The Lambert W function, also known as the omega function or the product logarithm, is a special function in mathematics that represents the solution to the equation z = w * exp(w), where w is a complex number. It is denoted by W(z) or W_n(z), where n is an integer representing the branch of the function.
The Lambert W function has a wide range of applications in various fields including mathematics, physics, engineering, and finance. It is particularly useful in solving equations that involve exponential and logarithmic terms since it combines both these functions.
The function is named after Johann Heinrich Lambert, who first introduced the concept in the mid-18th century. Lambert W function is an important tool in the study of transcendental equations and has connections to many other mathematical functions such as exponential, logarithmic, and trigonometric functions.
The Lambert W function is defined as the inverse function of the function f(w) = w * exp(w). In other words, if z = f(w), then w = W(z). The function has several branches and is multivalued for some values of z. The different branches of the function are represented by W_n(z), where n is an integer.
The Lambert W function has been extensively studied and various properties and formulas related to it have been derived. It plays a significant role in solving equations and optimizing functions that involve exponential and logarithmic terms.
