4 Elementary Functions Logarithm, Exponential, Powers 4.12 Generalized Logarithms and Exponentials 4.14 Definitions and Periodicity

§4.13 Lambert $\mathop{W\/}\nolimits$ -Function

Notes:: To verify the radius of convergence of the series (4.13.6) map the plane of $\mathop{W\/}\nolimits$ onto the plane of via $t=(-2v)^{{1/2}}$ , where $v=\mathop{W\/}\nolimits+\mathop{\ln\/}\nolimits\mathop{W\/}\nolimits+1-i\pi$ . Then $\mathop{W\/}\nolimits$ is analytic at , and its nearest singularities to the origin are located at $t=2\sqrt{\pi}e^{{\pm\pi i/4}}$ . Figure 4.13.1 was produced at NIST.
Defines:: $\mathop{\mathrm{Wp}\/}\nolimits\!\left(x\right)$ : principal branch of Lambert $\mathop{W\/}\nolimits$ -function and $\mathop{\mathrm{Wm}\/}\nolimits\!\left(x\right)$ : nonprincipal branch of Lambert $\mathop{W\/}\nolimits$ -function
Keywords:: Lambert $\mathop{W\/}\nolimits$ -function
Referenced by:: §26.7(iv)
Permalink:: http://dlmf.nist.gov/4.13

The Lambert $\mathop{W\/}\nolimits$ -function $\mathop{W\/}\nolimits\!\left(x\right)$ is the solution of the equation

4.13.1 $We^{W}=x.$

Defines:: $\mathop{W\/}\nolimits\!\left(x\right)$ : Lambert $\mathop{W\/}\nolimits$ -function
Symbols:: : base of exponential function and : real variable
Permalink:: http://dlmf.nist.gov/4.13.E1
Encodings:: TeX, pMML, png

On the -interval $[0,\infty)$ there is one real solution, and it is nonnegative and increasing. On the -interval there are two real solutions, one increasing and the other decreasing. We call the solution for which $\mathop{W\/}\nolimits\!\left(x\right)\geq\mathop{W\/}\nolimits\!\left(-1/e\right)$ the principal branch and denote it by $\mathop{\mathrm{Wp}\/}\nolimits\!\left(x\right)$ . The other solution is denoted by $\mathop{\mathrm{Wm}\/}\nolimits\!\left(x\right)$ . See Figure 4.13.1.

Figure 4.13.1: Branches $\mathop{\mathrm{Wp}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathrm{Wm}\/}\nolimits\!\left(x\right)$ of the Lambert $\mathop{W\/}\nolimits$ -function.

and

denote the points

and

, respectively, on the

-axis.

Symbols:: $\mathop{W\/}\nolimits\!\left(x\right)$ : Lambert $\mathop{W\/}\nolimits$ -function, $\mathop{\mathrm{Wm}\/}\nolimits\!\left(x\right)$ : nonprincipal branch of Lambert $\mathop{W\/}\nolimits$ -function, $\mathop{\mathrm{Wp}\/}\nolimits\!\left(x\right)$ : principal branch of Lambert $\mathop{W\/}\nolimits$ -function, : base of exponential function and : real variable
Keywords:: Lambert $\mathop{W\/}\nolimits$ -function
Referenced by:: §4.13, §4.13
Permalink:: http://dlmf.nist.gov/4.13.F1
Encodings:: pdf, png

Properties include:

4.13.2

$\mathop{\mathrm{Wp}\/}\nolimits\!\left(-1/e\right)=\mathop{\mathrm{Wm}\/}% \nolimits\!\left(-1/e\right)=-1,$

$\mathop{\mathrm{Wp}\/}\nolimits\!\left(0\right)=0,$

$\mathop{\mathrm{Wp}\/}\nolimits\!\left(e\right)=1.$

Symbols:: $\mathop{\mathrm{Wm}\/}\nolimits\!\left(x\right)$ : nonprincipal branch of Lambert $\mathop{W\/}\nolimits$ -function, $\mathop{\mathrm{Wp}\/}\nolimits\!\left(x\right)$ : principal branch of Lambert $\mathop{W\/}\nolimits$ -function and : base of exponential function
Permalink:: http://dlmf.nist.gov/4.13.E2
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

4.13.3

$U+\mathop{\ln\/}\nolimits U=x,$

$U=U(x)=\mathop{W\/}\nolimits\!\left(e^{x}\right).$

Symbols:: $\mathop{W\/}\nolimits\!\left(x\right)$ : Lambert $\mathop{W\/}\nolimits$ -function, : base of exponential function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : real variable
Permalink:: http://dlmf.nist.gov/4.13.E3
Encodings:: TeX, TeX, pMML, pMML, png, png

4.13.4 $\frac{d\mathop{W\/}\nolimits}{dx}=\frac{e^{{-\mathop{W\/}\nolimits}}}{1+% \mathop{W\/}\nolimits},$ $x\neq-\dfrac{1}{e}$ .

Symbols:: $\mathop{W\/}\nolimits\!\left(x\right)$ : Lambert $\mathop{W\/}\nolimits$ -function, $\frac{df}{dx}$ : derivative of with respect to , : base of exponential function and : real variable
Referenced by:: §4.45(iii)
Permalink:: http://dlmf.nist.gov/4.13.E4
Encodings:: TeX, pMML, png

4.13.5 $\mathop{\mathrm{Wp}\/}\nolimits\!\left(x\right)=\sum_{{n=1}}^{\infty}(-1)^{{n-% 1}}\frac{n^{{n-2}}}{(n-1)!}x^{n},$ $|x|<\dfrac{1}{e}$ .

Symbols:: $\mathop{\mathrm{Wp}\/}\nolimits\!\left(x\right)$ : principal branch of Lambert $\mathop{W\/}\nolimits$ -function, : base of exponential function, : : factorial, : integer and : real variable
Permalink:: http://dlmf.nist.gov/4.13.E5
Encodings:: TeX, pMML, png

4.13.6 $\mathop{W\/}\nolimits\!\left(-e^{{-1-(t^{2}/2)}}\right)=\sum_{{n=0}}^{\infty}(% -1)^{{n-1}}c_{n}t^{n},$ $|t|<2\sqrt{\pi}$ ,

Symbols:: $\mathop{W\/}\nolimits\!\left(x\right)$ : Lambert $\mathop{W\/}\nolimits$ -function, : base of exponential function, : integer and $c_{i}$ : coefficients
Referenced by:: §4.13, §4.45(iii)
Permalink:: http://dlmf.nist.gov/4.13.E6
Encodings:: TeX, pMML, png

where $t\geq 0$ for $\mathop{\mathrm{Wp}\/}\nolimits$ , $t\leq 0$ for $\mathop{\mathrm{Wm}\/}\nolimits$ ,

4.13.7 $c_{0}=1,c_{1}=1,c_{2}=\tfrac{1}{3},c_{3}=\tfrac{1}{36},c_{4}=-\tfrac{1}{270},$

Symbols:: $c_{i}$ : coefficients
Permalink:: http://dlmf.nist.gov/4.13.E7
Encodings:: TeX, pMML, png

4.13.8 $c_{n}=\frac{1}{n+1}\left(c_{{n-1}}-\sum_{{k=2}}^{{n-1}}kc_{k}c_{{n+1-k}}\right),$ $n\geq 2$ ,

Symbols:: : integer, : integer and $c_{i}$ : coefficients
Permalink:: http://dlmf.nist.gov/4.13.E8
Encodings:: TeX, pMML, png

and

4.13.9 $1\cdot 3\cdot 5\cdots(2n+1)c_{{2n+1}}=g_{n},$

Symbols:: : integer, $c_{i}$ : coefficients and $g_{n}$ : Unknown defined in GA
Permalink:: http://dlmf.nist.gov/4.13.E9
Encodings:: TeX, pMML, png

where $g_{n}$ is defined in §5.11(i).

As $x\to+\infty$

4.13.10 $\mathop{\mathrm{Wp}\/}\nolimits\!\left(x\right)=\xi-\mathop{\ln\/}\nolimits\xi% +\frac{\mathop{\ln\/}\nolimits\xi}{\xi}+\frac{(\mathop{\ln\/}\nolimits\xi)^{2}% }{2\xi^{2}}-\frac{\mathop{\ln\/}\nolimits\xi}{\xi^{2}}+\mathop{O\/}\nolimits\!% \left(\frac{(\mathop{\ln\/}\nolimits\xi)^{3}}{\xi^{3}}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{Wp}\/}\nolimits\!\left(x\right)$ : principal branch of Lambert $\mathop{W\/}\nolimits$ -function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : real variable
Referenced by:: §4.45(iii)
Permalink:: http://dlmf.nist.gov/4.13.E10
Encodings:: TeX, pMML, png

where $\xi=\mathop{\ln\/}\nolimits x$ . As $x\to 0-$

4.13.11 $\mathop{\mathrm{Wm}\/}\nolimits\!\left(x\right)=-\eta-\mathop{\ln\/}\nolimits% \eta-\frac{\mathop{\ln\/}\nolimits\eta}{\eta}-\frac{(\mathop{\ln\/}\nolimits% \eta)^{2}}{2\eta^{2}}-\frac{\mathop{\ln\/}\nolimits\eta}{\eta^{2}}+\mathop{O\/% }\nolimits\!\left(\frac{(\mathop{\ln\/}\nolimits\eta)^{3}}{\eta^{3}}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{Wm}\/}\nolimits\!\left(x\right)$ : nonprincipal branch of Lambert $\mathop{W\/}\nolimits$ -function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real variable and $\eta$
Permalink:: http://dlmf.nist.gov/4.13.E11
Encodings:: TeX, pMML, png

where $\eta=\mathop{\ln\/}\nolimits\!\left(-1/x\right)$ .

For the foregoing results and further information see Borwein and Corless (1999), Corless et al. (1996), de Bruijn (1961, pp. 25–28), Olver (1997b, pp. 12–13), and Siewert and Burniston (1973).

For integral representations of all branches of the Lambert $\mathop{W\/}\nolimits$ -function see Kheyfits (2004).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 4.12 Generalized Logarithms and Exponentials 4.14 Definitions and Periodicity

§4.13 Lambert -Function

§4.13 Lambert $\mathop{W\/}\nolimits$ -Function