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 ≥ −1/
e and demand
w≥−1, then a single valued function
W0(
x) is defined, whose graph is shown. We have
W0(0) = 0 and
W0(−1/
e) = −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'(
t) =
a y(
t − 1).
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 of
W0 around 0 can be found using the
Lagrange inversion theorem and is given by
The
radius of convergence is 1/
e, 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 (-∞, -1/
e]; this holomorphic function is also called the
prinicipal branch of the Lambert
W function.
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
References: