Elementary Functions - 4.13 Lambert -Function

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.13.E1 W ⁒ e W = x π‘Š superscript 𝑒 π‘Š π‘₯ {\displaystyle{\displaystyle We^{W}=x}}
We^{W} = x		 

W*exp(W) = x

W*Exp[W] == x
Failure Failure
Failed [30 / 30]
Result: -.263026030+2.030302705*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, x = 1.5}


Result: .736973970+2.030302705*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, x = .5}


Result: -.763026030+2.030302705*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, x = 2}


Result: -2.096603674+.1092863076*I
Test Values: {W = -1/2+1/2*I*3^(1/2), x = 1.5}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-0.2630260306572938, 2.0303027048207967]
Test Values: {Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}


Result: Complex[0.7369739693427062, 2.0303027048207967]
Test Values: {Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]}

... skip entries to safe data
4.13#Ex1 Wp ⁑ ( - 1 / e ) = Wm ⁑ ( - 1 / e ) Lambert-Wp 1 𝑒 Lambert-Wm 1 𝑒 {\displaystyle{\displaystyle\mathrm{Wp}\left(-1/e\right)=\mathrm{Wm}\left(-1/e% \right)}}
\LambertWp@{-1/e} = \LambertWm@{-1/e}		 

LambertW(0, - 1/exp(1)) = LambertW(-1, - 1/exp(1))

ProductLog[0, - 1/E] == ProductLog[-1, - 1/E]
Successful Successful - Successful [Tested: 1]
4.13#Ex1 Wm ⁑ ( - 1 / e ) = - 1 Lambert-Wm 1 𝑒 1 {\displaystyle{\displaystyle\mathrm{Wm}\left(-1/e\right)=-1}}
\LambertWm@{-1/e} = -1		 

LambertW(-1, - 1/exp(1)) = - 1

ProductLog[-1, - 1/E] == - 1
Successful Successful - Successful [Tested: 1]
4.13#Ex2 Wp ⁑ ( 0 ) = 0 Lambert-Wp 0 0 {\displaystyle{\displaystyle\mathrm{Wp}\left(0\right)=0}}
\LambertWp@{0} = 0		 

LambertW(0, 0) = 0

ProductLog[0, 0] == 0
Successful Successful - Successful [Tested: 1]
4.13#Ex3 Wp ⁑ ( e ) = 1 Lambert-Wp 𝑒 1 {\displaystyle{\displaystyle\mathrm{Wp}\left(e\right)=1}}
\LambertWp@{e} = 1		 

LambertW(0, exp(1)) = 1

ProductLog[0, E] == 1
Successful Successful - Successful [Tested: 1]
4.13#Ex4 U + ln ⁑ U = x π‘ˆ π‘ˆ π‘₯ {\displaystyle{\displaystyle U+\ln U=x}}
U+\ln@@{U} = x		 

U + ln(U) = x

U + Log[U] == x
Failure Failure
Failed [30 / 30]
Result: -.6339745958+1.023598776*I
Test Values: {U = 1/2*3^(1/2)+1/2*I, x = 1.5}


Result: .3660254042+1.023598776*I
Test Values: {U = 1/2*3^(1/2)+1/2*I, x = .5}


Result: -1.133974596+1.023598776*I
Test Values: {U = 1/2*3^(1/2)+1/2*I, x = 2}


Result: -2.000000000+2.960420506*I
Test Values: {U = -1/2+1/2*I*3^(1/2), x = 1.5}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-0.6339745962155613, 1.0235987755982987]
Test Values: {Rule[U, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}


Result: Complex[0.3660254037844387, 1.0235987755982987]
Test Values: {Rule[U, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]}

... skip entries to safe data
4.13#Ex5 U = U ⁒ ( x ) π‘ˆ π‘ˆ π‘₯ {\displaystyle{\displaystyle U=U(x)}}
U = U(x)		 

U = U*(x)

U == U*(x)
Failure Failure
Failed [30 / 30]
Result: -.4330127020-.2500000000*I
Test Values: {U = 1/2*3^(1/2)+1/2*I, x = 1.5}


Result: .4330127020+.2500000000*I
Test Values: {U = 1/2*3^(1/2)+1/2*I, x = .5}


Result: -.8660254040-.5000000000*I
Test Values: {U = 1/2*3^(1/2)+1/2*I, x = 2}


Result: .2500000000-.4330127020*I
Test Values: {U = -1/2+1/2*I*3^(1/2), x = 1.5}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-0.4330127018922193, -0.24999999999999994]
Test Values: {Rule[U, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}


Result: Complex[0.43301270189221935, 0.24999999999999997]
Test Values: {Rule[U, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]}

... skip entries to safe data
4.13#Ex5 U ⁒ ( x ) = W ⁑ ( e x ) π‘ˆ π‘₯ Lambert-W superscript 𝑒 π‘₯ {\displaystyle{\displaystyle U(x)=W\left(e^{x}\right)}}
U(x) = \LambertW@{e^{x}}		 

U(x) = LambertW(exp(x))

U[x] == ProductLog[Exp[x]]
Failure Failure
Failed [30 / 30]
Result: .34078386e-1+.7500000000*I
Test Values: {U = 1/2*3^(1/2)+1/2*I, x = 1.5}


Result: -.3332359062+.2500000000*I
Test Values: {U = 1/2*3^(1/2)+1/2*I, x = .5}


Result: .174905209+1.*I
Test Values: {U = 1/2*3^(1/2)+1/2*I, x = 2}


Result: -2.014959720+1.299038106*I
Test Values: {U = -1/2+1/2*I*3^(1/2), x = 1.5}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[0.0340783855511575, 0.7499999999999999]
Test Values: {Rule[U, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}


Result: Complex[-0.333235906269531, 0.24999999999999997]
Test Values: {Rule[U, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]}

... skip entries to safe data
4.13.E5 Wp ⁑ ( x ) = βˆ‘ n = 1 ∞ ( - 1 ) n - 1 ⁒ n n - 2 ( n - 1 ) ! ⁒ x n Lambert-Wp π‘₯ superscript subscript 𝑛 1 superscript 1 𝑛 1 superscript 𝑛 𝑛 2 𝑛 1 superscript π‘₯ 𝑛 {\displaystyle{\displaystyle\mathrm{Wp}\left(x\right)=\sum_{n=1}^{\infty}(-1)^% {n-1}\frac{n^{n-2}}{(n-1)!}x^{n}}}
\LambertWp@{x} = \sum_{n=1}^{\infty}(-1)^{n-1}\frac{n^{n-2}}{(n-1)!}x^{n}		 | x | < 1 e π‘₯ 1 𝑒 {\displaystyle{\displaystyle|x|<\dfrac{1}{e}}} 
LambertW(0, x) = sum((- 1)^(n - 1)*((n)^(n - 2))/(factorial(n - 1))*(x)^(n), n = 1..infinity)

ProductLog[0, x] == Sum[(- 1)^(n - 1)*Divide[(n)^(n - 2),(n - 1)!]*(x)^(n), {n, 1, Infinity}, GenerateConditions->None]
Failure Successful Error Successful [Tested: 0]
4.13.E6 W ⁑ ( - e - 1 - ( t 2 / 2 ) ) = βˆ‘ n = 0 ∞ ( - 1 ) n - 1 ⁒ c n ⁒ t n Lambert-W superscript 𝑒 1 superscript 𝑑 2 2 superscript subscript 𝑛 0 superscript 1 𝑛 1 subscript 𝑐 𝑛 superscript 𝑑 𝑛 {\displaystyle{\displaystyle W\left(-e^{-1-(t^{2}/2)}\right)=\sum_{n=0}^{% \infty}(-1)^{n-1}c_{n}t^{n}}}
\LambertW@{-e^{-1-(t^{2}/2)}} = \sum_{n=0}^{\infty}(-1)^{n-1}c_{n}t^{n}		 | t | < 2 ⁒ Ο€ 𝑑 2 πœ‹ {\displaystyle{\displaystyle|t|<2\sqrt{\pi}}} 
LambertW(- exp(- 1 -((t)^(2)/2))) = sum((- 1)^(n - 1)* c[n]*(t)^(n), n = 0..infinity)

ProductLog[- Exp[- 1 -((t)^(2)/2)]] == Sum[(- 1)^(n - 1)* Subscript[c, n]*(t)^(n), {n, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [60 / 60]
Result: Float(infinity)+Float(infinity)*I
Test Values: {t = -1.5, c[n] = 1/2*3^(1/2)+1/2*I}


Result: Float(infinity)+Float(infinity)*I
Test Values: {t = -1.5, c[n] = -1/2+1/2*I*3^(1/2)}


Result: Float(infinity)+Float(infinity)*I
Test Values: {t = -1.5, c[n] = 1/2-1/2*I*3^(1/2)}


Result: Float(infinity)+Float(infinity)*I
Test Values: {t = -1.5, c[n] = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [60 / 60]
Result: Plus[-0.13696418431579768, Times[-1.0, NSum[Times[Power[-1.5, n], Power[-1, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[Subscript[c, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[-0.13696418431579768, Times[-1.0, NSum[Times[Power[-1.5, n], Power[-1, Plus[-1, n]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[Subscript[c, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.13.E7 c 0 = 1 , c 1 subscript 𝑐 0 1 subscript 𝑐 1 {\displaystyle{\displaystyle c_{0}=1,c_{1}}}
c_{0} = 1,c_{1}		 

c[0] = 1; c[1]
 
Subscript[c, 0] == 1
 Subscript[c, 1]
 Skipped - no semantic math Skipped - no semantic math - -
4.13.E8 c n = 1 n + 1 ⁒ ( c n - 1 - βˆ‘ k = 2 n - 1 k ⁒ c k ⁒ c n + 1 - k ) subscript 𝑐 𝑛 1 𝑛 1 subscript 𝑐 𝑛 1 superscript subscript π‘˜ 2 𝑛 1 π‘˜ subscript 𝑐 π‘˜ subscript 𝑐 𝑛 1 π‘˜ {\displaystyle{\displaystyle c_{n}=\frac{1}{n+1}\left(c_{n-1}-\sum_{k=2}^{n-1}% kc_{k}c_{n+1-k}\right)}}
c_{n} = \frac{1}{n+1}\left(c_{n-1}-\sum_{k=2}^{n-1}kc_{k}c_{n+1-k}\right)		 n β‰₯ 2 𝑛 2 {\displaystyle{\displaystyle n\geq 2}} 
c[n] = (1)/(n + 1)*(c[n - 1]- sum(k*c[k]*c[n + 1 - k], k = 2..n - 1))
 
Subscript[c, n] == Divide[1,n + 1]*(Subscript[c, n - 1]- Sum[k*Subscript[c, k]*Subscript[c, n + 1 - k], {k, 2, n - 1}, GenerateConditions->None])
 Skipped - no semantic math Skipped - no semantic math - -
4.13.E9 1 β‹… 3 β‹… 5 ⁒ β‹― ⁒ ( 2 ⁒ n + 1 ) ⁒ c 2 ⁒ n + 1 = g n β‹… 1 3 5 β‹― 2 𝑛 1 subscript 𝑐 2 𝑛 1 subscript 𝑔 𝑛 {\displaystyle{\displaystyle 1\cdot 3\cdot 5\cdots(2n+1)c_{2n+1}=g_{n}}}
1\cdot 3\cdot 5\cdots(2n+1)c_{2n+1} = g_{n}		 

1 * 3 * 5*(2*n + 1)*c[2*n + 1] = g[n]
 
1 * 3 * 5*(2*n + 1)*Subscript[c, 2*n + 1] == Subscript[g, n]
 Skipped - no semantic math Skipped - no semantic math - -
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