Lambertderived

Lambertderived is a term used in mathematics to describe expressions, equations, or methods that arise from, or can be solved by, using the Lambert W function. It characterizes results that are reduced to a Lambert W form or obtained through substitutions that invoke W.

Origin and naming: The Lambert W function is the inverse of the function f(w) = w e^w and was introduced in the 18th century by Johann Heinrich Lambert to solve equations of the type x e^x = a. In modern usage, the function was popularized and named alongside its notation by Corless, Gonnet, Hare, and Leaf in a 1996 survey, and the adjective Lambertderived reflects techniques built on this function.

Definition and scope: A problem is considered Lambertderived when it can be transformed into a form solvable by the Lambert W function, typically equations that reduce to something proportional to w e^w = c after a substitution. The resulting solutions often involve W with appropriate branches.

Example: Solve x e^{2x} = 6. Set y = 2x to obtain (y/2) e^{y} = 6, which implies y e^{y} = 12, hence y = W(12) and x = W(12)/2.

Branches and limitations: Lambert W is multivalued; real solutions may require selecting W_0 (principal) or W_{-1} branches depending on the argument. Lambertderived methods must account for branch structure and domain constraints.

Applications: Lambertderived methods appear in physics, chemistry, and engineering, including solving delay differential equations, reaction kinetics, growth models, and control problems where exponential terms appear.

See also: Lambert W function, transcendental equation, inverse function, productlog.