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

§32.11 Asymptotic Approximations for Real Variables

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Keywords:: Painlevé transcendents, asymptotic approximations, real variables
Permalink:: http://dlmf.nist.gov/32.11
See also:: Annotations for Ch.32

§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

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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
See also:: Annotations for §32.11 and Ch.32

There are solutions of (32.2.1) such that

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

ⓘ

Symbols:: $o\left(\NVar{x}\right)$ : order less than, $\sin\NVar{z}$ : sine function, : real, $\phi(z)$ : function, : constant and $\theta_{0}$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(i), §32.11 and Ch.32

where

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

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Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, : real, $\phi(z)$ : function and : constant
Referenced by:: §32.11(i)
Permalink:: http://dlmf.nist.gov/32.11.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(i), §32.11 and Ch.32

and and $\theta_{0}$ are constants.

There are also solutions of (32.2.1) such that

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

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Symbols:: $\sim$ : asymptotic equality and : real
Permalink:: http://dlmf.nist.gov/32.11.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(i), §32.11 and Ch.32

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

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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
See also:: Annotations for §32.11 and Ch.32

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

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

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Symbols:: : real
Referenced by:: §32.11(ii), §32.11(iii)
Permalink:: http://dlmf.nist.gov/32.11.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(ii), §32.11 and Ch.32

with boundary condition

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

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Symbols:: : real
Referenced by:: §32.11(ii)
Permalink:: http://dlmf.nist.gov/32.11.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(ii), §32.11 and Ch.32

Any nontrivial real solution of (32.11.4) that satisfies (32.11.5) is asymptotic to $k\operatorname{Ai}\left(x\right)$ , for some nonzero real , where $\operatorname{Ai}$ 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 $k\operatorname{Ai}\left(x\right)$ as $x\to+\infty$ .

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

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

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Symbols:: $o\left(\NVar{x}\right)$ : order less than, $\sin\NVar{z}$ : sine function, : real, : real, $\phi(z)$ : function, : constant and $\theta_{0}$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(ii), §32.11 and Ch.32

where

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

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Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, : real, $\phi(z)$ : function and : constant
Permalink:: http://dlmf.nist.gov/32.11.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(ii), §32.11 and Ch.32

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

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, : real and : constant
Permalink:: http://dlmf.nist.gov/32.11.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(ii), §32.11 and Ch.32

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

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $\operatorname{sign}\NVar{x}$ : sign of, : real, : constant and $\theta_{0}$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(ii), §32.11 and Ch.32

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

If , then

32.11.10 $w_{k}(x)\sim\operatorname{sign}\left(k\right)\sqrt{\tfrac{1}{2}|x|},$ $x\to-\infty$ .

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Symbols:: $\sim$ : asymptotic equality, $\operatorname{sign}\NVar{x}$ : sign of, : real and : real
Permalink:: http://dlmf.nist.gov/32.11.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(ii), §32.11 and Ch.32

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

32.11.11 $w_{k}(x)\sim\operatorname{sign}\left(k\right)(x-c_{0})^{-1},$ $x\to c_{0}+$ .

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Symbols:: $\sim$ : asymptotic equality, $\operatorname{sign}\NVar{x}$ : sign of, : real, : real and $c_{0}$ : pole
Permalink:: http://dlmf.nist.gov/32.11.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(ii), §32.11 and Ch.32

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

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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
See also:: Annotations for §32.11 and Ch.32

Replacement of by in (32.11.4) gives

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

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Symbols:: : real
Referenced by:: §32.11(iii)
Permalink:: http://dlmf.nist.gov/32.11.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(iii), §32.11 and Ch.32

Any nontrivial real solution of (32.11.12) satisfies

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, : real, $\chi$ : constant, $\phi(z)$ : function and : constant
Permalink:: http://dlmf.nist.gov/32.11.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(iii), §32.11 and Ch.32

where

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

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Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, : real, $\phi(z)$ : function and : constant
Permalink:: http://dlmf.nist.gov/32.11.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(iii), §32.11 and Ch.32

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

In the case when

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

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, : integer, $\chi$ : constant and : constant
Permalink:: http://dlmf.nist.gov/32.11.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(iii), §32.11 and Ch.32

with $n\in\mathbb{Z}$ , we have

32.11.16 $w(x)\sim k\operatorname{Ai}\left(x\right),$ $x\to+\infty$ ,

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\sim$ : asymptotic equality, : real and : real
Permalink:: http://dlmf.nist.gov/32.11.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(iii), §32.11 and Ch.32

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

32.11.17 $d^{2}=\pi^{-1}\ln\left(1+k^{2}\right),$ $\operatorname{sign}\left(k\right)=(-1)^{n}$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{sign}\NVar{x}$ : sign of, : integer, : real and : constant
Permalink:: http://dlmf.nist.gov/32.11.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(iii), §32.11 and Ch.32

In the generic case

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

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, : integer, $\chi$ : constant and : constant
Permalink:: http://dlmf.nist.gov/32.11.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(iii), §32.11 and Ch.32

we have

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\cos\NVar{z}$ : 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
See also:: Annotations for §32.11(iii), §32.11 and Ch.32

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

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

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Symbols:: $\ln\NVar{z}$ : 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
See also:: Annotations for §32.11(iii), §32.11 and Ch.32

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

32.11.21 $\sigma=-\operatorname{sign}\left(\Im s\right),$

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Symbols:: $\Im$ : imaginary part, $\operatorname{sign}\NVar{x}$ : sign of, $\sigma$ : real constant and
Permalink:: http://dlmf.nist.gov/32.11.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(iii), §32.11 and Ch.32

32.11.22 $\rho^{2}=\pi^{-1}\ln\left((1+|s|^{2})/|2\Im s|\right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\Im$ : imaginary part, $\ln\NVar{z}$ : principal branch of logarithm function, $\rho$ : real constant and
Permalink:: http://dlmf.nist.gov/32.11.E22
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(iii), §32.11 and Ch.32

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

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $\rho$ : real constant, $\theta$ : real constant and
Permalink:: http://dlmf.nist.gov/32.11.E23
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(iii), §32.11 and Ch.32

where

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

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Defines:: (locally)
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $\chi$ : constant and : constant
Permalink:: http://dlmf.nist.gov/32.11.E24
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(iii), §32.11 and Ch.32

§32.11(iv) Third Painlevé Equation

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Notes:: See McCoy et al. (1977), Fokas et al. (2006, Chapter 14).
Permalink:: http://dlmf.nist.gov/32.11.iv
See also:: Annotations for §32.11 and Ch.32

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

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

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : asymptotic equality, $\mathrm{e}$ : base of natural logarithm, : real and $\nu$ : parameter
Referenced by:: §32.11(iv)
Permalink:: http://dlmf.nist.gov/32.11.E25
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(iv), §32.11 and Ch.32

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

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

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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
See also:: Annotations for §32.11(iv), §32.11 and Ch.32

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

32.11.27 $\sigma=(2/\pi)\operatorname{arcsin}\left(\pi\lambda\right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arcsin}\NVar{z}$ : arcsine function and $\sigma$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E27
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(iv), §32.11 and Ch.32

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

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\nu$ : parameter, : constant and $\sigma$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E28
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(iv), §32.11 and Ch.32

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

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Keywords:: Painlevé transcendents, asymptotic approximations, real variables
Notes:: See Bassom et al. (1992), Its and Kapaev (1998).
Referenced by:: §32.13(ii), §32.3(iii), Erratum (V1.0.7) for Other
Permalink:: http://dlmf.nist.gov/32.11.v
Correction (effective with 1.0.7):: The reference Wong and Zhang (2009a), formerly incorrectly placed in §32.13(ii), has been moved to the end of this subsection.
See also:: Annotations for §32.11 and Ch.32

Consider $\mbox{P}_{\mbox{\scriptsize IV}}$ with $\alpha=2\nu+1$ $(\in\mathbb{R})$ 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
See also:: Annotations for §32.11(v), §32.11 and Ch.32

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
See also:: Annotations for §32.11(v), §32.11 and Ch.32

Any nontrivial solution of (32.11.29) that satisfies (32.11.30) is asymptotic to $h{U}^{2}\left(-\nu-\frac{1}{2},\sqrt{2}x\right)$ 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 $h{U}^{2}\left(-\nu-\frac{1}{2},\sqrt{2}x\right)$ as $x\to+\infty$ . Here denotes the parabolic cylinder function (§12.2).

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

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

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter and $\nu$ : parameter
Permalink:: http://dlmf.nist.gov/32.11.E31
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(v), §32.11 and Ch.32

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

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

ⓘ

Symbols:: $\sim$ : asymptotic equality, $\exp\NVar{z}$ : exponential function, : integer and : real
Permalink:: http://dlmf.nist.gov/32.11.E32
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(v), §32.11 and Ch.32

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

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

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\sin\NVar{z}$ : sine function, : real, $\phi(z)$ : function, : constant and $\theta_{0}$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E33
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(v), §32.11 and Ch.32

where

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

ⓘ

Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, : real, $\phi(z)$ : function and : constant
Permalink:: http://dlmf.nist.gov/32.11.E34
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(v), §32.11 and Ch.32

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

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

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, : constant and $\mu$
Permalink:: http://dlmf.nist.gov/32.11.E35
Encodings:: TeX, pMML, png
See also:: Annotations for §32.11(v), §32.11 and Ch.32

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

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