Lambert's W function
In mathematics, Lambert's W function, named after Johann Heinrich Lambert, also called the Omega function, is the inverse function of f(w) = w.ew for complex numbers w; where ew is the exponential function.
This means that for every complex number z, we have
- W(z) eW(z) = z
Since the function f
is not injective
in (−∞, 0), the function W
is multivalued in [−1/e
, 0). If we restrict to real arguments x
and demand w
≥−1, then a single valued function W0
) is defined, whose graph is shown. We have W0
(0) = 0 and W0
) = −1.
The Lambert W
function cannot be expressed in terms of elementary functions
. It is useful in combinatorics
, for instance in the enumeration of trees
. It can be used to solve various equations involving exponentials and also occurs in the solution of time-delayed differential equations, such as y
) = a y
By implicit differentiation, one can show that W satisfies the differential equation
- z (1 + W) dW/dz = W for z ≠ −1/e.
The Taylor series
around 0 can be found using the Lagrange inversion theorem
and is given by
The radius of convergence
, as may be seen by the ratio test
. The function defined by this series can be extended to a holomorphic function
defined on all complex numbers except the real interval
]; this holomorphic function is also called the prinicipal branch
of the Lambert W
Many equations involving exponentials can be solved using the W function. The general strategy is to move all instances of the unknown to one side of the equation and make it look like x ex, at which point the W function provides the solution. For instance, to solve the equation 2t = 5t, we divide by 2t to get 1 = 5t e-ln(2)t, then divide by 5 and multiply by -ln(2) to get -ln(2)/5 = -ln(2)t e-ln(2)t. Now application of the W function yields −ln(2)t = W(−ln(2)/5), i.e. t = −W(−ln(2)/5) / ln(2).
Similar techniques show that has solution .
The function W(x), and many expressions involving W(x), can be integrated using the substitution w = W(x), i.e. x = w ew:
See also: Omega constant