Built-in Mathematica Symbol

ProductLog

ProductLog[z]
gives the principal solution for w in

.

ProductLog[k, z]
gives the k

solution.

MORE INFORMATION

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] for integer has a branch cut discontinuity from to 0.

EXAMPLES

CLOSE ALL

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:

TraditionalForm formatting:

Generalizations & Extensions (3)

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:

Properties & Relations (5)

Compositions with the inverse function may need PowerExpand:

Use FullSimplify to simplify expressions containing ProductLog:

Solve a transcendental equation:

Integrals:

Possible Issues (2)

Generically

:

On branch cuts machine-precision inputs can give numerically wrong answers:

Use arbitrary-precision arithmetic to get correct results:

Neat Examples (2)

Nested derivatives:

Nested integrals: