W0

W0 is a designation used in several scientific contexts, with meanings that depend on the discipline. The most common usages concern mathematics and astrophysics, where W0 denotes a specific, widely used quantity or function.

In mathematics, W0 denotes the principal branch of the Lambert W function. The Lambert W function is the inverse of the transcendental relation f(W) = W e^W. The principal branch W0 is defined for complex numbers z with z ≥ -1/e and satisfies W0(z) e^{W0(z)} = z. The values include W0(0) = 0 and W0(1) ≈ 0.5671432904, the omega constant. The function has other branches, noted as Wk for integers k ≠ 0, most notably W_{-1} defined on [-1/e, 0). The Lambert W function is used to solve equations of the form z = W e^W and appears in various areas of physics, chemistry, and combinatorics.

In astrophysics, W0 denotes the dimensionless central potential in King models, which are mathematical descriptions of self-gravitating stellar systems such as globular clusters. W0 determines the concentration and the shape of the model's density profile; higher W0 values correspond to more centrally concentrated systems with a steeper outer cutoff. Typical applications use W0 values that span a range reflecting different evolutionary states and mass distributions.

The notation W0 can also appear in other fields as a label for a principal quantity, a branch index, or a baseline parameter. When encountered, it generally signals a zero-subscript or principal variant of a broader family of related quantities.