The Lambert function
This functionality does not run in MATLAB.
lambertW(x) lambertW(k, x)
For integer k, the values represent the solutions of the equation y ey = x.
lambertW is the inverse function of .
In the complex plane, the equation y ey = x has a countably infinite number of solutions. They are represented by lambertW(k, x) with k ranging over the integers.
For all real x ≥ 0, the equation has exactly one real solution. It is represented by y=lambertW(x) or, equivalently, y=lambertW(0, x).
For all real x in the range 0 > x, there are exactly two real solutions. The larger one is represented by y=lambertW(x), the smaller one by y=lambertW(-1, x).
Exactly one real solution lambertW(0, -exp(-1))= lambertW(-1, -exp(-1))= -1 exists for .
For , lambertW(k, x) takes no real value.
The values lambertW(-1, 0)=- infinity and lambertW(0, 0)=0 are implemented. Further, the result y is returned for some exact arguments of the form . For floating-point arguments a floating-point value is returned. For all other arguments, unevaluated function calls are returned.
The float attributes are kernel functions, i.e., floating-point evaluation is fast.
When called with a floating-point argument, the function is sensitive to the environment variable DIGITS which determines the numerical working precision.
We demonstrate some calls with exact and symbolic input data:
lambertW(-3), lambertW(-1, -5/2), lambertW(1/2), lambertW(5, I), lambertW(3, 1 + I), lambertW(-1, x + 1)
Some exact values are found:
lambertW(-1, -exp(-1)), lambertW(-1, -2*exp(-2)), lambertW(-1, -3/2*exp(-3/2)), lambertW(exp(1)), lambertW(2*exp(2)), lambertW(5/2*exp(5/2)), lambertW(1, (3+4*I)*exp(3+4*I))
Floating point values are computed for floating-point arguments:
lambertW(-1, -0.3), lambertW(2000.0)
lambertW(-3, -0.277), lambertW(1, 2345.6)
diff(lambertW(k, x), x)
float(ln(3 + lambertW(sqrt(PI))))
series(lambertW(x), x = 0); series(lambertW(x), x = -1/exp(1), 3); series(lambertW(-1, x), x = -1/exp(1), 3);
An arithmetical expression, the "argument"
An arithmetical expression representing an integer, the "branch"