§32.2 Differential Equations

Permalink:: http://dlmf.nist.gov/32.2

§32.2(i) Introduction
§32.2(ii) Renormalizations
§32.2(iii) Alternative Forms
§32.2(iv) Elliptic Form
§32.2(v) Symmetric Forms
§32.2(vi) Coalescence Cascade

§32.2(i) Introduction

Notes:: See Hille (1976, pp. 439–444), Ince (1926, Chapter XIV), Kruskal and Clarkson (1992).
Keywords:: Painlevé equations, Painlevé property, Painlevé transcendents, branch point, isolated essential singularity, locally analytic, pole, singularities, transcendental functions
Permalink:: http://dlmf.nist.gov/32.2.i

The six Painlevé equations $\mbox{P}_{{\mbox{\scriptsize I}}}$ – $\mbox{P}_{{\mbox{\scriptsize VI}}}$ are as follows:

32.2.1 $\frac{{d}^{2}w}{{dz}^{2}}=6w^{2}+z,$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ and $z$ : real
Referenced by:: §32.11(i), §32.11(i)
Permalink:: http://dlmf.nist.gov/32.2.E1
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32.2.2 $\frac{{d}^{2}w}{{dz}^{2}}=2w^{3}+zw+\alpha,$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : real and $\alpha$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E2
Encodings:: TeX, pMML, png

32.2.3 $\frac{{d}^{2}w}{{dz}^{2}}=\frac{1}{w}\left(\frac{dw}{dz}\right)^{2}-\frac{1}{z}\frac{dw}{dz}+\frac{\alpha w^{2}+\beta}{z}+\gamma w^{3}+\frac{\delta}{w},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E3
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32.2.4 $\frac{{d}^{2}w}{{dz}^{2}}=\frac{1}{2w}\left(\frac{dw}{dz}\right)^{2}+\frac{3}{2}w^{3}+4zw^{2}+2(z^{2}-\alpha)w+\frac{\beta}{w},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : real, $\alpha$ : arbitrary constant and $\beta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E4
Encodings:: TeX, pMML, png

32.2.5 $\frac{{d}^{2}w}{{dz}^{2}}=\left(\frac{1}{2w}+\frac{1}{w-1}\right)\left(\frac{dw}{dz}\right)^{2}-\frac{1}{z}\frac{dw}{dz}+\frac{(w-1)^{2}}{z^{2}}\left(\alpha w+\frac{\beta}{w}\right)+\frac{\gamma w}{z}+\frac{\delta w(w+1)}{w-1},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E5
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32.2.6 $\frac{{d}^{2}w}{{dz}^{2}}=\frac{1}{2}\left(\frac{1}{w}+\frac{1}{w-1}+\frac{1}{w-z}\right)\left(\frac{dw}{dz}\right)^{2}-\left(\frac{1}{z}+\frac{1}{z-1}+\frac{1}{w-z}\right)\frac{dw}{dz}+\frac{w(w-1)(w-z)}{z^{2}(z-1)^{2}}\left(\alpha+\frac{\beta z}{w^{2}}+\frac{\gamma(z-1)}{(w-1)^{2}}+\frac{\delta z(z-1)}{(w-z)^{2}}\right),$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
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with $\alpha$ , $\beta$ , $\gamma$ , and $\delta$ arbitrary constants. The solutions of $\mbox{P}_{{\mbox{\scriptsize I}}}$ – $\mbox{P}_{{\mbox{\scriptsize VI}}}$ are called the Painlevé transcendents. The six equations are sometimes referred to as the Painlevé transcendents, but in this chapter this term will be used only for their solutions.

Let

32.2.7 $\frac{{d}^{2}w}{{dz}^{2}}=F\left(z,w,\frac{dw}{dz}\right),$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ and $z$ : real
Permalink:: http://dlmf.nist.gov/32.2.E7
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be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $\ifrac{dw}{dz}$ , and is locally analytic in $z$ , that is, analytic except for isolated singularities in $\Complex$ . In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions. An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however.

There are fifty equations with the Painlevé property. They are distinct modulo Möbius (bilinear) transformations

32.2.8

$W(\zeta)=\frac{a(z)w+b(z)}{c(z)w+d(z)},$

$\zeta=\phi(z),$

Symbols:: $z$ : real, $\phi(z)$ : analytic function, $W(\zeta)$ : bilinear transformation, $a(z)$ : analytic function, $b(z)$ : analytic function, $c(z)$ : analytic function and $d(z)$ : analytic function
Permalink:: http://dlmf.nist.gov/32.2.E8
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in which $a(z)$ , $b(z)$ , $c(z)$ , $d(z)$ , and $\phi(z)$ are locally analytic functions. The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of $\mbox{P}_{{\mbox{\scriptsize I}}}$ – $\mbox{P}_{{\mbox{\scriptsize VI}}}$ .

For arbitrary values of the parameters $\alpha$ , $\beta$ , $\gamma$ , and $\delta$ , the general solutions of $\mbox{P}_{{\mbox{\scriptsize I}}}$ – $\mbox{P}_{{\mbox{\scriptsize VI}}}$ are transcendental, that is, they cannot be expressed in closed-form elementary functions. However, for special values of the parameters, equations $\mbox{P}_{{\mbox{\scriptsize II}}}$ – $\mbox{P}_{{\mbox{\scriptsize VI}}}$ have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF.

§32.2(ii) Renormalizations

Keywords:: Painlevé equations
Referenced by:: §32.10(iii), §32.10(v), §32.8(iii), §32.8(v), §32.9(i)
Permalink:: http://dlmf.nist.gov/32.2.ii

If $\gamma\delta\neq 0$ in $\mbox{P}_{{\mbox{\scriptsize III}}}$ , then set $\gamma=1$ and $\delta=-1$ , without loss of generality, by rescaling $w$ and $z$ if necessary. If $\gamma=0$ and $\alpha\delta\neq 0$ in $\mbox{P}_{{\mbox{\scriptsize III}}}$ , then set $\alpha=1$ and $\delta=-1$ , without loss of generality. Lastly, if $\delta=0$ and $\beta\gamma\neq 0$ , then set $\beta=-1$ and $\gamma=1$ , without loss of generality.

If $\delta\neq 0$ in $\mbox{P}_{{\mbox{\scriptsize V}}}$ , then set $\delta=-\tfrac{1}{2}$ , without loss of generality.

§32.2(iii) Alternative Forms

Keywords:: Painlevé equations, nonlinear harmonic oscillator
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.2.iii

In $\mbox{P}_{{\mbox{\scriptsize III}}}$ , if $w(z)=\zeta^{{-1/2}}u(\zeta)$ with $\zeta=z^{2}$ , then

32.2.9 $\frac{{d}^{2}u}{{d\zeta}^{2}}=\frac{1}{u}\left(\frac{du}{d\zeta}\right)^{2}-\frac{1}{\zeta}\frac{du}{d\zeta}+\frac{u^{2}(\alpha+\gamma u)}{4\zeta^{2}}+\frac{\beta}{4\zeta}+\frac{\delta}{4u},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E9
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which is known as $\mbox{P}^{{\prime}}_{{\mbox{\scriptsize III}}}$ .

In $\mbox{P}_{{\mbox{\scriptsize III}}}$ , if $w(z)=\mathop{\exp\/}\nolimits\!\left(-iu(z)\right)$ , $\beta=-\alpha$ , and $\delta=-\gamma$ , then

32.2.10 $\frac{{d}^{2}u}{{dz}^{2}}+\frac{1}{z}\frac{du}{dz}=\frac{2\alpha}{z}\mathop{\sin\/}\nolimits u+2\gamma\mathop{\sin\/}\nolimits\!\left(2u\right).$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : real, $\alpha$ : arbitrary constant and $\gamma$ : arbitrary constant
Referenced by:: §32.13(ii)
Permalink:: http://dlmf.nist.gov/32.2.E10
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In $\mbox{P}_{{\mbox{\scriptsize IV}}}$ , if $w(z)=2\sqrt{2}(u(\zeta))^{2}$ with $\zeta=\sqrt{2}z$ and $\alpha=2\nu+1$ , then

32.2.11 $\frac{{d}^{2}u}{{d\zeta}^{2}}=3u^{5}+2\zeta u^{3}+\left(\tfrac{1}{4}\zeta^{2}-\nu-\tfrac{1}{2}\right)u+\frac{\beta}{32u^{3}}.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\nu$ and $\beta$ : arbitrary constant
Referenced by:: §32.3(iii)
Permalink:: http://dlmf.nist.gov/32.2.E11
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When $\beta=0$ this is a nonlinear harmonic oscillator.

In $\mbox{P}_{{\mbox{\scriptsize V}}}$ , if $w(z)=(\mathop{\coth\/}\nolimits u(\zeta))^{2}$ with $\zeta=\mathop{\ln\/}\nolimits z$ , then

32.2.12 $\frac{{d}^{2}u}{{d\zeta}^{2}}=-\frac{\alpha\mathop{\cosh\/}\nolimits u}{2(\mathop{\sinh\/}\nolimits u)^{3}}-\frac{\beta\mathop{\sinh\/}\nolimits u}{2(\mathop{\cosh\/}\nolimits u)^{3}}-\tfrac{1}{4}\gamma e^{\zeta}\mathop{\sinh\/}\nolimits\!\left(2u\right)-\tfrac{1}{8}\delta e^{{2\zeta}}\mathop{\sinh\/}\nolimits\!\left(4u\right).$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E12
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See also Okamoto (1987c), McCoy et al. (1977), Bassom et al. (1992), Bassom et al. (1995), and Takasaki (2001).

§32.2(iv) Elliptic Form

Keywords:: Painlevé equations
Permalink:: http://dlmf.nist.gov/32.2.iv

$\mbox{P}_{{\mbox{\scriptsize VI}}}$ can be written in the form

32.2.13 $z(1-z)I\left(\int _{{\infty}}^{w}\frac{dt}{\sqrt{t(t-1)(t-z)}}\right)=\sqrt{w(w-1)(w-z)}\*\left(\alpha+\frac{\beta z}{w^{2}}+\frac{\gamma(z-1)}{(w-1)^{2}}+(\delta-\tfrac{1}{2})\frac{z(z-1)}{(w-z)^{2}}\right),$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $z$ : real, $\alpha$ : arbitrary constant, $I$ : operator, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E13
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where

32.2.14 $I=z(1-z)\frac{{d}^{2}}{{dz}^{2}}+(1-2z)\frac{d}{dz}-\frac{1}{4}.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : real and $I$ : operator
Permalink:: http://dlmf.nist.gov/32.2.E14
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See Fuchs (1907), Painlevé (1906), Gromak et al. (2002, §42); also Manin (1998).

§32.2(v) Symmetric Forms

Notes:: See Adler (1994); also Noumi (2004, pp. 13–23).
Keywords:: Painlevé equations
Permalink:: http://dlmf.nist.gov/32.2.v

Let

32.2.15

$\frac{df_{1}}{dz}+f_{1}(f_{2}-f_{3})+2\mu _{1}=0,$

$\frac{df_{2}}{dz}+f_{2}(f_{3}-f_{1})+2\mu _{2}=0,$

$\frac{df_{3}}{dz}+f_{3}(f_{1}-f_{2})+2\mu _{3}=0,$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : real, $f_{{j}}(z)$ : solutions and $\mu _{j}$ : constants
Permalink:: http://dlmf.nist.gov/32.2.E15
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where $\mu _{1}$ , $\mu _{2}$ , $\mu _{3}$ are constants, $f_{1}$ , $f_{2}$ , $f_{3}$ are functions of $z$ , with

32.2.16 $\mu _{1}+\mu _{2}+\mu _{3}=1,$

Symbols:: $\mu _{j}$ : constants
Permalink:: http://dlmf.nist.gov/32.2.E16
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32.2.17 $f_{1}(z)+f_{2}(z)+f_{3}(z)+2z=0.$

Symbols:: $z$ : real and $f_{{j}}(z)$ : solutions
Permalink:: http://dlmf.nist.gov/32.2.E17
Encodings:: TeX, pMML, png

Then $w(z)=f_{1}(z)$ satisfies $\mbox{P}_{{\mbox{\scriptsize IV}}}$ with

32.2.18 $(\alpha,\beta)=(\mu _{3}-\mu _{2},-2\mu _{1}^{2}).$

Symbols:: $(a,b)$ : open interval, $\alpha$ : arbitrary constant, $\mu _{j}$ : constants and $\beta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E18
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See Noumi and Yamada (1998).

Next, let

32.2.19

$z\frac{df_{1}}{dz}=f_{1}f_{3}(f_{2}-f_{4})+(\tfrac{1}{2}-\mu _{3})f_{1}+\mu _{1}f_{3},$

$z\frac{df_{2}}{dz}=f_{2}f_{4}(f_{3}-f_{1})+(\tfrac{1}{2}-\mu _{4})f_{2}+\mu _{2}f_{4},$

$z\frac{df_{3}}{dz}=f_{3}f_{1}(f_{4}-f_{2})+(\tfrac{1}{2}-\mu _{1})f_{3}+\mu _{3}f_{1},$

$z\frac{df_{4}}{dz}=f_{4}f_{2}(f_{1}-f_{3})+(\tfrac{1}{2}-\mu _{2})f_{4}+\mu _{4}f_{2},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : real, $f_{{j}}(z)$ : solutions and $\mu _{j}$ : constants
Permalink:: http://dlmf.nist.gov/32.2.E19
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where $\mu _{1}$ , $\mu _{2}$ , $\mu _{3}$ , $\mu _{4}$ are constants, $f_{1}$ , $f_{2}$ , $f_{3}$ , $f_{4}$ are functions of $z$ , with

32.2.20 $\mu _{1}+\mu _{2}+\mu _{3}+\mu _{4}=1,$

Symbols:: $\mu _{j}$ : constants
Permalink:: http://dlmf.nist.gov/32.2.E20
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32.2.21 $f_{1}(z)+f_{3}(z)=\sqrt{z},$

Symbols:: $z$ : real and $f_{{j}}(z)$ : solutions
Permalink:: http://dlmf.nist.gov/32.2.E21
Encodings:: TeX, pMML, png

32.2.22 $f_{2}(z)+f_{4}(z)=\sqrt{z}.$

Symbols:: $z$ : real and $f_{{j}}(z)$ : solutions
Permalink:: http://dlmf.nist.gov/32.2.E22
Encodings:: TeX, pMML, png

Then $w(z)=1-(\sqrt{z}/f_{1}(z))$ satisfies $\mbox{P}_{{\mbox{\scriptsize V}}}$ with

32.2.23 $(\alpha,\beta,\gamma,\delta)=(\tfrac{1}{2}\mu _{1}^{2},-\tfrac{1}{2}\mu _{3}^{2},\mu _{4}-\mu _{2},-\tfrac{1}{2}).$

Symbols:: $\alpha$ : arbitrary constant, $\mu _{j}$ : constants, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E23
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§32.2(vi) Coalescence Cascade

Notes:: See Ince (1926, pp. 345–346), Iwasaki et al. (1991, pp. 125–126).
Keywords:: Painlevé equations
Permalink:: http://dlmf.nist.gov/32.2.vi

$\mbox{P}_{{\mbox{\scriptsize I}}}$ – $\mbox{P}_{{\mbox{\scriptsize V}}}$ are obtained from $\mbox{P}_{{\mbox{\scriptsize VI}}}$ by a coalescence cascade:

32.2.24 $\begin{array}[]{ccccccc}\mbox{$\mbox{P}_{{\mbox{\scriptsize VI}}}$}&\longrightarrow&\mbox{$\mbox{P}_{{\mbox{\scriptsize V}}}$}&\longrightarrow&\mbox{$\mbox{P}_{{\mbox{\scriptsize IV}}}$}\\ &&\big\downarrow&&\big\downarrow\\ &&\mbox{$\mbox{P}_{{\mbox{\scriptsize III}}}$}&\longrightarrow&\mbox{$\mbox{P}_{{\mbox{\scriptsize II}}}$}&\longrightarrow&\mbox{$\mbox{P}_{{\mbox{\scriptsize I}}}$}\end{array}$

Permalink:: http://dlmf.nist.gov/32.2.E24
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For example, if in $\mbox{P}_{{\mbox{\scriptsize II}}}$

32.2.25 $w(z;\alpha)=\epsilon W(\zeta)+\frac{1}{\epsilon^{5}},$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant and $W(\zeta)$ : bilinear transformation
Permalink:: http://dlmf.nist.gov/32.2.E25
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32.2.26

$z=\epsilon^{2}\zeta-\frac{6}{\epsilon^{{10}}},$

$\alpha=\frac{4}{\epsilon^{{15}}},$

Symbols:: $z$ : real and $\alpha$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E26
Encodings:: TeX, TeX, pMML, pMML, png, png

then

32.2.27 $\frac{{d}^{2}W}{{d\zeta}^{2}}=6W^{2}+\zeta+\epsilon^{6}(2W^{3}+\zeta W);$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ and $W(\zeta)$ : bilinear transformation
Permalink:: http://dlmf.nist.gov/32.2.E27
Encodings:: TeX, pMML, png

thus in the limit as $\epsilon\to 0$ , $W(\zeta)$ satisfies $\mbox{P}_{{\mbox{\scriptsize I}}}$ with $z=\zeta$ .

If in $\mbox{P}_{{\mbox{\scriptsize III}}}$

32.2.28 $w(z;\alpha,\beta,\gamma,\delta)=1+2\epsilon W(\zeta;a),$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant, $\delta$ : arbitrary constant and $W(\zeta)$ : bilinear transformation
Permalink:: http://dlmf.nist.gov/32.2.E28
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32.2.29

$z=1+\epsilon^{2}\zeta,$

$\alpha=-\tfrac{1}{2}\epsilon^{{-6}},$

$\beta=\tfrac{1}{2}\epsilon^{{-6}}+2a\epsilon^{{-3}},$

$\gamma=-\delta=\tfrac{1}{4}\epsilon^{{-6}},$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E29
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then as $\epsilon\to 0$ , $W(\zeta;a)$ satisfies $\mbox{P}_{{\mbox{\scriptsize II}}}$ with $z=\zeta$ , $\alpha=a$ .

If in $\mbox{P}_{{\mbox{\scriptsize IV}}}$

32.2.30 $w(z;\alpha,\beta)=2^{{2/3}}\epsilon^{{-1}}W(\zeta;a)+\epsilon^{{-3}},$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant and $W(\zeta)$ : bilinear transformation
Permalink:: http://dlmf.nist.gov/32.2.E30
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32.2.31

$z=2^{{-2/3}}\epsilon\zeta-\epsilon^{{-3}},$

$\alpha=-2a-\tfrac{1}{2}\epsilon^{{-6}},$

$\beta=-\tfrac{1}{2}\epsilon^{{-12}},$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant and $\beta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E31
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

then as $\epsilon\to 0$ , $W(\zeta;a)$ satisfies $\mbox{P}_{{\mbox{\scriptsize II}}}$ with $z=\zeta$ , $\alpha=a$ .

If in $\mbox{P}_{{\mbox{\scriptsize V}}}$

32.2.32 $w(z;\alpha,\beta,\gamma,\delta)=1+\epsilon\zeta W(\zeta;a,b,c,d),$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant, $\delta$ : arbitrary constant and $W(\zeta)$ : bilinear transformation
Permalink:: http://dlmf.nist.gov/32.2.E32
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32.2.33

$z=\zeta^{2},$

$\alpha=\tfrac{1}{4}a\epsilon^{{-1}}+\tfrac{1}{8}c\epsilon^{{-2}},$

$\beta=-\tfrac{1}{8}c\epsilon^{{-2}},$

$\gamma=\tfrac{1}{4}\epsilon b,$

$\delta=\tfrac{1}{8}\epsilon^{2}d,$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E33
Encodings:: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png

then as $\epsilon\to 0$ , $W(\zeta;a,b,c,d)$ satisfies $\mbox{P}_{{\mbox{\scriptsize III}}}$ with $z=\zeta$ , $\alpha=a$ , $\beta=b$ , $\gamma=c$ , $\delta=d$ .

If in $\mbox{P}_{{\mbox{\scriptsize V}}}$

32.2.34 $w(z;\alpha,\beta,\gamma,\delta)=\tfrac{1}{2}\sqrt{2}\epsilon W(\zeta;a,b),$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant, $\delta$ : arbitrary constant and $W(\zeta)$ : bilinear transformation
Permalink:: http://dlmf.nist.gov/32.2.E34
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32.2.35

$z=1+\sqrt{2}\epsilon\zeta,$

$\alpha=\tfrac{1}{2}\epsilon^{{-4}},$

$\beta=\tfrac{1}{4}b,$

$\gamma=-\epsilon^{{-4}},$

$\delta=a\epsilon^{{-2}}-\tfrac{1}{2}\epsilon^{{-4}},$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E35
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then as $\epsilon\to 0$ , $W(\zeta;a,b)$ satisfies $\mbox{P}_{{\mbox{\scriptsize IV}}}$ with $z=\zeta$ , $\alpha=a$ , $\beta=b$ .

Lastly, if in $\mbox{P}_{{\mbox{\scriptsize VI}}}$

32.2.36 $w(z;\alpha,\beta,\gamma,\delta)=W(\zeta;a,b,c,d),$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant, $\delta$ : arbitrary constant and $W(\zeta)$ : bilinear transformation
Permalink:: http://dlmf.nist.gov/32.2.E36
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32.2.37

$z=1+\epsilon\zeta,$

$\gamma=c\epsilon^{{-1}}-d\epsilon^{{-2}},$

$\delta=d\epsilon^{{-2}},$

Symbols:: $z$ : real, $\gamma$ : arbitrary constant and $\delta$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/32.2.E37
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

then as $\epsilon\to 0$ , $W(\zeta;a,b,c,d)$ satisfies $\mbox{P}_{{\mbox{\scriptsize V}}}$ with $z=\zeta$ , $\alpha=a$ , $\beta=b$ , $\gamma=c$ , $\delta=d$ .