Lambert W function

The Lambert W function is the multivalued inverse of the map x ↦ xeˣ, defined by W(z)e^{W(z)} = z. Its branches describe the distinct solutions of this equation.

Lambert W function branches: The branches are the distinct analytic inverse functions solving W(z)e^{W(z)} = z. They account for the multiple values that make the inverse multivalued.
Exponential equation: An exponential equation is an equation in which an unknown occurs in an exponent. Rearranging many such equations produces the defining form u eᵘ = c.
Exponential function: The exponential function maps x to eˣ and extends naturally from real to complex inputs. Multiplying this function by its input creates the map that Lambert W inverts.
Johann Heinrich Lambert: Johann Heinrich Lambert was an eighteenth-century mathematician and scientist who studied equations involving exponentials. His work included equations closely related to the one defining the function named for him.
Principal branch: The principal branch W₀ is the branch of Lambert W that is real for real z ≥ −1/e. It supplies the commonly used real-valued inverse on the function’s increasing segment.
Wright omega function: The Wright omega function solves ω + ln ω = z using a branch-aware convention. It provides a related way to evaluate equations that Lambert W expresses through a logarithm.
Inverse function: An inverse function reverses the input-output relationship of a function on a domain where it is one-to-one. Lambert W extends ordinary inversion to a map that is not globally one-to-one.
Leonhard Euler: Leonhard Euler was an eighteenth-century mathematician whose work shaped analysis, number theory, and mechanics. Euler studied equations of the form xᵅ = eˣ that connect to Lambert W.
Branch point: A branch point is a point around which analytic continuation can change a function’s value. At z = −1/e, the real branches meet and the inverse develops a branch point.
Delay differential equation: A delay differential equation relates a function’s derivative to values of the function at earlier times. Characteristic roots of linear delay equations can be expressed using different Lambert W branches.

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