime}+(4(\theta_{0}+\theta_{1})\psi-b)x^{-1},$		(3.2)

where $b=b(x)$ is as defined in Section 2 by using the Lagrangian $a_{\phi,\mathrm{Lag}}.$

(2) $\psi_{0}(x)$ and $b_{0}(x)$ are bounded in $S(\phi,t_{\infty},\kappa_{0},\delta_{0})$ and fulfil

	$\displaystyle 4(\psi^{\prime}_{0})^{2}$	$\displaystyle=(1-\psi_{0}^{2})(A_{\phi}-\psi_{0}^{2}),$		(3.3)
	$\displaystyle b^{\prime}_{0}$	$\displaystyle=-2(A_{\phi}-\psi_{0}^{2})+4\psi_{0}^{\prime}$		(3.4)

[10], which at least formally approximates system (3.1), (3.2).

From Proposition 3.1 with (3.3), it follows that $2\psi^{\prime}(x)=2(1+h^{\prime}(x))\psi^{\prime}_{0}(x+h(x))=(1+h^{\prime})\sqrt{(1-\psi_{0}(x+h)^{2})(A_{\phi}-\psi_{0}(x+h)^{2})}=(1+h^{\prime})\sqrt{(1-\psi^{2})(A_{\phi}-\psi^{2})}.$ Then (3.1) becomes $(1+h^{\prime})^{2}=1-2F_{1}(\psi,b)x^{-1}+2F_{2}(\psi)x^{-2}$, which yields

$$h^{\prime}=-F_{1}(\psi,b)x^{-1}+(F_{2}(\psi)-\tfrac{1}{2}F_{1}(\psi,b)^{2})x^{-2}+O(x^{-3})$$

(3.5)

in $\check{S}(\phi,t_{\infty},\kappa_{0},\delta_{0}),$ where

$$F_{1}(\psi,b)=\frac{4(\theta_{0}+\theta_{1})\psi-b}{2(A_{\phi}-\psi^{2})},\quad F_{2}(\psi)=\frac{2(2(\theta_{0}-\theta_{1})\theta_{\infty}\psi+(\theta_{0}-\theta_{1})^{2}+\theta_{\infty}^{2})}{(1-\psi^{2})(A_{\phi}-\psi^{2})}.$$

Using $\psi=\psi_{0}+\psi^{\prime}_{0}h+O(h^{2}),$ we have

	$\displaystyle h^{\prime}=$	$\displaystyle-F_{1}(\psi_{0},b)x^{-1}+(F_{2}(\psi_{0})-\tfrac{1}{2}F_{1}(\psi_{0},b)^{2})x^{-2}$
		$\displaystyle-(F_{1})_{\psi}(\psi_{0},b)\psi^{\prime}_{0}hx^{-1}+O(x^{-1}(|x^{-1}|+|h|)^{2}).$		(3.6)

In what follows we suppose that, for a positive number $\mu\leq 1$,

$$h(x)\ll x^{-\mu}\phantom{---------}$$

(3.7)

in $\check{S}_{\mathrm{cut}}(\phi,t_{\infty},\kappa_{0},\delta_{0})$. By Proposition 3.1, estimate (3.7) is true if, say, $\mu=1/9.$

Let $\{x_{\nu}\}\subset\check{S}_{\mathrm{cut}}(\phi,t_{\infty},\kappa_{0},\delta_{0})$ be a given sequence such that $|x_{1}|<\cdots<|x_{\nu}|<\cdots,$ $|x_{\nu}|\to\infty$. Then, by (3.2) and (3.4)

	$\displaystyle b(x)-b(x_{\nu})=\int^{x}_{x_{\nu}}(4\psi^{\prime}-2(A_{\phi}-\psi^{2}))d\xi+\int^{x}_{x_{\nu}}(4(\theta_{0}+\theta_{1})\psi-b)\frac{d\xi}{\xi},$
	$\displaystyle b_{0}(x)-b_{0}(x_{\nu})=\int^{x}_{x_{\nu}}(4\psi^{\prime}_{0}-2(A_{\phi}-\psi_{0}^{2}))d\xi,$

from which we derive, for $x\in\check{S}_{\mathrm{cut}}(\phi,t_{\infty},\kappa_{0},\delta_{0})$ with $|x|<|x_{\nu}|$,

	$\displaystyle b(x)-b_{0}(x)-$	$\displaystyle(b(x_{\nu})-b_{0}(x_{\nu}))=4(\psi(x)-\psi_{0}(x)-(\psi(x_{\nu})-\psi_{0}(x_{\nu})))$
		$\displaystyle+2\int^{x}_{x_{\nu}}(\psi^{2}-\psi_{0}^{2})d\xi+2\int^{x}_{x_{\nu}}(A_{\phi}-\psi^{2})F_{1}(\psi,b)\frac{d\xi}{\xi}.$		(3.8)

In this equality, by (3.7) and Proposition 3.1,

	$\displaystyle\psi(x)-\psi_{0}(x)-(\psi(x_{\nu})-\psi_{0}(x_{\nu}))\ll|h(x)|+|h(x_{\nu})|\ll|x^{-\mu}|+|x_{\nu}^{-\mu}|,$
	$\displaystyle b(x_{\nu})-b_{0}(x_{\nu})\ll x_{\nu}^{-2/9+\varepsilon}.$

Furthermore,

	$\displaystyle 2\int^{x}_{x_{\nu}}(\psi^{2}-\psi_{0}^{2})d\xi$	$\displaystyle=2\int^{x}_{x_{\nu}}\Bigl{(}(\psi_{0}^{2})^{\prime}h+\frac{(\psi_{0}^{2})^{\prime\prime}}{2}h^{2}+\cdots+\frac{(\psi_{0}^{2})^{(p)}}{p!}h^{p}+O(h^{p+1})\Bigr{)}d\xi$
		$\displaystyle=-2\int^{x}_{x_{\nu}}\Bigl{(}\psi_{0}^{2}+(\psi_{0}^{2})^{\prime}h+\cdots+\frac{(\psi_{0}^{2})^{(p-1)}}{(p-1)!}h^{p-1}\Bigr{)}h^{\prime}d\xi+O(|x^{-\mu}|+|x_{\nu}^{-\mu}|),$

if $-(p+1)\mu+1\leq-\mu,$ i.e. $-p\mu+1\leq 0$; and by (3.5) and (3.7),

	$\displaystyle 2\int^{x}_{x_{\nu}}(A_{\phi}-\psi^{2})F_{1}(\psi,b)\frac{d\xi}{\xi}=$	$\displaystyle-2\int^{x}_{x_{\nu}}(A_{\phi}-\psi^{2})(h^{\prime}+O(\xi^{-2}))d\xi$
	$\displaystyle=$	$\displaystyle-2\int^{x}_{x_{\nu}}(A_{\phi}-\psi^{2})h^{\prime}d\xi+O(|x^{-1}|+|x_{\nu}^{-1}|),$

where

		$\displaystyle-2\int^{x}_{x_{\nu}}(A_{\phi}-\psi^{2})h^{\prime}d\xi$
	$\displaystyle=$	$\displaystyle 2\int^{x}_{x_{\nu}}\Bigl{(}\psi_{0}^{2}+(\psi_{0}^{2})^{\prime}h+\cdots+\frac{(\psi_{0}^{2})^{(p-1)}}{(p-1)!}h^{p-1}\Bigr{)}h^{\prime}d\xi+O(|x^{-\mu}|+|x_{\nu}^{-\mu}|),$

since $h^{\prime}\ll\xi^{-1}$ by (3.6) and (3.7). Insert these quantities with $p$ such that $-p\mu+1\leq 0$ into (3.8). Under the passage to the limit $x_{\nu}\to\infty$, we arrive at the estimate

$$b(x)-b_{0}(x)\ll x^{-\mu}$$

(3.9)

in $\check{S}_{\mathrm{cut}}(\phi,t_{\infty},\kappa_{0},\delta_{0}).$ Then equation (3.6) is written in the form

	$\displaystyle h^{\prime}$	$\displaystyle=-F_{1}(\psi_{0},b)x^{-1}+(F_{2}(\psi_{0})-\tfrac{1}{2}F_{1}(\psi_{0},b_{0})^{2})x^{-2}-(F_{1})_{\psi}(\psi_{0},b_{0})\psi_{0}^{\prime}hx^{-1}+O(x^{-1-2\mu})$
		$\displaystyle=-F_{1}(\psi_{0},b_{0})x^{-1}+O(x^{-1-\mu}).$		(3.10)

For any sequence $\{x_{\nu}\}\subset\check{S}_{\mathrm{cut}}(\phi,t_{\infty},\kappa_{0},\delta_{0})$, integration of this yields

		$\displaystyle h(x)-h(x_{\nu})$
	$\displaystyle=$	$\displaystyle-\int^{x}_{x_{\nu}}F_{1}(\psi_{0},b)\frac{d\xi}{\xi}+\int^{x}_{x_{\nu}}(F_{2}(\psi_{0})-\tfrac{1}{2}F_{1}(\psi_{0},b_{0})^{2})\frac{d\xi}{\xi^{2}}-\mathcal{I}_{0}+O(|x^{-2\mu}|+|x_{\nu}^{-2\mu}|)$

with

	$\displaystyle\mathcal{I}_{0}$	$\displaystyle=\int^{x}_{x_{\nu}}(F_{1})_{\psi}(\psi_{0},b_{0})\psi_{0}^{\prime}h\frac{d\xi}{\xi}$
		$\displaystyle=\int^{x}_{x_{\nu}}\Bigl{(}F_{1}(\psi_{0},0)_{\xi}-\Bigl{(}\frac{1}{2(A_{\phi}-\psi_{0}^{2})}\Bigr{)}_{\!\xi}b_{0}\Bigr{)}h\frac{d\xi}{\xi}$
		$\displaystyle=\int^{x}_{x_{\nu}}\Bigl{(}F_{1}(\psi_{0},0)F_{1}(\psi_{0},b_{0})-\frac{b_{0}F_{1}(\psi_{0},b_{0})-b_{0}^{\prime}h\xi}{2(A_{\phi}-\psi_{0}^{2})}\Bigr{)}\frac{d\xi}{\xi^{2}}+O(|x^{-1-\mu}|+|x_{\nu}^{-1-\mu}|)$
		$\displaystyle=\int^{x}_{x_{\nu}}F_{1}(\psi_{0},b_{0})^{2}\frac{d\xi}{\xi^{2}}+\frac{1}{2}\int^{x}_{x_{\nu}}\frac{b_{0}^{\prime}h}{A_{\phi}-\psi_{0}^{2}}\frac{d\xi}{\xi}+O(|x^{-1-\mu}|+|x_{\nu}^{-1-\mu}|),$

where the third line is due to integration by parts. Hence we have, in $\check{S}_{\mathrm{cut}}(\phi,t_{\infty},\kappa_{0},\delta_{0})$,

	$\displaystyle h(x)=$	$\displaystyle-\int^{x}_{\infty}F_{1}(\psi_{0},b)\frac{d\xi}{\xi}$
		$\displaystyle+\int^{x}_{\infty}\Bigl{(}F_{2}(\psi_{0})-\frac{3}{2}F_{1}(\psi_{0},b_{0})^{2}\Bigr{)}\frac{d\xi}{\xi^{2}}-\frac{1}{2}\int^{x}_{\infty}\frac{b_{0}^{\prime}h}{A_{\phi}-\psi_{0}^{2}}\frac{d\xi}{\xi}+O(x^{-2\mu}),$		(3.11)

in which the convergence of $\int^{x}_{\infty}F_{1}(\psi_{0},b)\xi^{-1}d\xi$ is guaranteed by the absolute convergence of the remaining two integrals.