32 Painlevé Transcendents Properties 32.10 Special Function Solutions 32.12 Asymptotic Approximations for Complex Variables

§32.11 Asymptotic Approximations for Real Variables

Keywords:: Painlevé transcendents
Permalink:: http://dlmf.nist.gov/32.11

§32.11(i) First Painlevé Equation
§32.11(ii) Second Painlevé Equation
§32.11(iii) Modified Second Painlevé Equation
§32.11(iv) Third Painlevé Equation
§32.11(v) Fourth Painlevé Equation

§32.11(i) First Painlevé Equation

Notes:: See Bender and Orszag (1978, pp. 158–166), Holmes and Spence (1984). For (32.11.2) see Qin and Lu (2008).
Referenced by:: §32.3(i)
Permalink:: http://dlmf.nist.gov/32.11.i

There are solutions of (32.2.1) such that

32.11.1 $w(x)=-\sqrt{\tfrac{1}{6}|x|}+d|x|^{{-1/8}}\mathop{\sin\/}\nolimits\!\left(\phi% (x)-\theta_{0}\right)+\mathop{o\/}\nolimits\!\left(|x|^{{-1/8}}\right),$ $x\to-\infty$ ,

Symbols:: $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\mathop{\sin\/}\nolimits z$ : sine function, : real, $\phi(z)$ : function, : constant and $\theta_{0}$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E1
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where

32.11.2 $\phi(x)=(24)^{{1/4}}\left(\tfrac{4}{5}|x|^{{5/4}}-\tfrac{5}{8}d^{2}\mathop{\ln% \/}\nolimits|x|\right),$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real, $\phi(z)$ : function and : constant
Referenced by:: §32.11(i)
Permalink:: http://dlmf.nist.gov/32.11.E2
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and and $\theta_{0}$ are constants.

There are also solutions of (32.2.1) such that

32.11.3 $w(x)\sim\sqrt{\tfrac{1}{6}|x|},$ $x\to-\infty$ .

Symbols:: $\sim$ : asymptotic equality and : real
Permalink:: http://dlmf.nist.gov/32.11.E3
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Next, for given initial conditions and $w^{{\prime}}(0)=k$ , with real, has at least one pole on the real axis. There are two special values of , $k_{1}$ and $k_{2}$ , with the properties $-0.45142\;8<k_{1}<-0.45142\;7$ , $1.85185\;3<k_{2}<1.85185\;5$ , and such that:

(a)

If $k<k_{1}$ , then for $x_{0}<x<0$ , where $x_{0}$ is the first pole on the negative real axis.
(b)

If $k_{1}<k<k_{2}$ , then oscillates about, and is asymptotic to, $-\sqrt{\tfrac{1}{6}|x|}$ as $x\to-\infty$ .
(c)

If $k_{2}<k$ , then changes sign once, from positive to negative, as passes from $x_{0}$ to 0.

For illustration see Figures 32.3.1 to 32.3.4, and for further information see Joshi and Kitaev (2005), Joshi and Kruskal (1992), Kapaev (1988), Kapaev and Kitaev (1993), and Kitaev (1994).

§32.11(ii) Second Painlevé Equation

Notes:: See Ablowitz and Segur (1977), Fokas et al. (2006, Chapter 10), Hastings and McLeod (1980).
Referenced by:: §2.8(iii), §32.3(ii)
Permalink:: http://dlmf.nist.gov/32.11.ii

Consider the special case of $\mbox{P}_{{\mbox{\scriptsize II}}}$ with $\alpha=0$ :

32.11.4 $w^{{\prime\prime}}=2w^{3}+xw,$

Symbols:: : real
Referenced by:: §32.11(ii), §32.11(iii)
Permalink:: http://dlmf.nist.gov/32.11.E4
Encodings:: TeX, pMML, png

with boundary condition

32.11.5 $w(x)\to 0,$ $x\to+\infty$ .

Symbols:: : real
Referenced by:: §32.11(ii)
Permalink:: http://dlmf.nist.gov/32.11.E5
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Any nontrivial real solution of (32.11.4) that satisfies (32.11.5) is asymptotic to $k\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ , for some nonzero real , where $\mathop{\mathrm{Ai}\/}\nolimits$ denotes the Airy function (§9.2). Conversely, for any nonzero real , there is a unique solution $w_{k}(x)$ of (32.11.4) that is asymptotic to $k\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ as $x\to+\infty$ .

If , then $w_{k}(x)$ exists for all sufficiently large as $x\to-\infty$ , and

32.11.6 $w_{k}(x)=d|x|^{{-1/4}}\mathop{\sin\/}\nolimits\!\left(\phi(x)-\theta_{0}\right% )+\mathop{o\/}\nolimits\!\left(|x|^{{-1/4}}\right),$

Symbols:: $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\mathop{\sin\/}\nolimits z$ : sine function, : real, : real, $\phi(z)$ : function, : constant and $\theta_{0}$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E6
Encodings:: TeX, pMML, png

where

32.11.7 $\phi(x)=\tfrac{2}{3}|x|^{{3/2}}-\tfrac{3}{4}d^{2}\mathop{\ln\/}\nolimits|x|,$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real, $\phi(z)$ : function and : constant
Permalink:: http://dlmf.nist.gov/32.11.E7
Encodings:: TeX, pMML, png

and $(\neq 0)$ , $\theta_{0}$ are real constants. Connection formulas for and $\theta_{0}$ are given by

32.11.8 $d^{2}=-\pi^{{-1}}\mathop{\ln\/}\nolimits\!\left(1-k^{2}\right),$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real and : constant
Permalink:: http://dlmf.nist.gov/32.11.E8
Encodings:: TeX, pMML, png

32.11.9 $\theta_{0}=\tfrac{3}{2}d^{2}\mathop{\ln\/}\nolimits 2+\mathop{\mathrm{ph}\/}% \nolimits\mathop{\Gamma\/}\nolimits\!\left(1-\tfrac{1}{2}id^{2}\right)+\tfrac{% 1}{4}\pi(1-2\mathop{\mathrm{sign}\/}\nolimits\!\left(k\right)),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathrm{sign}\/}\nolimits x$ : sign of , : real, : constant and $\theta_{0}$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E9
Encodings:: TeX, pMML, png

where $\mathop{\Gamma\/}\nolimits$ is the gamma function (§5.2(i)), and the branch of the $\mathop{\mathrm{ph}\/}\nolimits$ function is immaterial.

If , then

32.11.10 $w_{k}(x)\sim\mathop{\mathrm{sign}\/}\nolimits\!\left(k\right)\sqrt{\tfrac{1}{2% }|x|},$ $x\to-\infty$ .

Symbols:: $\mathop{\mathrm{sign}\/}\nolimits x$ : sign of , $\sim$ : asymptotic equality, : real and : real
Permalink:: http://dlmf.nist.gov/32.11.E10
Encodings:: TeX, pMML, png

If , then $w_{k}(x)$ has a pole at a finite point $x=c_{0}$ , dependent on , and

32.11.11 $w_{k}(x)\sim\mathop{\mathrm{sign}\/}\nolimits\!\left(k\right)(x-c_{0})^{{-1}},$ $x\to c_{0}+$ .

Symbols:: $\mathop{\mathrm{sign}\/}\nolimits x$ : sign of , $\sim$ : asymptotic equality, : real, : real and $c_{0}$ : pole
Permalink:: http://dlmf.nist.gov/32.11.E11
Encodings:: TeX, pMML, png

For illustration see Figures 32.3.5 and 32.3.6, and for further information see Ablowitz and Clarkson (1991), Bassom et al. (1998), Clarkson and McLeod (1988), Deift and Zhou (1995), Segur and Ablowitz (1981), and Suleĭmanov (1987). For numerical studies see Miles (1978, 1980) and Rosales (1978).

§32.11(iii) Modified Second Painlevé Equation

Notes:: See Its et al. (1994), Its and Kapaev (1987), Deift and Zhou (1995), Fokas et al. (2006, Chapter 9).
Referenced by:: §2.8(iii)
Permalink:: http://dlmf.nist.gov/32.11.iii

Replacement of by in (32.11.4) gives

32.11.12 $w^{{\prime\prime}}=-2w^{3}+xw.$

Symbols:: : real
Referenced by:: §32.11(iii)
Permalink:: http://dlmf.nist.gov/32.11.E12
Encodings:: TeX, pMML, png

Any nontrivial real solution of (32.11.12) satisfies

32.11.13 $w(x)=d|x|^{{-1/4}}\mathop{\sin\/}\nolimits\!\left(\phi(x)-\chi\right)+\mathop{% O\/}\nolimits\!\left(|x|^{{-5/4}}\mathop{\ln\/}\nolimits|x|\right),$ $x\to-\infty$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\sin\/}\nolimits z$ : sine function, : real, $\chi$ : constant, $\phi(z)$ : function and : constant
Permalink:: http://dlmf.nist.gov/32.11.E13
Encodings:: TeX, pMML, png

where

32.11.14 $\phi(x)=\tfrac{2}{3}|x|^{{3/2}}+\tfrac{3}{4}d^{2}\mathop{\ln\/}\nolimits|x|,$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real, $\phi(z)$ : function and : constant
Permalink:: http://dlmf.nist.gov/32.11.E14
Encodings:: TeX, pMML, png

with $(\neq 0)$ and $\chi$ arbitrary real constants.

In the case when

32.11.15 $\chi+\tfrac{3}{2}d^{2}\mathop{\ln\/}\nolimits 2-\tfrac{1}{4}\pi-\mathop{% \mathrm{ph}\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}id^{2}% \right)=n\pi,$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : integer, $\chi$ : constant and : constant
Permalink:: http://dlmf.nist.gov/32.11.E15
Encodings:: TeX, pMML, png

with $n\in\Integer$ , we have

32.11.16 $w(x)\sim k\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right),$ $x\to+\infty$ ,

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\sim$ : asymptotic equality, : real and : real
Permalink:: http://dlmf.nist.gov/32.11.E16
Encodings:: TeX, pMML, png

where is a nonzero real constant. The connection formulas for are

32.11.17 $d^{2}=\pi^{{-1}}\mathop{\ln\/}\nolimits\!\left(1+k^{2}\right),$ $\mathop{\mathrm{sign}\/}\nolimits\!\left(k\right)=(-1)^{n}$ .

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{sign}\/}\nolimits x$ : sign of , : integer, : real and : constant
Permalink:: http://dlmf.nist.gov/32.11.E17
Encodings:: TeX, pMML, png

In the generic case

32.11.18 $\chi+\tfrac{3}{2}d^{2}\mathop{\ln\/}\nolimits 2-\tfrac{1}{4}\pi-\mathop{% \mathrm{ph}\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}id^{2}% \right)\neq n\pi,$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : integer, $\chi$ : constant and : constant
Permalink:: http://dlmf.nist.gov/32.11.E18
Encodings:: TeX, pMML, png

we have

32.11.19 $w(x)=\sigma\sqrt{\tfrac{1}{2}x}+\sigma\rho(2x)^{{-1/4}}\mathop{\cos\/}% \nolimits\!\left(\psi(x)+\theta\right)+\mathop{O\/}\nolimits\!\left(x^{{-1}}% \right),$ $x\to+\infty$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\cos\/}\nolimits z$ : cosine function, : real, $\sigma$ : real constant, $\rho$ : real constant, $\theta$ : real constant and $\psi(z)$ : function
Permalink:: http://dlmf.nist.gov/32.11.E19
Encodings:: TeX, pMML, png

where $\sigma$ , $\rho$ , and $\theta$ are real constants, and

32.11.20 $\psi(x)=\tfrac{2}{3}\sqrt{2}x^{{3/2}}-\tfrac{3}{2}\rho^{2}\mathop{\ln\/}% \nolimits x.$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real, $\rho$ : real constant and $\psi(z)$ : function
Permalink:: http://dlmf.nist.gov/32.11.E20
Encodings:: TeX, pMML, png

The connection formulas for $\sigma$ , $\rho$ , and $\theta$ are

32.11.21 $\sigma=-\mathop{\mathrm{sign}\/}\nolimits\!\left(\imagpart{s}\right),$

Symbols:: $\imagpart{}$ : imaginary part, $\mathop{\mathrm{sign}\/}\nolimits x$ : sign of , $\sigma$ : real constant and
Permalink:: http://dlmf.nist.gov/32.11.E21
Encodings:: TeX, pMML, png

32.11.22 $\rho^{2}=\pi^{{-1}}\mathop{\ln\/}\nolimits\!\left((1+|s|^{2})/|2\imagpart{s}|% \right),$

Symbols:: $\imagpart{}$ : imaginary part, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\rho$ : real constant and
Permalink:: http://dlmf.nist.gov/32.11.E22
Encodings:: TeX, pMML, png

32.11.23 $\theta=-\tfrac{3}{4}\pi-\tfrac{7}{2}\rho^{2}\mathop{\ln\/}\nolimits{2}+\mathop% {\mathrm{ph}\/}\nolimits\!\left(1+s^{2}\right)+\mathop{\mathrm{ph}\/}\nolimits% \mathop{\Gamma\/}\nolimits\!\left(i\rho^{2}\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\rho$ : real constant, $\theta$ : real constant and
Permalink:: http://dlmf.nist.gov/32.11.E23
Encodings:: TeX, pMML, png

where

32.11.24 $s=\left(\mathop{\exp\/}\nolimits\!\left(\pi d^{2}\right)-1\right)^{{1/2}}\*% \mathop{\exp\/}\nolimits\!\left(i\left(\tfrac{3}{2}d^{2}\mathop{\ln\/}% \nolimits 2-\tfrac{1}{4}\pi+\chi-\mathop{\mathrm{ph}\/}\nolimits\mathop{\Gamma% \/}\nolimits\!\left(\tfrac{1}{2}id^{2}\right)\right)\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\chi$ : constant, and : constant
Permalink:: http://dlmf.nist.gov/32.11.E24
Encodings:: TeX, pMML, png

§32.11(iv) Third Painlevé Equation

Notes:: See McCoy et al. (1977), Fokas et al. (2006, Chapter 14).
Permalink:: http://dlmf.nist.gov/32.11.iv

For $\mbox{P}_{{\mbox{\scriptsize III}}}$ , with $\alpha=-\beta=2\nu$ $(\in\Real)$ and $\gamma=-\delta=1$ ,

32.11.25 $w(x)-1\sim-\lambda\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)2^{% {-2\nu}}x^{{-\nu-(1/2)}}e^{{-2x}},$ $x\to+\infty$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : base of exponential function, $\sim$ : asymptotic equality, : real and $\nu$ : parameter
Referenced by:: §32.11(iv)
Permalink:: http://dlmf.nist.gov/32.11.E25
Encodings:: TeX, pMML, png

where $\lambda$ is an arbitrary constant such that $-1/\pi<\lambda<1/\pi$ , and

32.11.26 $w(x)\sim Bx^{\sigma},$ $x\to 0$ ,

Symbols:: $\sim$ : asymptotic equality, : real, : constant and $\sigma$ : constant
Referenced by:: §32.11(iv)
Permalink:: http://dlmf.nist.gov/32.11.E26
Encodings:: TeX, pMML, png

where and $\sigma$ are arbitrary constants such that $B\neq 0$ and $|\realpart{\sigma}|<1$ . The connection formulas relating (32.11.25) and (32.11.26) are

32.11.27 $\sigma=(2/\pi)\mathop{\mathrm{arcsin}\/}\nolimits\!\left(\pi\lambda\right),$

Symbols:: $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function and $\sigma$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E27
Encodings:: TeX, pMML, png

32.11.28 $B=2^{{-2\sigma}}\frac{{\mathop{\Gamma\/}\nolimits^{{2}}}\!\left(\tfrac{1}{2}(1% -\sigma)\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}(1+\sigma)+\nu% \right)}{{\mathop{\Gamma\/}\nolimits^{{2}}}\!\left(\tfrac{1}{2}(1+\sigma)% \right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}(1-\sigma)+\nu\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\nu$ : parameter, : constant and $\sigma$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E28
Encodings:: TeX, pMML, png

See also Abdullaev (1985), Novokshënov (1985), Its and Novokshënov (1986), Kitaev (1987), Bobenko (1991), Bobenko and Its (1995), Tracy and Widom (1997), and Kitaev and Vartanian (2004).

§32.11(v) Fourth Painlevé Equation

Notes:: See Bassom et al. (1992), Its and Kapaev (1998).
Keywords:: Painlevé transcendents
Referenced by:: §32.3(iii)
Permalink:: http://dlmf.nist.gov/32.11.v

Consider $\mbox{P}_{{\mbox{\scriptsize IV}}}$ with $\alpha=2\nu+1$ $(\in\Real)$ and $\beta=0$ , that is,

32.11.29 $w^{{\prime\prime}}=\frac{(w^{{\prime}})^{2}}{2w}+\frac{3}{2}w^{3}+4xw^{2}+2(x^% {2}-2\nu-1)w,$

Symbols:: : real and $\nu$ : parameter
Referenced by:: §32.11(v)
Permalink:: http://dlmf.nist.gov/32.11.E29
Encodings:: TeX, pMML, png

and with boundary condition

32.11.30 $w(x)\to 0,$ $x\to+\infty$ .

Symbols:: : real
Referenced by:: §32.11(v)
Permalink:: http://dlmf.nist.gov/32.11.E30
Encodings:: TeX, pMML, png

Any nontrivial solution of (32.11.29) that satisfies (32.11.30) is asymptotic to $h{\mathop{U\/}\nolimits^{{2}}}\!\left(-\nu-\frac{1}{2},\sqrt{2}x\right)$ as $x\to+\infty$ , where $(\neq 0)$ is a constant. Conversely, for any $(\neq 0)$ there is a unique solution $w_{h}(x)$ of (32.11.29) that is asymptotic to $h{\mathop{U\/}\nolimits^{{2}}}\!\left(-\nu-\frac{1}{2},\sqrt{2}x\right)$ as $x\to+\infty$ . Here $\mathop{U\/}\nolimits$ denotes the parabolic cylinder function (§12.2).

Now suppose $x\to-\infty$ . If $0\leq h<h^{{*}}$ , where

32.11.31 $h^{{*}}=\ifrac{1}{\left(\pi^{{1/2}}\mathop{\Gamma\/}\nolimits\!\left(\nu+1% \right)\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function and $\nu$ : parameter
Permalink:: http://dlmf.nist.gov/32.11.E31
Encodings:: TeX, pMML, png

then $w_{h}(x)$ has no poles on the real axis. Furthermore, if $\nu=n=0,1,2,\dots$ , then

32.11.32 $w_{h}(x)\sim h2^{n}x^{{2n}}\mathop{\exp\/}\nolimits\!\left(-x^{2}\right),$ $x\to-\infty$ .

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $\sim$ : asymptotic equality, : integer and : real
Permalink:: http://dlmf.nist.gov/32.11.E32
Encodings:: TeX, pMML, png

Alternatively, if $\nu$ is not zero or a positive integer, then

32.11.33 $w_{h}(x)=-\tfrac{2}{3}x+\tfrac{4}{3}d\sqrt{3}\mathop{\sin\/}\nolimits\!\left(% \phi(x)-\theta_{0}\right)+\mathop{O\/}\nolimits\!\left(x^{{-1}}\right),$ $x\to-\infty$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\sin\/}\nolimits z$ : sine function, : real, $\phi(z)$ : function, : constant and $\theta_{0}$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E33
Encodings:: TeX, pMML, png

where

32.11.34 $\phi(x)=\tfrac{1}{3}\sqrt{3}x^{2}-\tfrac{4}{3}d^{2}\sqrt{3}\mathop{\ln\/}% \nolimits\!\left(\sqrt{2}|x|\right),$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real, $\phi(z)$ : function and : constant
Permalink:: http://dlmf.nist.gov/32.11.E34
Encodings:: TeX, pMML, png

and and $\theta_{0}$ are real constants. Connection formulas for and $\theta_{0}$ are given by

32.11.35 $d^{2}=-\tfrac{1}{4}\sqrt{3}\pi^{{-1}}\mathop{\ln\/}\nolimits\!\left(1-|\mu|^{2% }\right),$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : constant and $\mu$
Permalink:: http://dlmf.nist.gov/32.11.E35
Encodings:: TeX, pMML, png

32.11.36 $\theta_{0}=\tfrac{1}{3}d^{2}\sqrt{3}\mathop{\ln\/}\nolimits 3+\tfrac{2}{3}\pi% \nu+\tfrac{7}{12}\pi+\mathop{\mathrm{ph}\/}\nolimits\mu+\mathop{\mathrm{ph}\/}% \nolimits\mathop{\Gamma\/}\nolimits\!\left(-\tfrac{2}{3}i\sqrt{3}d^{2}\right),$