Lambert W function - MATLAB lambertw (2024)

FAQs

What is the Lambert W function equal to? ›

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) = we^w, where w is any complex number and e^w is the exponential 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.

The branch corresponding to k = 0 is the principal branch. In mathematical notation, the k is usually a subscript, i.e W₀(z) is the principal branch of the Lambert W function. For real z ≥ -1/e it returns a real value.

S = solve( eqn , var ) solves the equation eqn for the variable var . If you do not specify var , the symvar function determines the variable to solve for. For example, solve(x + 1 == 2, x) solves the equation x + 1 = 2 for x.

Create the transfer function G ( s ) = s s 2 + 3 s + 2 : num = [1 0]; den = [1 3 2]; G = tf(num,den); num and den are the numerator and denominator polynomial coefficients in descending powers of s. For example, den = [1 3 2] represents the denominator polynomial s² + 3s + 2.

The Lambert W function W(z) is the inverse function of f(w) = wew, meaning that W(z)eW(z) = z holds for any z.

The Lambert W function, although not familiar to many engi- late 1700s. W(z) is the function that solves the equation As stated in [2], even if z is real, the branches other than p = 0 and p-1 are always complex. where, in general, z is a complex number.

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.

The Lambert W function is defined as the multivalued inverse of the function w↦we^w. It has a wide range of applications. We propose a new method to construct a high-precision analytical approximation of the two branches of W. The method is based on Padé approximation and Schröder's iteration.

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 real branch of the Lambert W function? ›

Figure 1 Real branches of the Lambert W function. The solid line is the principal branch W0; the dashed line is W−1, which is the only other branch that takes real values. The small filled circle at the branch point corresponds to the one in figure 2. The Wright ω function helps to solve the equation y +ln y = z.

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 ) . The following alternative definition is often used to solve Equations of type of y=f(x)^e^f(x):

The basic idea of principal functions is simple: Given a Riemann surface or a Riemannian space R, a neighborhood A of its ideal boundary, and a harmonic function s on A, the principal function problem consists in constructing a harmonic function p on all of R which imitates the behavior of s in A.

Th = THETA(TYPE,V,M) returns values of theta functions evaluated for corresponding values of argument V and parameter M. TYPE is a type of the theta function, there are four numbered types. The arrays V and M must be the same size (or either can be scalar).

T = taylor( f , var ) approximates f with the Taylor series expansion of f up to the fifth order at the point var = 0 . If you do not specify var , then taylor uses the default variable determined by symvar(f,1) . T = taylor( f , var , a ) approximates f with the Taylor series expansion of f at the point var = a .

x = square( t ) generates a square wave with period 2π for the elements of the time array t . square is similar to the sine function but creates a square wave with values of –1 and 1. x = square( t , duty ) generates a square wave with specified duty cycle duty .