朗伯W函数

$W_{0}(x)$ 的图像， $-{\frac {1}{e}}\leq x\leq 4$

朗伯W函数（英語：Lambert W function，又称为欧米加函数或乘积对数），是 $f(w)=we^{w}$ 的反函数，其中 $e^{w}$ 是指数函数， $w$ 是任意复数。对于任何复数 $z$ ，都有：

z=W(z)e^{W(z)}.

由于函数 $f$ 不是单射，因此函数 $W$ 是多值的（除了0以外）。如果我们把 $x$ 限制为实数，并要求 $w$ 是实数，那么函数仅对于 $x\geq {\frac {1}{e}}$ 有定义，在 $(-{\frac {1}{e}},0)$ 内是多值的；如果加上 $w\geq -1$ 的限制，则定义了一个单值函数 $W_{0}(x)$ （见图）。我们有 $W_{0}(0)=0$ ， $W_{0}(-{\frac {1}{e}})=-1$ 。而在 $[-{\frac {1}{e}},0)$ 内的 $w\leq -1$ 分支，则记为 $W_{-1}(x)$ ，从 $W_{-1}(-{\frac {1}{e}})=-1$ 递减为 $W_{-1}(0^{-})=-\infty$ 。

朗伯 $W$ 函数不能用初等函数来表示。它在组合数学中有许多用途，例如树的计算。它可以用来解许多含有指数的方程，也出现在某些微分方程的解中，例如 $y(t)=ay(t-1)$ 。

微分和积分

朗伯 $W\,$ 函数的积分形式为

W(x)={\frac {x}{\pi }}\int _{0}^{\pi }{\frac {\left(1-v\cot v\right)^{2}+v^{2}}{x+v\csc v\cdot e^{-v\cot v}}}{\rm {d}}v,|\arg \left(x\right)|<\pi \,

W(x)=\int _{-\infty }^{-{\frac {1}{e}}}{-{\frac {1}{\pi }}}\Im \left[{\frac {\rm {d}}{{\rm {d}}x}}W(x)\right]\ln \left(1-{\frac {z}{x}}\right){\rm {d}}x\,

若 $x\not \in \left[-{\frac {1}{e}},0\right],k\in {\mathbb {Z} }\,$ ,若 $x\in \left(-{\frac {1}{e}},0\right),k=1,\pm 2,\pm 3,...\,$

W_{k}(x)=1+\left(\ln x-1+2k\pi {\rm {i}}\right)e^{{\frac {\rm {i}}{2\pi }}\int _{0}^{\infty }\ln {\frac {t-\ln t+\ln x+(2k+1)\pi {\rm {i}}}{t-\ln t+\ln x+(2k-1)\pi {\rm {i}}}}\cdot {\frac {{\rm {d}}t}{t+1}}}=1+\left(\ln x-1+2k\pi {\rm {i}}\right)e^{{\frac {\rm {i}}{2\pi }}\int _{0}^{\infty }\ln {\frac {\left(t-\ln t+\ln x\right)^{2}+\left(4k^{2}-1\right)\pi ^{2}+2\pi \left(t-\ln t+\ln x\right){\rm {i}}}{\left(t-\ln t+\ln x\right)^{2}+\left(2k-1\right)^{2}\pi ^{2}}}\cdot {\frac {{\rm {d}}t}{t+1}}}\,

把被积函数的实部和虚部分离出来：

W_{k}(x)=1+\left(\ln x-1+2k\pi {\rm {i}}\right)e^{{\frac {\rm {i}}{2\pi }}\int _{0}^{\infty }\left[{\frac {1}{2}}\ln {\frac {\left(t-\ln t+\ln x\right)^{2}+\left(2k+1\right)^{2}\pi ^{2}}{\left(t-\ln t+\ln x\right)^{2}+\left(2k-1\right)^{2}\pi ^{2}}}+{\rm {i}}\arctan {\frac {2\pi \left(t-\ln t+\ln x\right)}{\left(t-\ln t+\ln x\right)^{2}+\left(4k^{2}-1\right)\pi ^{2}}}\right]\cdot {\frac {{\rm {d}}t}{t+1}}}

{}_{W_{k}(x)=1+{\frac {\left(\ln x-1\right)\cos {\frac {1}{4\pi }}\int _{0}^{\infty }\ln {\frac {\left(t-\ln t+\ln x\right)^{2}+\left(2k+1\right)^{2}\pi ^{2}}{\left(t-\ln t+\ln x\right)^{2}+\left(2k-1\right)^{2}\pi ^{2}}}\cdot {\frac {{\rm {d}}t}{t+1}}-2k\pi \sin {\frac {1}{4\pi }}\int _{0}^{\infty }\ln {\frac {\left(t-\ln t+\ln x\right)^{2}+\left(2k+1\right)^{2}\pi ^{2}}{\left(t-\ln t+\ln x\right)^{2}+\left(2k-1\right)^{2}\pi ^{2}}}\cdot {\frac {{\rm {d}}t}{t+1}}+{\rm {i}}\left[\left(\ln x-1\right)\sin {\frac {1}{4\pi }}\int _{0}^{\infty }\ln {\frac {\left(t-\ln t+\ln x\right)^{2}+\left(2k+1\right)^{2}\pi ^{2}}{\left(t-\ln t+\ln x\right)^{2}+\left(2k-1\right)^{2}\pi ^{2}}}\cdot {\frac {{\rm {d}}t}{t+1}}+2k\pi \cos {\frac {1}{4\pi }}\int _{0}^{\infty }\ln {\frac {\left(t-\ln t+\ln x\right)^{2}+\left(2k+1\right)^{2}\pi ^{2}}{\left(t-\ln t+\ln x\right)^{2}+\left(2k-1\right)^{2}\pi ^{2}}}\cdot {\frac {{\rm {d}}t}{t+1}}\right]}{e^{{\frac {1}{2\pi }}\int _{0}^{\infty }\arctan {\frac {2\pi \left(t-\ln t+\ln x\right)}{\left(t-\ln t+\ln x\right)^{2}+\left(4k^{2}-1\right)\pi ^{2}}}\cdot {\frac {\rm {{d}t}}{t+1}}}}}}

设 $W_{k}(x)=u+v{\rm {i}},x=t+s{\rm {i}}$ ，则有 $\left(u+v{\rm {i}}\right)e^{u+v{\rm {i}}}=t+s{\rm {i}}$ ，展开分离出实部和虚部，

$e^{u}\left(u\cos v-v\sin v\right)=t,e^{u}\left(u\sin v+v\cos v\right)=s$ ,当 $s=0$ 时，易知 $u=-v\cot v$

{}_{W_{k}(x)={\frac {\left(1-\ln x\right)\sin {\frac {1}{4\pi }}\int _{0}^{\infty }\ln {\frac {\left(t-\ln t+\ln x\right)^{2}+\left(2k+1\right)^{2}\pi ^{2}}{\left(t-\ln t+\ln x\right)^{2}+\left(2k-1\right)^{2}\pi ^{2}}}\cdot {\frac {{\rm {d}}t}{t+1}}-2k\pi \cos {\frac {1}{4\pi }}\int _{0}^{\infty }\ln {\frac {\left(t-\ln t+\ln x\right)^{2}+\left(2k+1\right)^{2}\pi ^{2}}{\left(t-\ln t+\ln x\right)^{2}+\left(2k-1\right)^{2}\pi ^{2}}}\cdot {\frac {{\rm {d}}t}{t+1}}}{e^{{\frac {1}{2\pi }}\int _{0}^{\infty }\arctan {\frac {2\pi \left(t-\ln t+\ln x\right)}{\left(t-\ln t+\ln x\right)^{2}+\left(4k^{2}-1\right)\pi ^{2}}}\cdot {\frac {\rm {{d}t}}{t+1}}}}}\cot {\frac {\left(\ln x-1\right)\sin {\frac {1}{4\pi }}\int _{0}^{\infty }\ln {\frac {\left(t-\ln t+\ln x\right)^{2}+\left(2k+1\right)^{2}\pi ^{2}}{\left(t-\ln t+\ln x\right)^{2}+\left(2k-1\right)^{2}\pi ^{2}}}\cdot {\frac {{\rm {d}}t}{t+1}}+2k\pi \cos {\frac {1}{4\pi }}\int _{0}^{\infty }\ln {\frac {\left(t-\ln t+\ln x\right)^{2}+\left(2k+1\right)^{2}\pi ^{2}}{\left(t-\ln t+\ln x\right)^{2}+\left(2k-1\right)^{2}\pi ^{2}}}\cdot {\frac {{\rm {d}}t}{t+1}}}{e^{{\frac {1}{2\pi }}\int _{0}^{\infty }\arctan {\frac {2\pi \left(t-\ln t+\ln x\right)}{\left(t-\ln t+\ln x\right)^{2}+\left(4k^{2}-1\right)\pi ^{2}}}\cdot {\frac {\rm {{d}t}}{t+1}}}}}+{\frac {\left(\ln x-1\right)\sin {\frac {1}{4\pi }}\int _{0}^{\infty }\ln {\frac {\left(t-\ln t+\ln x\right)^{2}+\left(2k+1\right)^{2}\pi ^{2}}{\left(t-\ln t+\ln x\right)^{2}+\left(2k-1\right)^{2}\pi ^{2}}}\cdot {\frac {{\rm {d}}t}{t+1}}+2k\pi \cos {\frac {1}{4\pi }}\int _{0}^{\infty }\ln {\frac {\left(t-\ln t+\ln x\right)^{2}+\left(2k+1\right)^{2}\pi ^{2}}{\left(t-\ln t+\ln x\right)^{2}+\left(2k-1\right)^{2}\pi ^{2}}}\cdot {\frac {{\rm {d}}t}{t+1}}}{e^{{\frac {1}{2\pi }}\int _{0}^{\infty }\arctan {\frac {2\pi \left(t-\ln t+\ln x\right)}{\left(t-\ln t+\ln x\right)^{2}+\left(4k^{2}-1\right)\pi ^{2}}}\cdot {\frac {\rm {{d}t}}{t+1}}}}}{\rm {i}},}

W_{0}(x)=1+\left(\ln x-1\right)e^{-{\frac {1}{\pi }}\int _{0}^{\infty }\arg \left(t-\ln t+\ln x+\pi {\rm {i}}\right)\cdot {\frac {\rm {{d}t}}{t+1}}},x>0

若 $x>{\frac {1}{e}}$ ，上式还可化为 $W_{0}(x)=1+\left(\ln x-1\right)e^{-{\frac {1}{\pi }}\int _{0}^{\infty }\arctan {\frac {\pi }{t-\ln t+\ln x}}\cdot {\frac {\rm {{d}t}}{t+1}}}$

由隐函数的求导法则，朗伯 $W\,$ 函数满足以下的微分方程：

z\left[1+W(z)\right]{\frac {\rm {d}}{{\rm {d}}z}}W(z)=W(z)

，

z\neq -{\frac {1}{e}}\,,

因此：

{\frac {\rm {d}}{{\rm {d}}z}}W(z)={\frac {W(z)}{z\left[1+W(z)\right]}}

，

z\neq -{\frac {1}{e}}\,.

函数 $W(x)\,$ ，以及许多含有 $W(x)\,$ 的表达式，都可以用 $w=W(x)\,$ 的变量代换来积分，也就是说 $x=we^{w}\,$

\int W(x){\rm {d}}x=x\left[W(x)+{\frac {1}{W(x)}}-1\right]+C

\int _{0}^{1}W(x){\rm {d}}x=\Omega +{\frac {1}{\Omega }}-2\approx 0.330366

其中 $\Omega$ 為欧米加常数。

性质

$1\,$ 、 $z^{z^{z^{z^{z^{.^{.^{.}}}}}}}=\lim _{n\to \infty }(z\upuparrows n)=-{\frac {W(-\ln z)}{\ln z}}$ ，

其中 $\upuparrows$ 是高德納箭號表示法。

$2\,$ 、若 $z>0\,$ ，则 $\ln W(z)=\ln z-W(z)\,$

泰勒级数

$W_{0}\,$ 在 $x=0\,$ 的泰勒级数如下：

W_{0}(x)=\sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}\ x^{n}=x-x^{2}+{\frac {3}{2}}x^{3}-{\frac {8}{3}}x^{4}+{\frac {125}{24}}x^{5}-\cdots

收敛半径为 ${\frac {1}{e}}\,$ 。

加法定理

W(x)+W(y)=W\left[{\frac {xy}{W(x)}}+{\frac {xy}{W(y)}}\right]\,

x>0,y>0\,

複數值

實部

\Re \left[W(x+y{\rm {i}})\right]=\sum _{k=1}^{\infty }{\frac {(-k)^{k-1}}{k!}}{\sqrt {(x^{2}+y^{2})^{k}}}\cos \left(k\arctan {\frac {x}{y}}\right)\,

,

x^{2}+y^{2}<{\frac {1}{e^{2}}}\,

虛部

\Im \left[W(x+y{\rm {i}})\right]=\sum _{k=1}^{\infty }{\frac {(-k)^{k-1}}{k!}}{\sqrt {(x^{2}+y^{2})^{k}}}\sin \left(k\arctan {\frac {x}{y}}\right)\,

,

x^{2}+y^{2}<{\frac {1}{e^{2}}}\,

模長

|W(x+y{\rm {i}})|=W({\sqrt {x+y}})\,

模角

\arg \left[W(x+y{\rm {i}})\right]=\sum _{k=1}^{\infty }{\frac {(-k)^{k-1}}{k!}}\arctan \left[\cot(k\arctan {\frac {x}{y}})\right]\,

,

x^{2}+y^{2}<{\frac {1}{e^{2}}}\,

共軛值

{\overline {W(x+y{\rm {i}})}}=\sum _{k=1}^{\infty }{\frac {(-k)^{k-1}}{k!}}{\sqrt {(x^{2}+y^{2})^{k}}}\left[\cos \left(k\arctan {\frac {x}{y}}\right)-{\rm {i}}\sin \left(k\arctan {\frac {x}{y}}\right)\right]\,

,

x^{2}+y^{2}<{\frac {1}{e^{2}}}\,

特殊值

W\left(-{\frac {\pi }{2}}\right)={\frac {\pi }{2}}i

W\left(-{\frac {\ln 2}{2}}\right)=-\ln 2

W\left(-{1 \over e}\right)=-1

W\left(1\right)=\Omega ={\frac {1}{\int _{-\infty }^{\infty }{\frac {{\rm {d}}x}{(e^{x}-x)^{2}+\pi ^{2}}}}}-1\approx 0.56714329\dots \,

（欧米加常数）

W(e)=1\,

W(e^{e+1})=e\,

W\left({\frac {1}{e^{1-{\frac {1}{e}}}}}\right)={\frac {1}{e}}

W({\pi }e^{\pi })=\pi

W(k{\ln k})={\ln k}

(k>0)

W({\rm {i}}\pi )=-{\rm {i}}\pi

W(-{\rm {i}}\pi )={\rm {i}}\pi

W({\rm {i}}\cos 1-\sin 1)={\rm {i}}

W(-{\frac {3}{2}}{\pi })=-{\frac {3}{2}}{\pi }{\rm {i}}

W(-{\frac {\sqrt[{7}]{8}}{7}}{\ln 2})=-{\frac {32}{7}}{\ln 2}

W(-{\frac {\sqrt {3}}{54}}{\ln 3})=-{\frac {9}{2}}{\ln 3}

W(-{\frac {\ln 2}{4}})=-4{\ln 2}

W\left(-1\right)={\frac {e^{{\frac {1}{2\pi }}\int _{0}^{\infty }{1 \over t+1}\arctan {2\pi  \over t-\ln t}{\rm {d}}t}-\cos \left[{\frac {1}{4\pi }}\int _{0}^{\infty }{1 \over t+1}\ln {\left(t-\ln t\right)^{2} \over 4\pi ^{2}+\left(t-\ln t\right)^{2}}{\rm {d}}t\right]+\pi \sin \left[{\frac {1}{4\pi }}\int _{0}^{\infty }{1 \over t+1}\ln {\left(t-\ln t\right)^{2} \over 4\pi ^{2}+\left(t-\ln t\right)^{2}}{\rm {d}}t\right]-{\rm {i}}\left\{\pi \cos \left[{\frac {1}{4\pi }}\int _{0}^{\infty }{1 \over t+1}\ln {\left(t-\ln t\right)^{2} \over 4\pi ^{2}+\left(t-\ln t\right)^{2}}{\rm {d}}t\right]+\sin \left[{\frac {1}{4\pi }}\int _{0}^{\infty }{1 \over t+1}\ln {\left(t-\ln t\right)^{2} \over 4\pi ^{2}+\left(t-\ln t\right)^{2}}{\rm {d}}t\right]\right\}}{e^{{\frac {1}{2\pi }}\int _{0}^{\infty }{1 \over t+1}\arctan {2\pi  \over t-\ln t}{\rm {d}}t}}}\approx -0.31813-1.33723{\rm {i}}

W(-{\frac {\ln k}{k}})=-\ln k

W\left[-{\frac {\ln(x+1)}{x(x+1)^{\frac {1}{x}}}}\right]=-{\frac {x+1}{x}}\ln(x+1)>,-1<x<0

应用

许多含有指数的方程都可以用 $W\,$ 函数来解出。一般的方法是把未知数都移到方程的一侧，并设法化为 $Y=Xe^{X}\,$ 的形式。

例子

例子1

2^{t}=5t\,

\Rightarrow 1={\frac {5t}{2^{t}}}\,

\Rightarrow 1=5t\,e^{-t\ln 2}\,

\Rightarrow {\frac {1}{5}}=t\,e^{-t\ln 2}\,

\Rightarrow -{\frac {\ln 2}{5}}=(-\,t\,\ln 2)\,e^{-t\ln 2}\,

\Rightarrow -t\ln 2=W_{k}\left(-{\frac {\ln 2}{5}}\right)\,

\Rightarrow t=-{\frac {W_{k}\left(-{\frac {\ln 2}{5}}\right)}{\ln 2}}\,

更一般地，以下的方程

Q^{ax+b}=cx+d\,

其中

Q>0\land Q\neq 1\land c\neq 0

两边同乘: ${\frac {a}{c}}$ ，

得到： ${\frac {a}{c}}Q^{ax+b}=ax+{\frac {ad}{c}}\,$

同除以： $Q^{ax}\,$ ，

得到： ${\frac {a}{c}}Q^{b}=\left(ax+{\frac {ad}{c}}\right)Q^{-ax}\,$

同除： $Q^{\frac {ad}{c}}\,$ ，

${\frac {a}{c}}Q^{b-{\frac {ad}{c}}}=\left(ax+{\frac {ad}{c}}\right)Q^{-\left(ax+{\frac {ad}{c}}\right)}\,$

可以用变量代换

令 $t=ax+{\frac {ad}{c}}$

化为

tQ^{-t}={\frac {a}{c}}Q^{b-{\frac {ad}{c}}}

即： $t\left(e^{\ln Q}\right)^{-t}={\frac {a}{c}}Q^{b-{\frac {ad}{c}}}$

同乘： ${\ln Q}\,$

得出

t{\ln Q}\cdot e^{-t\ln Q}={\ln Q}\cdot {\frac {a}{c}}Q^{b-{\frac {ad}{c}}}

故 $t{\ln Q}=-W_{k}\left(-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\right)$

带入 $t=ax+{\frac {ad}{c}}$

为

\left(ax+{\frac {ad}{c}}\right){\ln Q}=-W_{k}\left(-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\right)

因此最终的解为

x=-{\frac {W_{k}\left(-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\right)}{a\ln Q}}-{\frac {d}{c}}

若辅助方程： $xe^{x}=-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}$ 中，

-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\in \left(-\infty ,-{\frac {1}{e}}\right)

,

辅助方程无实数解，原方程亦无实解；

若： $-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\in \left\{-{\frac {1}{e}}\right\}\cup \mathbf {[} 0,+\infty )$ ,

辅助方程有一实数解，原方程有一实解：

x=-{\frac {W_{k}\left(-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\right)}{a\ln Q}}-{\frac {d}{c}}

若: $-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\in \left(-{\frac {1}{e}},0\right)$ ,

辅助方程有二实解，设为 $W\left(-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\right)$ ，

${\rm {W}}_{-1}\left(-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\right)$ ，

为

$x_{1}=-{\frac {W\left(-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\right)}{a\ln Q}}-{\frac {d}{c}}$

$x_{2}=-{\frac {{\rm {W}}_{-1}\left(-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\right)}{a\ln Q}}-{\frac {d}{c}}$

例子2

用类似的方法，可知以下方程的解

x^{x}={\mathrm {t} }\,,

为

x={\frac {\ln {\rm {t}}}{W(\ln {\rm {t}})}}\,

或

x=\exp \left(W_{k}\left[\ln({\rm {t}})\right]\right).

例子3

以下方程的解

x\log _{b}{x}=a\,

具有形式

x={\frac {a{\ln b}}{W_{k}\left(a{\ln b}\right)}}

例子4: $x^{a}-b^{x}=0\,$; $a>0\,$ : $b>0\,$ : $x>0\,$

取对数，

a\ln x=x\ln b\,

{\frac {\ln x}{x}}={\frac {\ln b}{a}}\,

e^{\frac {\ln x}{x}}=e^{\frac {\ln b}{a}}\,

x^{\frac {1}{x}}=b^{\frac {1}{a}}\,

取倒数，

\left({\frac {1}{x}}\right)^{\frac {1}{x}}=b^{-{\frac {1}{a}}}\,

{\frac {1}{x}}=-{\frac {\ln b}{aW\left(-{\frac {1}{a}}\ln b\right)}}\,

最终解为 : $x=-{\frac {a}{\ln b}}W_{k}\left(-{\frac {\ln b}{a}}\right)\,$

例子5: $(ax+b)^{n}=u^{cx+d}\,$

两边开 $n\,$ 次方并除以 $a\,$ 得

$x+{\frac {b}{a}}={\frac {u^{{\frac {c}{n}}x+{\frac {d}{n}}}}{a}}\left(\cos {\frac {2k\pi }{n}}+{\rm {i}}\sin {\frac {2k\pi }{n}}\right)\,$

令 $u=e^{\ln u}\,$ ，

化为

$x+{\frac {b}{a}}={\frac {e^{{\frac {c\ln u}{n}}x+{\frac {d\ln u}{n}}}}{a}}\left(\cos {\frac {2k\pi }{n}}+{\rm {i}}\sin {\frac {2k\pi }{n}}\right)\,$

两边同乘

$-{\frac {c\ln u}{n}}u^{-{\frac {c}{n}}x-{\frac {cb}{na}}}\,$ ，

$\left(-{\frac {c\ln u}{n}}x-{\frac {cb\ln u}{na}}\right)e^{-{\frac {c\ln u}{n}}x-{\frac {cb\ln u}{na}}}=-{\frac {c\ln u}{na}}u^{{\frac {d}{n}}-{\frac {cb}{na}}}\left(\cos {\frac {2k\pi }{n}}+{\rm {i}}\sin {\frac {2k\pi }{n}}\right)\,$

最终得

$x_{k}=-{\frac {n}{c\ln u}}W_{k}\left[-{\frac {c\ln u}{na}}u^{{\frac {d}{n}}-{\frac {cb}{na}}}\left(\cos {\frac {2k\pi }{n}}+{\rm {i}}\sin {\frac {2k\pi }{n}}\right)\right]-{\frac {b}{a}}\,$

$k\in {\mathbb {Z} }\,$

一般化

標準的 Lambert W 函數可用來表示以下超越代數方程式的解：

e^{-cx}=a_{o}(x-r)~~\quad \qquad \qquad \qquad \qquad (1)

其中 a₀, c 與 r 為實常數。

其解為 ${\textstyle x=r+{\tfrac {W\left({\frac {ce^{-cr}}{a_{o}}}\right)}{c}}}$

Lambert W 函數之一般化^[1]^[2]^[3]包括:

一項在低維空間內廣義相對論與量子力學的應用（量子引力），實際上一種以前未知的連結於此二區域中，如 “Journal of Classical and Quantum Gravity”^[4]所示其 (1) 的右邊式現為二維多項式 x：

e^{-cx}=a_{o}(x-r_{1})(x-r_{2})~~\qquad \qquad (2)

其中 r₁ 和 r₂ 是不同實常數，為二維多項式的根。於此函數解有單一引數 x 但 r_i 和 a_o 為函數的參數。如此一來，此一般式類似於 “hypergeometric”（超几何分布）函數與 “Meijer G“，但屬於不同類函數。當 r₁ = r₂，(2)的兩方可分解為 (1) 因此其解簡化為標準 W 函數。(2)式代表著 “dilaton”（軸子）場的方程，可據此推導線性，雙體重力問題 1+1 維（一空間維與一時間維）當兩不等（靜止）質量，以及，量子力學的特徵能Delta位勢阱給不等電位於一維空間。

量子力學的一特例特徵能的分析解三體問題，亦即（三維）氢分子離子。^[5]於此 (1)（或 (2)）的右手邊現為無限級數多項式之比於 x：

e^{-cx}=a_{o}{\frac {\prod _{i=1}^{\infty }(x-r_{i})}{\prod _{i=1}^{\infty }(x-s_{i})}}\qquad \qquad \qquad (3)

其中 r_i 與 s_i 是相異實常數而 x 是特徵能和內核距離R之函數。式 (3) 與其特例表示於 (1) 和 (2) 是與一更大類型延遲微分方程。由于哈代的“虚假导数”概念，多根的特殊情况得以解决^[6]。

Lambert "W" 函數於基礎物理問題之應用並未完全即使標準情況如 (1) 最近在原子，分子，與光學物理領域可見^[7]以及黎曼假设的 Keiper-Li 准则 ^[8]

图象

朗伯W函数在复平面上的图像
z = Re(W₀(x + i y))
z = Im(W₀(x + i y))

计算

W函数可以用以下的递推关系算出：

w_{j+1}=w_{j}-{\frac {w_{j}e^{w_{j}}-z}{e^{w_{j}}(w_{j}+1)-{\frac {(w_{j}+2)(w_{j}e^{w_{j}}-z)}{2w_{j}+2}}}}

参考来源

^ T.C. Scott and R.B. Mann, General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function, AAECC (Applicable Algebra in Engineering, Communication and Computing), vol. 17, no. 1, (April 2006), pp.41-47, [1]; Arxiv [2] （页面存档备份，存于互联网档案馆）
^ T.C. Scott, G. Fee and J. Grotendorst, "Asymptotic series of Generalized Lambert W Function" （页面存档备份，存于互联网档案馆）, SIGSAM, vol. 47, no. 3, (September 2013), pp. 75-83
^ T.C. Scott, G. Fee, J. Grotendorst and W.Z. Zhang, "Numerics of the Generalized Lambert W Function" （页面存档备份，存于互联网档案馆）, SIGSAM, vol. 48, no. 2, (June 2014), pp. 42-56
^ P.S. Farrugia, R.B. Mann, and T.C. Scott, N-body Gravity and the Schrödinger Equation, Class. Quantum Grav. vol. 24, (2007), pp. 4647-4659, [3]; Arxiv [4] （页面存档备份，存于互联网档案馆）
^ T.C. Scott, M. Aubert-Frécon and J. Grotendorst, New Approach for the Electronic Energies of the Hydrogen Molecular Ion, Chem. Phys. vol. 324, (2006), pp. 323-338, [5] （页面存档备份，存于互联网档案馆）; Arxiv [6] （页面存档备份，存于互联网档案馆）
^ Aude Maignan, T.C. Scott, "Fleshing out the Generalized Lambert W Function", SIGSAM, vol. 50, no. 2, (June 2016), pp. 45-60
^ T.C. Scott, A. Lüchow, D. Bressanini and J.D. Morgan III, The Nodal Surfaces of Helium Atom Eigenfunctions, Phys. Rev. A 75, (2007), p. 060101, [7] （页面存档备份，存于互联网档案馆）
^ R.C. McPhedran, T.C Scott and Aude Maignan, "The Keiper-Li Criterion for the Riemann Hypothesis and Generalized Lambert Functions", ACM Commun. Comput. Algebra, vol. 57, no. 3, (December 2023), pp. 85-110

外部链接

MathWorld （页面存档备份，存于互联网档案馆）

[1] T.C. Scott and R.B. Mann, General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function, AAECC (Applicable Algebra in Engineering, Communication and Computing), vol. 17, no. 1, (April 2006), pp.41-47, [1]; Arxiv [2] （页面存档备份，存于互联网档案馆）

[2] T.C. Scott, G. Fee and J. Grotendorst, "Asymptotic series of Generalized Lambert W Function" （页面存档备份，存于互联网档案馆）, SIGSAM, vol. 47, no. 3, (September 2013), pp. 75-83

[3] T.C. Scott, G. Fee, J. Grotendorst and W.Z. Zhang, "Numerics of the Generalized Lambert W Function" （页面存档备份，存于互联网档案馆）, SIGSAM, vol. 48, no. 2, (June 2014), pp. 42-56

[4] P.S. Farrugia, R.B. Mann, and T.C. Scott, N-body Gravity and the Schrödinger Equation, Class. Quantum Grav. vol. 24, (2007), pp. 4647-4659, [3]; Arxiv [4] （页面存档备份，存于互联网档案馆）

[5] T.C. Scott, M. Aubert-Frécon and J. Grotendorst, New Approach for the Electronic Energies of the Hydrogen Molecular Ion, Chem. Phys. vol. 324, (2006), pp. 323-338, [5] （页面存档备份，存于互联网档案馆）; Arxiv [6] （页面存档备份，存于互联网档案馆）

[6] Aude Maignan, T.C. Scott, "Fleshing out the Generalized Lambert W Function", SIGSAM, vol. 50, no. 2, (June 2016), pp. 45-60

[7] T.C. Scott, A. Lüchow, D. Bressanini and J.D. Morgan III, The Nodal Surfaces of Helium Atom Eigenfunctions, Phys. Rev. A 75, (2007), p. 060101, [7] （页面存档备份，存于互联网档案馆）

[8] R.C. McPhedran, T.C Scott and Aude Maignan, "The Keiper-Li Criterion for the Riemann Hypothesis and Generalized Lambert Functions", ACM Commun. Comput. Algebra, vol. 57, no. 3, (December 2023), pp. 85-110

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