gives the principal solution for w in .
gives the k solution.
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The solutions are ordered according to their imaginary parts.
- For , ProductLog[z] is real.
- ProductLog[z] satisfies the differential equation .
- For certain special arguments, ProductLog automatically evaluates to exact values.
- ProductLog can be evaluated to arbitrary numerical precision.
- ProductLog automatically threads over lists.
- ProductLog[z] has a branch cut discontinuity in the complex z plane running from to .
- ProductLog[k,z] allows k to be any integer, with corresponding to the principal solution.
- ProductLog[k,z] for integer has a branch cut discontinuity from to 0.
- ProductLog can be used with Interval and CenteredInterval objects. »
Examplesopen allclose all
Basic Examples (6)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Asymptotic expansions at Infinity:
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
ProductLog threads elementwise over lists and matrices:
ProductLog can be used with Interval and CenteredInterval objects:
Specific Values (4)
Values of ProductLog at fixed points:
Find a value of x for which the ProductLog[x]=0.5 using FindRoot:
Plot the ProductLog function:
Function Properties (10)
ProductLog is defined for all real values from the interval [-,∞):
ProductLog is defined for all complex values:
The two-argument form requires that be an integer and :
ProductLog is not an analytic function:
ProductLog is increasing on its real domain:
ProductLog is injective:
ProductLog is not surjective:
ProductLog is neither non-negative nor non-positive:
ProductLog has both singularity and discontinuity in (-∞,-]:
ProductLog is concave on its real domain:
Compute the indefinite integral using Integrate:
Definite integral of ProductLog:
Series Expansions (5)
Find the Taylor expansions using Series:
Plots of the first three approximations around :
The general term in the series expansion using SeriesCoefficient:
Find the series expansion at Infinity:
Find series expansions at branch points and branch cuts:
The series expansion at infinity contains nested logarithms:
Function Identities and Simplifications (2)
ProductLog gives the solution for the following equation:
Generalizations & Extensions (3)
Solve an equation in terms of ProductLog:
Plot the real and imaginary parts of ProductLog:
Plot the Riemann surface of ProductLog:
Compare the exact result with explicit iterations for :
Determine the number of labeled unrooted trees from the generating function:
Solve Lotka–Volterra equations:
Find the frequency of the maximum of the blackbody spectrum:
Solve the Haissinski equation:
Properties & Relations (5)
Compositions with the inverse function may need PowerExpand:
Use FullSimplify to simplify expressions containing ProductLog:
Solve a transcendental equation:
Possible Issues (2)
Wolfram Research (1996), ProductLog, Wolfram Language function, https://reference.wolfram.com/language/ref/ProductLog.html (updated 2022).
Wolfram Language. 1996. "ProductLog." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/ProductLog.html.
Wolfram Language. (1996). ProductLog. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ProductLog.html