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# ProductLog

 ProductLog[z]gives the principal solution for w in . ProductLog[k, z]gives the k solution.
• Mathematical function, suitable for both symbolic and numerical manipulation.
• The solutions are ordered according to their imaginary parts.
• 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[z] has a branch cut discontinuity in the complex z plane running from to . ProductLog[k, z] for integer has a branch cut discontinuity from to 0.
 Scope   (7)
Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
The precision of the output can be much lower than the precision of the input:
Simple exact values are generated automatically:
ProductLog threads element-wise over lists:
Find series expansions at branch points and branch cuts:
The series expansion at infinity contains nested logarithms:
Evaluate numerically on different sheets of the Riemann surface:
Find series expansions at branch points and branch cuts:
The branch points and branch cuts are different for :
 Applications   (9)
Plot the real and imaginary parts of ProductLog:
Plot the Riemann surface of ProductLog:
Calculate the limit of :
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:
Equipotential curves of a plate capacitor:
Compositions with the inverse function may need PowerExpand:
Use FullSimplify to simplify expressions containing ProductLog:
Solve a transcendental equation:
Integrals:
Generically :
On branch cuts machine-precision inputs can give numerically wrong answers:
Use arbitrary-precision arithmetic to get correct results:
Nested derivatives:
Nested integrals:
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