朗伯W函数

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 ) = w e w {\displaystyle f(w)=we^{w}} 的反函数，其中 e w {\displaystyle e^{w}} 是指数函数， w {\displaystyle w} 是任意复数。对于任何复数 z {\displaystyle z} ，都有： z = W ( z ) e W ( z ) . {\displaystyle z=W(z)e^{W(z)}.} 由于函数 f {\displaystyle f} 不是单射，因此函数 W {\displaystyle W} 是多值的（除了0以外）。如果我们把 x {\displaystyle x} 限制为实数，并要求 w {\displaystyle w} 是实数，那么函数仅对于 x ≥ 1 e {\displaystyle x\geq {\frac {1}{e}}} 有定义，在 ( − 1 e , 0 ) {\displaystyle (-{\frac {1}{e}},0)} 内是多值的；如果加上 w ≥ − 1 {\displaystyle w\geq -1} 的限制，则定义了一个单值函数 W 0 ( x ) {\displaystyle W_{0}(x)} （见图）。我们有 W 0 ( 0 ) = 0 {\displaystyle W_{0}(0)=0} ， W 0 ( − 1 e ) = − 1 {\displaystyle W_{0}(-{\frac {1}{e}})=-1} 。而在 [ − 1 e , 0 ) {\displaystyle [-{\frac {1}{e}},0)} 内的 w ≤ − 1 {\displaystyle w\leq -1} 分支，则记为 W − 1 ( x ) {\displaystyle W_{-1}(x)} ，从 W − 1 ( − 1 e ) = − 1 {\displaystyle W_{-1}(-{\frac {1}{e}})=-1} 递减为 W − 1 ( 0 − ) = − ∞ {\displaystyle W_{-1}(0^{-})=-\infty } 。 朗伯 W {\displaystyle 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 ) = a y ( t − 1 ) {\displaystyle y(t)=ay(t-1)} 。 朗伯 W {\displaystyle W\,} 函数的积分形式为 W ( x ) = x π ∫ 0 π ( 1 − v cot ⁡ v ) 2 + v 2 x + v csc ⁡ v ⋅ e − v cot ⁡ v d v , | arg ⁡ ( x ) | < π {\displaystyle 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 ) = ∫ − ∞ − 1 e − 1 π ℑ [ d d x W ( x ) ] ln ⁡ ( 1 − z x ) d x {\displaystyle 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 ∉ [ − 1 e , 0 ] , k ∈ Z {\displaystyle x\not \in \left[-{\frac {1}{e}},0\right],k\in {\mathbb {Z} }\,} ,若 x ∈ ( − 1 e , 0 ) , k = 1 , ± 2 , ± 3 , . . . {\displaystyle x\in \left(-{\frac {1}{e}},0\right),k=1,\pm 2,\pm 3,...\,} W k ( x ) = 1 + ( ln ⁡ x − 1 + 2 k π i ) e i 2 π ∫ 0 ∞ ln ⁡ t − ln ⁡ t + ln ⁡ x + ( 2 k + 1 ) π i t − ln ⁡ t + ln ⁡ x + ( 2 k − 1 ) π i ⋅ d t t + 1 = 1 + ( ln ⁡ x − 1 + 2 k π i ) e i 2 π ∫ 0 ∞ ln ⁡ ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 4 k 2 − 1 ) π 2 + 2 π ( t − ln ⁡ t + ln ⁡ x ) i ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k − 1 ) 2 π 2 ⋅ d t t + 1 {\displaystyle 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 + ( ln ⁡ x − 1 + 2 k π i ) e i 2 π ∫ 0 ∞ [ 1 2 ln ⁡ ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k + 1 ) 2 π 2 ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k − 1 ) 2 π 2 + i arctan ⁡ 2 π ( t − ln ⁡ t + ln ⁡ x ) ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 4 k 2 − 1 ) π 2 ] ⋅ d t t + 1 {\displaystyle 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 + ( ln ⁡ x − 1 ) cos ⁡ 1 4 π ∫ 0 ∞ ln ⁡ ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k + 1 ) 2 π 2 ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k − 1 ) 2 π 2 ⋅ d t t + 1 − 2 k π sin ⁡ 1 4 π ∫ 0 ∞ ln ⁡ ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k + 1 ) 2 π 2 ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k − 1 ) 2 π 2 ⋅ d t t + 1 + i [ ( ln ⁡ x − 1 ) sin ⁡ 1 4 π ∫ 0 ∞ ln ⁡ ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k + 1 ) 2 π 2 ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k − 1 ) 2 π 2 ⋅ d t t + 1 + 2 k π cos ⁡ 1 4 π ∫ 0 ∞ ln ⁡ ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k + 1 ) 2 π 2 ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k − 1 ) 2 π 2 ⋅ d t t + 1 ] e 1 2 π ∫ 0 ∞ arctan ⁡ 2 π ( t − ln ⁡ t + ln ⁡ x ) ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 4 k 2 − 1 ) π 2 ⋅ d t t + 1 {\displaystyle {}_{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 i , x = t + s i {\displaystyle W_{k}(x)=u+v{\rm {i}},x=t+s{\rm {i}}} ，则有 ( u + v i ) e u + v i = t + s i {\displaystyle \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 ( u cos ⁡ v − v sin ⁡ v ) = t , e u ( u sin ⁡ v + v cos ⁡ v ) = s {\displaystyle 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 {\displaystyle s=0} 时，易知 u = − v cot ⁡ v {\displaystyle u=-v\cot v} W k ( x ) = ( 1 − ln ⁡ x ) sin ⁡ 1 4 π ∫ 0 ∞ ln ⁡ ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k + 1 ) 2 π 2 ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k − 1 ) 2 π 2 ⋅ d t t + 1 − 2 k π cos ⁡ 1 4 π ∫ 0 ∞ ln ⁡ ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k + 1 ) 2 π 2 ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k − 1 ) 2 π 2 ⋅ d t t + 1 e 1 2 π ∫ 0 ∞ arctan ⁡ 2 π ( t − ln ⁡ t + ln ⁡ x ) ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 4 k 2 − 1 ) π 2 ⋅ d t t + 1 cot ⁡ ( ln ⁡ x − 1 ) sin ⁡ 1 4 π ∫ 0 ∞ ln ⁡ ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k + 1 ) 2 π 2 ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k − 1 ) 2 π 2 ⋅ d t t + 1 + 2 k π cos ⁡ 1 4 π ∫ 0 ∞ ln ⁡ ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k + 1 ) 2 π 2 ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k − 1 ) 2 π 2 ⋅ d t t + 1 e 1 2 π ∫ 0 ∞ arctan ⁡ 2 π ( t − ln ⁡ t + ln ⁡ x ) ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 4 k 2 − 1 ) π 2 ⋅ d t t + 1 + ( ln ⁡ x − 1 ) sin ⁡ 1 4 π ∫ 0 ∞ ln ⁡ ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k + 1 ) 2 π 2 ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k − 1 ) 2 π 2 ⋅ d t t + 1 + 2 k π cos ⁡ 1 4 π ∫ 0 ∞ ln ⁡ ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k + 1 ) 2 π 2 ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 2 k − 1 ) 2 π 2 ⋅ d t t + 1 e 1 2 π ∫ 0 ∞ arctan ⁡ 2 π ( t − ln ⁡ t + ln ⁡ x ) ( t − ln ⁡ t + ln ⁡ x ) 2 + ( 4 k 2 − 1 ) π 2 ⋅ d t t + 1 i , {\displaystyle {}_{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 + ( ln ⁡ x − 1 ) e − 1 π ∫ 0 ∞ arg ⁡ ( t − ln ⁡ t + ln ⁡ x + π i ) ⋅ d t t + 1 , x > 0 {\displaystyle 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 > 1 e {\displaystyle x>{\frac {1}{e}}} ，上式还可化为 W 0 ( x ) = 1 + ( ln ⁡ x − 1 ) e − 1 π ∫ 0 ∞ arctan ⁡ π t − ln ⁡ t + ln ⁡ x ⋅ d t t + 1 {\displaystyle 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 {\displaystyle W\,} 函数满足以下的微分方程： z [ 1 + W ( z ) ] d d z W ( z ) = W ( z ) {\displaystyle z\left[1+W(z)\right]{\frac {\rm {d}}{{\rm {d}}z}}W(z)=W(z)} ， z ≠ − 1 e , {\displaystyle z\neq -{\frac {1}{e}}\,,} 因此： d d z W ( z ) = W ( z ) z [ 1 + W ( z ) ] {\displaystyle {\frac {\rm {d}}{{\rm {d}}z}}W(z)={\frac {W(z)}{z\left[1+W(z)\right]}}} ， z ≠ − 1 e . {\displaystyle z\neq -{\frac {1}{e}}\,.} 函数 W ( x ) {\displaystyle W(x)\,} ，以及许多含有 W ( x ) {\displaystyle W(x)\,} 的表达式，都可以用 w = W ( x ) {\displaystyle w=W(x)\,} 的变量代换来积分，也就是说 x = w e w {\displaystyle x=we^{w}\,} ∫ W ( x ) d x = x [ W ( x ) + 1 W ( x ) − 1 ] + C {\displaystyle \int W(x){\rm {d}}x=x\left[W(x)+{\frac {1}{W(x)}}-1\right]+C} ∫ 0 1 W ( x ) d x = Ω + 1 Ω − 2 ≈ 0.330366 {\displaystyle \int _{0}^{1}W(x){\rm {d}}x=\Omega +{\frac {1}{\Omega }}-2\approx 0.330366} 其中 Ω {\displaystyle \Omega } 為欧米加常数。 1 {\displaystyle 1\,} 、 z z z z z . . . = lim n → ∞ ( z ⇈ n ) = − W ( − ln ⁡ z ) ln ⁡ z {\displaystyle z^{z^{z^{z^{z^{.^{.^{.}}}}}}}=\lim _{n\to \infty }(z\upuparrows n)=-{\frac {W(-\ln z)}{\ln z}}} ， 其中 ⇈ {\displaystyle \upuparrows } 是高德納箭號表示法。 2 {\displaystyle 2\,} 、若 z > 0 {\displaystyle z>0\,} ，则 ln ⁡ W ( z ) = ln ⁡ z − W ( z ) {\displaystyle \ln W(z)=\ln z-W(z)\,} W 0 {\displaystyle W_{0}\,} 在 x = 0 {\displaystyle x=0\,} 的泰勒级数如下： W 0 ( x ) = ∑ n = 1 ∞ ( − n ) n − 1 n !   x n = x − x 2 + 3 2 x 3 − 8 3 x 4 + 125 24 x 5 − ⋯ {\displaystyle 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 1 e {\displaystyle {\frac {1}{e}}\,} 。 W ( x ) + W ( y ) = W [ x y W ( x ) + x y W ( y ) ] {\displaystyle W(x)+W(y)=W\left[{\frac {xy}{W(x)}}+{\frac {xy}{W(y)}}\right]\,} x > 0 , y > 0 {\displaystyle x>0,y>0\,} 實部 ℜ [ W ( x + y i ) ] = ∑ k = 1 ∞ ( − k ) k − 1 k ! ( x 2 + y 2 ) k cos ⁡ ( k arctan ⁡ x y ) {\displaystyle \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 < 1 e 2 {\displaystyle x^{2}+y^{2}<{\frac {1}{e^{2}}}\,} 虛部 ℑ [ W ( x + y i ) ] = ∑ k = 1 ∞ ( − k ) k − 1 k ! ( x 2 + y 2 ) k sin ⁡ ( k arctan ⁡ x y ) {\displaystyle \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 < 1 e 2 {\displaystyle x^{2}+y^{2}<{\frac {1}{e^{2}}}\,} 模長 | W ( x + y i ) | = W ( x + y ) {\displaystyle |W(x+y{\rm {i}})|=W({\sqrt {x+y}})\,} 模角 arg ⁡ [ W ( x + y i ) ] = ∑ k = 1 ∞ ( − k ) k − 1 k ! arctan ⁡ [ cot ⁡ ( k arctan ⁡ x y ) ] {\displaystyle \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 < 1 e 2 {\displaystyle x^{2}+y^{2}<{\frac {1}{e^{2}}}\,} 共軛值 W ( x + y i ) ¯ = ∑ k = 1 ∞ ( − k ) k − 1 k ! ( x 2 + y 2 ) k [ cos ⁡ ( k arctan ⁡ x y ) − i sin ⁡ ( k arctan ⁡ x y ) ] {\displaystyle {\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 < 1 e 2 {\displaystyle x^{2}+y^{2}<{\frac {1}{e^{2}}}\,} W ( − π 2 ) = π 2 i {\displaystyle W\left(-{\frac {\pi }{2}}\right)={\frac {\pi }{2}}i} W ( − ln ⁡ 2 2 ) = − ln ⁡ 2 {\displaystyle W\left(-{\frac {\ln 2}{2}}\right)=-\ln 2} W ( − 1 e ) = − 1 {\displaystyle W\left(-{1 \over e}\right)=-1} W ( 1 ) = Ω = 1 ∫ − ∞ ∞ d x ( e x − x ) 2 + π 2 − 1 ≈ 0.56714329 … {\displaystyle 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 {\displaystyle W(e)=1\,} W ( e e + 1 ) = e {\displaystyle W(e^{e+1})=e\,} W ( 1 e 1 − 1 e ) = 1 e {\displaystyle W\left({\frac {1}{e^{1-{\frac {1}{e}}}}}\right)={\frac {1}{e}}} W ( π e π ) = π {\displaystyle W({\pi }e^{\pi })=\pi } W ( k ln ⁡ k ) = ln ⁡ k {\displaystyle W(k{\ln k})={\ln k}} ( k > 0 ) {\displaystyle (k>0)} W ( i π ) = − i π {\displaystyle W({\rm {i}}\pi )=-{\rm {i}}\pi } W ( − i π ) = i π {\displaystyle W(-{\rm {i}}\pi )={\rm {i}}\pi } W ( i cos ⁡ 1 − sin ⁡ 1 ) = i {\displaystyle W({\rm {i}}\cos 1-\sin 1)={\rm {i}}} W ( − 3 2 π ) = − 3 2 π i {\displaystyle W(-{\frac {3}{2}}{\pi })=-{\frac {3}{2}}{\pi }{\rm {i}}} W ( − 8 7 7 ln ⁡ 2 ) = − 32 7 ln ⁡ 2 {\displaystyle W(-{\frac {\sqrt[{7}]{8}}{7}}{\ln 2})=-{\frac {32}{7}}{\ln 2}} W ( − 3 54 ln ⁡ 3 ) = − 9 2 ln ⁡ 3 {\displaystyle W(-{\frac {\sqrt {3}}{54}}{\ln 3})=-{\frac {9}{2}}{\ln 3}} W ( − ln ⁡ 2 4 ) = − 4 ln ⁡ 2 {\displaystyle W(-{\frac {\ln 2}{4}})=-4{\ln 2}} W ( − 1 ) = e 1 2 π ∫ 0 ∞ 1 t + 1 arctan ⁡ 2 π t − ln ⁡ t d t − cos ⁡ [ 1 4 π ∫ 0 ∞ 1 t + 1 ln ⁡ ( t − ln ⁡ t ) 2 4 π 2 + ( t − ln ⁡ t ) 2 d t ] + π sin ⁡ [ 1 4 π ∫ 0 ∞ 1 t + 1 ln ⁡ ( t − ln ⁡ t ) 2 4 π 2 + ( t − ln ⁡ t ) 2 d t ] − i { π cos ⁡ [ 1 4 π ∫ 0 ∞ 1 t + 1 ln ⁡ ( t − ln ⁡ t ) 2 4 π 2 + ( t − ln ⁡ t ) 2 d t ] + sin ⁡ [ 1 4 π ∫ 0 ∞ 1 t + 1 ln ⁡ ( t − ln ⁡ t ) 2 4 π 2 + ( t − ln ⁡ t ) 2 d t ] } e 1 2 π ∫ 0 ∞ 1 t + 1 arctan ⁡ 2 π t − ln ⁡ t d t ≈ − 0.31813 − 1.33723 i {\displaystyle 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 ( − ln ⁡ k k ) = − ln ⁡ k {\displaystyle W(-{\frac {\ln k}{k}})=-\ln k} W [ − ln ⁡ ( x + 1 ) x ( x + 1 ) 1 x ] = − x + 1 x ln ⁡ ( x + 1 ) > , − 1 < x < 0 {\displaystyle W\left[-{\frac {\ln(x+1)}{x(x+1)^{\frac {1}{x}}}}\right]=-{\frac {x+1}{x}}\ln(x+1)>,-1<x<0} 许多含有指数的方程都可以用 W {\displaystyle 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 = X e X {\displaystyle Y=Xe^{X}\,} 的形式。 例子1 2 t = 5 t {\displaystyle 2^{t}=5t\,} ⇒ 1 = 5 t 2 t {\displaystyle \Rightarrow 1={\frac {5t}{2^{t}}}\,} ⇒ 1 = 5 t e − t ln ⁡ 2 {\displaystyle \Rightarrow 1=5t\,e^{-t\ln 2}\,} ⇒ 1 5 = t e − t ln ⁡ 2 {\displaystyle \Rightarrow {\frac {1}{5}}=t\,e^{-t\ln 2}\,} ⇒ − ln ⁡ 2 5 = ( − t ln ⁡ 2 ) e − t ln ⁡ 2 {\displaystyle \Rightarrow -{\frac {\ln 2}{5}}=(-\,t\,\ln 2)\,e^{-t\ln 2}\,} ⇒ − t ln ⁡ 2 = W k ( − ln ⁡ 2 5 ) {\displaystyle \Rightarrow -t\ln 2=W_{k}\left(-{\frac {\ln 2}{5}}\right)\,} ⇒ t = − W k ( − ln ⁡ 2 5 ) ln ⁡ 2 {\displaystyle \Rightarrow t=-{\frac {W_{k}\left(-{\frac {\ln 2}{5}}\right)}{\ln 2}}\,} 更一般地，以下的方程 Q a x + b = c x + d {\displaystyle Q^{ax+b}=cx+d\,} 其中 Q > 0 ∧ Q ≠ 1 ∧ c ≠ 0 {\displaystyle Q>0\land Q\neq 1\land c\neq 0} 两边同乘: a c {\displaystyle {\frac {a}{c}}} ， 得到： a c Q a x + b = a x + a d c {\displaystyle {\frac {a}{c}}Q^{ax+b}=ax+{\frac {ad}{c}}\,} 同除以： Q a x {\displaystyle Q^{ax}\,} ， 得到： a c Q b = ( a x + a d c ) Q − a x {\displaystyle {\frac {a}{c}}Q^{b}=\left(ax+{\frac {ad}{c}}\right)Q^{-ax}\,} 同除： Q a d c {\displaystyle Q^{\frac {ad}{c}}\,} ， a c Q b − a d c = ( a x + a d c ) Q − ( a x + a d c ) {\displaystyle {\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 = a x + a d c {\displaystyle t=ax+{\frac {ad}{c}}} 化为 t Q − t = a c Q b − a d c {\displaystyle tQ^{-t}={\frac {a}{c}}Q^{b-{\frac {ad}{c}}}} 即： t ( e ln ⁡ Q ) − t = a c Q b − a d c {\displaystyle t\left(e^{\ln Q}\right)^{-t}={\frac {a}{c}}Q^{b-{\frac {ad}{c}}}} 同乘： ln ⁡ Q {\displaystyle {\ln Q}\,} 得出 t ln ⁡ Q ⋅ e − t ln ⁡ Q = ln ⁡ Q ⋅ a c Q b − a d c {\displaystyle 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 ( − a ln ⁡ Q c Q b − a d c ) {\displaystyle t{\ln Q}=-W_{k}\left(-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\right)} 带入 t = a x + a d c {\displaystyle t=ax+{\frac {ad}{c}}} 为 ( a x + a d c ) ln ⁡ Q = − W k ( − a ln ⁡ Q c Q b − a d c ) {\displaystyle \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 = − W k ( − a ln ⁡ Q c Q b − a d c ) a ln ⁡ Q − d c {\displaystyle 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： x e x = − a ln ⁡ Q c Q b − a d c {\displaystyle xe^{x}=-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}} 中， − a ln ⁡ Q c Q b − a d c ∈ ( − ∞ , − 1 e ) {\displaystyle -{\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： − a ln ⁡ Q c Q b − a d c ∈ { − 1 e } ∪ [ 0 , + ∞ ) {\displaystyle -{\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 = − W k ( − a ln ⁡ Q c Q b − a d c ) a ln ⁡ Q − d c {\displaystyle x=-{\frac {W_{k}\left(-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\right)}{a\ln Q}}-{\frac {d}{c}}} 若: − a ln ⁡ Q c Q b − a d c ∈ ( − 1 e , 0 ) {\displaystyle -{\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 ( − a ln ⁡ Q c Q b − a d c ) {\displaystyle W\left(-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\right)} ， W − 1 ( − a ln ⁡ Q c Q b − a d c ) {\displaystyle {\rm {W}}_{-1}\left(-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\right)} ， 为 x 1 = − W ( − a ln ⁡ Q c Q b − a d c ) a ln ⁡ Q − d c {\displaystyle x_{1}=-{\frac {W\left(-{\frac {a\ln Q}{c}}\,Q^{b-{\frac {ad}{c}}}\right)}{a\ln Q}}-{\frac {d}{c}}} x 2 = − W − 1 ( − a ln ⁡ Q c Q b − a d c ) a ln ⁡ Q − d c {\displaystyle 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 = t , {\displaystyle x^{x}={\mathrm {t} }\,,} 为 x = ln ⁡ t W ( ln ⁡ t ) {\displaystyle x={\frac {\ln {\rm {t}}}{W(\ln {\rm {t}})}}\,} 或 x = exp ⁡ ( W k [ ln ⁡ ( t ) ] ) . {\displaystyle 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 {\displaystyle x\log _{b}{x}=a\,} 具有形式 x = a ln ⁡ b W k ( a ln ⁡ b ) {\displaystyle x={\frac {a{\ln b}}{W_{k}\left(a{\ln b}\right)}}} 例子4 x a − b x = 0 {\displaystyle x^{a}-b^{x}=0\,} a > 0 {\displaystyle a>0\,}  : b > 0 {\displaystyle b>0\,}  : x > 0 {\displaystyle x>0\,} 取对数， a ln ⁡ x = x ln ⁡ b {\displaystyle a\ln x=x\ln b\,} ln ⁡ x x = ln ⁡ b a {\displaystyle {\frac {\ln x}{x}}={\frac {\ln b}{a}}\,} e ln ⁡ x x = e ln ⁡ b a {\displaystyle e^{\frac {\ln x}{x}}=e^{\frac {\ln b}{a}}\,} x 1 x = b 1 a {\displaystyle x^{\frac {1}{x}}=b^{\frac {1}{a}}\,} 取倒数， ( 1 x ) 1 x = b − 1 a {\displaystyle \left({\frac {1}{x}}\right)^{\frac {1}{x}}=b^{-{\frac {1}{a}}}\,} 1 x = − ln ⁡ b a W ( − 1 a ln ⁡ b ) {\displaystyle {\frac {1}{x}}=-{\frac {\ln b}{aW\left(-{\frac {1}{a}}\ln b\right)}}\,} 最终解为 : x = − a ln ⁡ b W k ( − ln ⁡ b a ) {\displaystyle x=-{\frac {a}{\ln b}}W_{k}\left(-{\frac {\ln b}{a}}\right)\,} 例子5 ( a x + b ) n = u c x + d {\displaystyle (ax+b)^{n}=u^{cx+d}\,} 两边开 n {\displaystyle n\,} 次方并除以 a {\displaystyle a\,} 得 x + b a = u c n x + d n a ( cos ⁡ 2 k π n + i sin ⁡ 2 k π n ) {\displaystyle 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 {\displaystyle u=e^{\ln u}\,} ， 化为 x + b a = e c ln ⁡ u n x + d ln ⁡ u n a ( cos ⁡ 2 k π n + i sin ⁡ 2 k π n ) {\displaystyle 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 − c ln ⁡ u n u − c n x − c b n a {\displaystyle -{\frac {c\ln u}{n}}u^{-{\frac {c}{n}}x-{\frac {cb}{na}}}\,} ， ( − c ln ⁡ u n x − c b ln ⁡ u n a ) e − c ln ⁡ u n x − c b ln ⁡ u n a = − c ln ⁡ u n a u d n − c b n a ( cos ⁡ 2 k π n + i sin ⁡ 2 k π n ) {\displaystyle \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 = − n c ln ⁡ u W k [ − c ln ⁡ u n a u d n − c b n a ( cos ⁡ 2 k π n + i sin ⁡ 2 k π n ) ] − b a {\displaystyle 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 ∈ Z {\displaystyle k\in {\mathbb {Z} }\,} 標準的 Lambert W 函數可用來表示以下超越代數方程式的解： e − c x = a o ( x − r )     ( 1 ) {\displaystyle 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 x = r + W ( c e − c r a o ) c {\textstyle x=r+{\tfrac {W\left({\frac {ce^{-cr}}{a_{o}}}\right)}{c}}} Lambert W 函數之一般化 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” 所示其 (1) 的右邊式現為二維多項式 x： e − c x = a o ( x − r 1 ) ( x − r 2 )     ( 2 ) {\displaystyle 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 ri 和 ao 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。於此 (1)（或 (2)）的右手邊現為無限級數多項式之比於 x： e − c x = a o ∏ i = 1 ∞ ( x − r i ) ∏ i = 1 ∞ ( x − s i ) ( 3 ) {\displaystyle 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 ri 與 si 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。 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 以及黎曼假设的 Keiper-Li 准则 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)) W函数可以用以下的递推关系算出： w j + 1 = w j − w j e w j − z e w j ( w j + 1 ) − ( w j + 2 ) ( w j e w j − z ) 2 w j + 2 {\displaystyle 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}}}}} 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 MathWorld （页面存档备份，存于互联网档案馆）