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Lambert W Function and Its Derivatives

Source: R/lambert_w_function.R
lambert_w_function.Rd

Functions to compute the Lambert W function and its derivatives for the principal branch (W_0) and the branch -1 (W_-_1).

The Lambert W function is the solution of:

$$W(z) \cdot e^{W(z)} = z$$

Branches:

The function has two real branches:

  • W_0 (Principal Branch):

    • lambert_w0(z): Returns the principal branch value

    • lambert_w0_prime(z): Returns the derivative of W_0

    • For z >= 0, there is a single real solution

  • W_-_1 (Secondary Branch):

    • lambert_wm1(z): Returns the -1 branch value

    • lambert_wm1_prime(z): Returns the derivative of W_-_1

    • Exists where two real solutions occur on (-1/e, 0)

    • As z approaches 0, W_-_1(z) approaches -Inf

Usage

lambert_w0(z)

lambert_wm1(z)

lambert_w0_prime(z)

lambert_wm1_prime(z)

Arguments

z

Argument of the Lambert W function

Value

A single numeric value with the computed Lambert W function or its derivative.

See also

Boost Documentation for more details on the mathematical background.

Examples

# Lambert W Function (Principal Branch)
lambert_w0(0.3)
#> [1] 0.2367553
# Lambert W Function (Branch -1)
lambert_wm1(-0.3)
#> [1] -1.781337
# Derivative of the Lambert W Function (Principal Branch)
lambert_w0_prime(0.3)
#> [1] 0.6381087
# Derivative of the Lambert W Function (Branch -1)
lambert_wm1_prime(-0.3)
#> [1] -7.599525