In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function , where is any complex number and is the exponential function. The function is named after Johann Lambert, who considered a related problem in 1758. Building on Lambert's work, Leonhard Euler described the W function per se in 1783.
For each integer there is one branch, denoted by , which is a complex-valued function of one complex argument. is known as the principal branch. These functions have the following property: if and are any complex numbers, then
holds if and only if
When dealing with real numbers only, the two branches and suffice: for real numbers and the equation
can be solved for only if ; yields if and the two values and if .
The Lambert W function's branches 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 (e.g. the maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions) and also occurs in the solution of delay differential equations, such as . In biochemistry, and in particular enzyme kinetics, an opened-form solution for the time-course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.
Terminology The notation convention chosen here (with and ) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth.
The name "product logarithm" can be understood as follows: since the inverse function of is termed the logarithm, it makes sense to call the inverse "function" of the product the "product logarithm". Like the complex logarithm, it is multivalued and thus W is described as a converse relation rather than inverse function. It is related to the omega constant, which is equal to .
History Lambert first considered the related Lambert's Transcendental Equation in 1758, which led to an article by Leonhard Euler in 1783 that discussed the special case of .
The equation Lambert considered was
Euler transformed this equation into the form
Both authors derived a series solution for their equations.
Once Euler had solved this equation, he considered the case . Taking limits, he derived the equation
He then put and obtained a convergent series solution for the resulting equation, expressing in terms of .
After taking derivatives with respect to and some manipulation, the standard form of the Lambert function is obtained.
In 1993, it was reported that the Lambert function provides an exact solution to the quantum-mechanical double-well Dirac delta function model for equal charges—a fundamental problem in physics. Prompted by this, Rob Corless and developers of the Maple computer algebra system realized that "the Lambert W function has been widely used in many fields, but because of differing notation and the absence of a standard name, awareness of the function was not as high as it should have been."
Another example where this function is found is in Michaelis–Menten kinetics. Although it was widely believed that the Lambert function cannot be expressed in terms of elementary (Liouvillian) functions, the first published proof did not appear until 2008.
Elementary properties, branches and range
There are countably many branches of the function, denoted by , for integer ; being the main (or principal) branch. is defined for all complex numbers z while with is defined for all non-zero z, with and for all .
The branch point for the principal branch is at , with the standard branch cut extending along the negative real axis to . This branch cut separates the principal branch from the two branches and . In all branches with , there is a branch point at and a branch cut is conventionally taken along the entire negative real axis.
The functions are all injective and their ranges are disjoint. The range of the entire multivalued function is the complex plane. The image of the real axis is the union of the real axis and the quadratrix of Hippias, the parametric curve .
Inverse
The range plot above also delineates the regions in the complex plane where the simple inverse relationship is true. implies that there exists an such that where depends upon the value of
For real values of the value of the integer changes abruptly when is at the branch cut of which means that except for where it is .
For complex define where and are real, and expressing in polar coordinates, it is seen that
For the branch cut for is the non-positive real axis, so that
and
For the branch cut for is the real axis with so that the inequality becomes or
Inside the regions bounded by the above, there are no discontinuous changes in and those regions specify where the function is simply invertible, i.e.
Transcendence
For each algebraic number , the numbers are transcendental. This can be proved as follows. Suppose that is algebraic. Then by the Lindemann–Weierstrass theorem we have is transcendental, but which is algebraic, giving a contradiction.
Calculus
Derivative By implicit differentiation, one can show that all branches of satisfy the differential equation
( is not differentiable for .) As a consequence, one arrives at the following formula for the derivative of W:
Using the identity , gives the following equivalent formula:
At the origin we have
The n-th derivative of is of the form:
Where is a polynomial function with coefficients defined in . If and only if is a root of , then is a root of the n-th derivative of .
Taking the derivative of the n-th derivative of yields:
Inductively proving the n-th derivative equation.
Integral The function , and many other expressions involving , can be integrated using the substitution , i.e. :
(The last equation is more common in the literature but is undefined at ). One consequence of this (using the fact that ) is the identity
Asymptotic expansions By the Lagrange inversion theorem, the Taylor series of the principal branch around is:
The radius of convergence is by the ratio test, and the function defined by the series can be extended to a holomorphic function defined on all complex numbers except a branch cut along the interval .
For large values , the real function is asymptotic to
where , , and is a non-negative Stirling number of the first kind. Keeping only the first two terms of the expansion,
The other real branch, , defined in the interval , has an approximation of the same form as approaches zero, in this case with and .
Integer and complex powers Integer powers of also admit simple Taylor (or Laurent) series expansions at zero:
More generally, for , the Lagrange inversion formula gives
which is, in general, a Laurent series of order . Equivalently, the latter can be written in the form of a Taylor expansion of powers of :
which holds for any and .
Bounds and inequalities A number of non-asymptotic bounds are known for the Lambert function.
Principal branch Hoorfar and Hassani showed that the following bound holds for :
Roberto Iacono and John P. Boyd enhanced the bounds for as follows:
Hoorfar and Hassani also showed the general bound
for every and , with equality only for . The bound allows many other bounds to be derived, such as taking which gives the bound
Bounds for the function for are obtained by Stewart.
Roig-Solvas and Sznaier further showed that the principal branch can be bounded above and below in the interval as:
Secondary branch The branch can be bounded as follows:
Identities
A few identities follow from the definition:
Since is not injective, it does not always hold that , much like with the inverse trigonometric functions. For fixed and , the equation has two real solutions in , one of which is of course . Then, for and , as well as for and , is the other solution.
Some other identities:
(which can be extended to other and if the correct branch is chosen).
Substituting in the definition: "Note: although one of the assumptions of the relevant lemma states that x must be > 1/e, inspection of said lemma reveals that this assumption is unused. The lower bound is in fact x > 0. The reason for the branch switch at e is simple: for x > 1 there are always two solutions, −ln x and another one that you'd get from the x on the other side of e that would feed the same value to W; these must crossover at x = e: Wn cannot distinguish a value of ln x/x from an x < e from the same value from the other x > e, so it cannot flip the order of its return values."
With Euler's iterated exponential :
Special values The following are special values of the principal branch:
(the omega constant)
Special values of the branch :
Representations The principal branch of the Lambert function can be represented by a proper integral, due to Poisson:
Another representation of the principal branch was found by Kalugin–Jeffrey–Corless: See Theorem 3.4, p. 821 of published version (p. 5 of preprint).
The following continued fraction representation also holds for the principal branch:
Also, if :
In turn, if , then
Other formulas
Definite integrals There are several useful definite integral formulas involving the principal branch of the function, including the following:
where denotes the gamma function.
The first identity can be found by writing the Gaussian integral in polar coordinates.
The second identity can be derived by making the substitution , which gives
Thus
The third identity may be derived from the second by making the substitution and the first can also be derived from the third by the substitution . Deriving its generalization, the fourth identity, is only slightly more involved and can be done by substituting, in turn, , , and , observing that one obtains two integrals matching the definition of the gamma function, and finally using the properties of the gamma function to collect terms and simplify.
Except for along the branch cut (where the integral does not converge), the principal branch of the Lambert function can be computed by the following integral:
where the two integral expressions are equivalent due to the symmetry of the integrand.
Indefinite integrals
Applications
Solving equations
Linear functions and exponentials
The Lambert function is used to solve equations in which the unknown quantity occurs both in the base and in the exponent, or both inside and outside of a logarithm. The strategy is to convert such an equation into one of the form and then to solve for using the function.
For example, the equation
(where is an unknown real number) can be solved by rewriting it as
This last equation has the desired form and the solutions for real are:
and thus:
Generally, the solution to
is:
where , , and are complex constants, with and not equal to zero, and the function is of any integer order.
Similarly, the general solution to the intersection of linear functions and exponentials
can be deduced to be:
where and must be non-zero, and other domain and branch considerations are necessary.
Super root
Along with the topic of the so called Sophomore's dream the tetration function became a well known function. Its inverse function is a special case of the so called super root and it can be determined and displayed as follows:
The power rule gives following expression:
The natural logarithm of that is taken:
The Lambert W function is used now:
And in the final step the second last equation will be divided by the last equation:
A calculation example is made:
Tree counting and combinatorics