Lambert W function - Knowino (2024)

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The Lambert W function is used in mathematics to solve equations in which the unknown appears both outside and inside an exponential function or a logarithm, such as 3x + 2 = ex or x = ln(4x). Such equations cannot otherwise, except in special cases, be solved explicitly in terms of algebraic operations, exponentials and logarithms.

Contents

  • 1 Definition
  • 2 Examples of use
  • 3 Calculus
  • 4 Numerical calculation
  • 5 History and application
  • 6 See also
  • 7 References

[edit] Definition

The Lambert W function is defined as the multivalued function W that satisfies

Lambert W function - Knowino (2)

for any complex number Lambert W function - Knowino (3). Equivalently, it may be defined as the inverse function of f(w) = wew. Thus, the equation C = wew is by definition solved by Lambert W function - Knowino (4). The multivaluedness of the Lambert W function means that there is generally more than one solution. The graph of the Lambert W function in the real numbers looks as follows:

The function has two real branches in the interval − 1 / e < x < 0 which join at x = − 1 / e. Concretely, this means that the equation x = wew has two real solutions if − 1 / e < x < 0. For example, if x = − 0.15 (which is about half way between − 1 / e and 0), there is one solution Lambert W function - Knowino (6) that lies on the blue graph and another solution Lambert W function - Knowino (7) that lies on the dashed red graph.

The single-valued function corresponding to the blue graph for Lambert W function - Knowino (8) is called the principal branch of the Lambert W function and is denoted by W0. The single-valued function corresponding to the dashed red graph for Lambert W function - Knowino (9) is called the negative branch, denoted by W − 1. The negative branch goes to Lambert W function - Knowino (10) as Lambert W function - Knowino (11) while the principal branch grows slowly but unboundedly as Lambert W function - Knowino (12).

The behavior of the Lambert W function can be understood by comparing it to the natural logarithm, the inverse of ew. For large negative or positive w, ew and wew grow similarly, so their respective inverse functions have similar asymptotes as well (except for the sign when w is negative). Multiplying the exponential by w deforms its graph around zero so that it is no longer monotone, and that is why the Lambert W function has two real branches: one for values on each side of the stationary point. The following pair of graphs shows the relation between the two exponential functions (left graph) and their inverse functions (right graph):

Like the complex logarithm, the Lambert W function has infinitely many complex branches; they are conventionally labeled Wk where k runs over all the integers. The details are discussed for instance in Corless et al.[1]

Besides W( − 1 / e) = − 1, the Lambert W function has the special values W0(0) = 0 and W0(e) = 1. The value Lambert W function - Knowino (14) is called the omega constant.

[edit] Examples of use

The Lambert W function solves any equation of the form C = xex — we may call this the canonical form. Many other types of equations can be solved in terms of the Lambert W function as well, by using the rules of exponentials and logarithms to rewrite them in the canonical form. For example, the equation xbx = a is solved by Lambert W function - Knowino (15) and ax = x + b is solved by

Lambert W function - Knowino (16)[1]

Valluri et al. give the following example of an equation from physics that can be solved with the Lambert W function.[2] Planck's law states that the radiation intensity I at wavelength λ from a black body at temperature T is given by

Lambert W function - Knowino (17)

where h is Planck's constant, k is Boltzmann's constant and c is the speed of light. Note that λ appears both inside and outside an exponential. Next, Wien's displacement law states that the maximum intensity is attained at the wavelength λmax = b / T where b is a quantity called Wien's displacement constant. Using the Lambert W function, we can give an explicit formula for b.

To derive Wien's displacement law, we wish to solve Lambert W function - Knowino (18). If we calculate the partial derivative, simplify, and perform the substitution x = hc / λkT, we obtain the equation (x − 5)ex = − 5. To write this in canonical form for application of the Lambert W function, we substitute w = x − 5 and multiply both sides of the resulting equation by e − 5. This leaves wew = − 5e − 5, with the nonzero solution w = W0( − 5e − 5). Substituting back the expression for x, we obtain Wien's displacement law with the value for Wien's constant given explicitly by

Lambert W function - Knowino (19)

[edit] Calculus

It is possible to do calculus with the Lambert W function much like with other elementary functions. Its properties can be derived from the familiar properties of exponentials and logarithms, although we have to use the theory of implicit functions since the Lambert W function is defined implicitly rather than by a direct formula.

Let us first consider limiting behavior. When discussing the graph of the Lambert W function, we said that its asymptotes are "similar" to those of the natural logarithm, since the exponential in wew dominates when w is large. A more precise statement is that

Lambert W function - Knowino (20)

By similar reasoning, the negative branch approaches the singularity at x = 0 logarithmically:

Lambert W function - Knowino (21)

However, the absolute difference between the Lambert W function and the natural logarithm, Lambert W function - Knowino (22), diverges to infinity as Lambert W function - Knowino (23). It is possible to give a more complicated asymptotic formula for the Lambert W function that eliminates this difference, in the form of an infinite series

Lambert W function - Knowino (24)

where

Lambert W function - Knowino (25)

and s(n,k) denotes a Stirling number of the first kind. The series is absolutely convergent for sufficiently large x.[3]

The Taylor series of the Lambert W function can be found by using the Lagrange inversion theorem to invert the series of wew. This way, we find that the principal branch of the Lambert W function has the Taylor series expansion

Lambert W function - Knowino (26)

around x = 0. Due to the singularity at x = − 1 / e, and as can be proved with the ratio test, the series converges for | x | < 1 / e. The Lambert W function further has the derivative

Lambert W function - Knowino (27)

which is infinite at the branch point x = − 1 / e. The indefinite integral is given by

Lambert W function - Knowino (28)

It is also possible to find integrals of more complicated expressions containing the Lambert W function. See Corless et al. for an overview.[1]

[edit] Numerical calculation

The value of W(z) can be calculated by using a regular root-finding algorithm to solve the equation wewz = 0 for w. Since the left-hand side has simple derivatives, Newton's method and Halley's method are both good choices. Halley's method amounts to iterating

Lambert W function - Knowino (29)

for Lambert W function - Knowino (30). The initial value w0 must be chosen carefully in order to end up on the correct branch. To calculate real values on the principal branch, any real number above − 1 will work, but faster convergence is obtained by taking the initial value from an interpolating function around 0 for small arguments and a few terms of the asymptotic series for large arguments. With a sufficiently accurate initial value, a few iterations of Halley's method will give a value that is correct to full precision in ordinary floating-point arithmetic.

Of course, we could use standard root-finding techniques to solve equations without going through the trouble of rewriting them in terms of the Lambert W function. From the numerical point of view, the advantage of using the Lambert W function is that highly optimized implementations that have been tested for correctness are available; further, those implementations allow the user to select a complex branch of choice.

[edit] History and application

Equations of the kind that can be solved analytically with the Lambert W function are common in mathematics and science, yet the utility of such a function was not realized until recently. The Lambert W function was introduced in the 1980's as a function in the Maple computer algebra system, whose interface required an explicit notation for solutions of equations. The function's history highlights the importance of good mathematical notation: due to previously not being recognized as a function in its own right, it had not been studied systematically, despite its most important properties requiring only elementary complex analysis. An account of the function and its history that helped popularize it is given in a 1996 paper by R. M. Corless et al. (with Donald Knuth a notable co-author).[1]

The basic theory behind the Lambert W function was investigated in 1779 by Leonhard Euler.[4] The Maple developers chose the name of Johann Heinrich Lambert instead of Euler's since Euler had referenced work by Lambert in his paper, and possibly because "naming yet another function after Euler would not be useful".[3]

Since its introduction, the Lambert W function has been applied to problems ranging from quantum physics to population dynamics to the complexity of algorithms. Cranmer[5] discusses the application of the Lambert W function in solar wind physics and writes in the conclusion: "The Lambert W function used in these solutions was defined and publicized only about a decade ago, but it has rapidly become a convenient tool for mathematical physicists. The elegance of explicit solutions to equations thought previously to be expressible only implicitly is clear, but there also are many practical benefits to having explicit solutions as well."

[edit] See also

[edit] References

  1. 1.0 1.1 1.2 1.3 Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. & Knuth, D. E. (1996). "On the Lambert W function". Adv. Computational Maths. 5, 329–359
  2. Valluri, S. R., Jeffrey, D. R. & Corless, R. M. (2000). "Some applications of the Lambert W function to physics". Can. J. Phys. 78, 823–831
  3. 3.0 3.1 Corless, R., Jeffrey, D. & Knuth, D. E. (1997). "A Sequence of Series for the Lambert W Function", ISSAC: Proceedings of the ACM SIGSAM International Symposium on Symbolic and Algebraic Computation (formerly SYMSAM, SYMSAC, EUROSAM, EUROCAL) (also sometimes in cooperation with the Symbolic and Algebraic Manipulation Groupe in Europe (SAME))
  4. Euler, L. (1779). "De serie Lambertina plurimisque eius insignibus proprietatibus". Originally published in Acta Academiae Scientarum Imperialis Petropolitinae 1779, 1783, 29–51. Also in Opera Omnia: Series 1, Volume 6, pp. 350 - 369. See E532 in The Euler Archive for a scanned copy.
  5. Cranmer, S. R. (2004). "New views of the solar wind with the Lambert W function". Am. J. Phys. 72, 1397
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Category: Mathematics

Lambert W function - Knowino (2024)

FAQs

What is the Lambert W function formula? ›

The Lambert W function W(x) represents the solutions y of the equation y e y = x for any complex number x . For complex x, the equation has an infinite number of solutions y = lambertW(k,x) where k ranges over all integers. For all real x ≥ 0, the equation has exactly one real solution y = lambertW(x) = lambertW(0,x).

What is the Lambert wave function? ›

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 f(w) = wew, where w is any complex number and ew is the exponential function.

What is the Lambert W function in R? ›

The Lambert W function is the inverse of x --> x e^x , with two real branches, W0 for x >= -1/e and W-1 for -1/e <= x < 0 . Here the principal branch is called lambertWp , tho other one lambertWp , computed for real x . The value is calculated using an iteration that stems from applying Halley's method.

What is the generalized Lambert W function? ›

The Lambert W function gives the solutions of a simple exponential polynomial. The generalized Lambert W function was defined by Mezö and Baricz, and has found applications in delay differential equations and physics.

What is the representation of the Lambert W function? ›

The Lambert W function is the many-valued analytic inverse of z(w)=we w . We use elementary complex analysis to derive closed-form representations of all of the branches of W through simple quadratures.

What is the Lambert W function for applications in physics? ›

The Lambert function and its possible applications in physics are presented. The actual numerical implementation in C++ consists of Halley's and Fritsch's iterations with initial approximations based on branch-point expansion, asymptotic series, rational fits, and continued-logarithm recursion.

What is the principle of the Lambert law? ›

Lambert's Law

This expression says that the absorbance of light in a hom*ogenous material/medium is directly proportional to the thickness of the material/medium.

What is Lambert's theorem commonly used for? ›

Hohmann Transfer Orbits: Lambert's theorem is applied in the planning of Hohmann transfer orbits, which are fuel-efficient trajectories used for transferring between two circular orbits.

Is Lambert W an elementary function? ›

Solutions to a wide variety of transcendental equations can be expressed in terms of the Lambert W function. The W function, also occurring frequently in many branches of science, is a non-elementary but now standard mathematical function implemented in all major technical computing systems.

What is the real branch of the Lambert W function? ›

W0(x) is the upper branch and W-1(x) is the lower branch. Transcendental functions are a fundamental building block of science and engineering.

What is the inverse of the Lambert W function? ›

The inverse of this function is called the Lambert W_{-1} function. It is represented by its maroon graph in the picture below. Mathematica's notation for W_0(x) function is ProductLog[x], or, equivalently ProductLog[0,x]. Mathematica's notation for W_{-1}(x) function is ProductLog[-1,x].

What is the Lambert W function in ecological and evolutionary models? ›

The Lambert W function expands the range of explicitly solvable models in ecology and evolution, appears in a surprising number of problems in mathematical biology and is therefore a useful addition to the toolkit of modellers in these fields.

What is the Lambert W function Quora? ›

The Lambert W function is known in mathematics as the inverse of f(x)=xex f ( x ) = x e x ; y=xex⇔x=W(y) y = x e x ⇔ x = W ( y ) .

What is the Lambertian formula? ›

In optics, Lambert's cosine law says that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle θ between the observer's line of sight and the surface normal; I = I0 cos θ.

What is the formula for the Lambert distance law? ›

Matter intervening in the path of a radiation beam that absorbs energy from that beam. Coefficient in Lambert's law, I = Ioexp(−μd), which describes the attenuation of the radiation penetrating an absorber.

What is the formula for shift function? ›

To shift, move, or translate horizontally, replace y = f(x) with y = f(x + c) (left by c) or y = f(x - c) (right by c).

What is the Beer Lambert concentration formula? ›

The Beer–Lambert law relates the absorption of light by a solution to the properties of the solution according to the following equation: A = εbc, where ε is the molar absorptivity of the absorbing species, b is the path length, and c is the concentration of the absorbing species.

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