Lambert W function
In mathematics, the Lambertian W - function (or Lambert -W function), and omega function, named after Johann Heinrich Lambert, the inverse function of
The exponential function. The Lambertian W function is usually denoted by. It is
- 3.1 generalizations
Properties
Since the function on the interval is not injective, has the Lambertian W function on the interval two functional branches and. With, however, generally referred to the upper one of the branches.
The W- function can not be expressed as an elementary function.
Mostly it is used in the field of combination, for example, for evaluation of trees or asymptotic determination of the Bell numbers.
The derivative function ( the upper Funktionsastes ) of the W- function can be found by means of the set on the derivative of the reverse function ( at the position -1 / E does not exist, the derivative):
The higher order derivatives have the form
The polynomials, which can be calculated from the following recursion formula:
Based arise so that the next three leads to:
A primitive function is obtained by substitution of the entire integrand:
By implicit differentiation, one can show that W satisfies the following differential equation:
The Taylor series of in is given by
The radius of convergence is.
Special values
Properties
Use outside of Combinatorics
The Lambertian W function can be used to equations of the type
To solve ( is any of correlated expression ).
The equation
Can be solved by means of the Lambert W function. The solution is
The infinite (infinite ) power tower
Can be brought to the convergent points of the wireless function in closed form, which allows the derivative:
Generalizations
Using the normal Lambertian W - function the exact solutions " transcendental algebraic " equations express the following form ( in x):
Wherein a0, c and r are real constants. The solution is. Generalizations of the Lambertian W function include:
- An application in the field of general relativity and quantum mechanics ( quantum gravity ) in lower dimensions, which demonstrated a previously unknown link between the two areas, see Journal of Classical and Quantum Gravity, where the right hand side of ( 1 ) is now a quadratic polynomial in x is:
- Analytical solutions of the energy eigenvalues for a special case of quantum-mechanical analogue of the Euler's three-body problem, namely, the ( three -dimensional ) hydrogen molecule ion. Now here is the right side of ( 1) (or ( 2) ) is the ratio of two polynomials of infinite order in x:
Numerical
A sequence of approximations to the wireless function can recursively using the relationship
Be calculated. Alternatively, the Newton's method may be used to solve the equation:
Table real function values
Upper branch:
Lower branch:
Other values can be easily calculated.
An approximation of large is