sqrtw2

sqrtw2 is the shorthand for the expression sqrt(w^2), the square root of the square of a variable w. In real-number contexts, this quantity equals the absolute value of w, denoted |w|, and is therefore always nonnegative. This makes sqrtw2 a convenient way to express a nonnegative magnitude derived from w.

For real w, the value is determined by the sign of w: if w is nonnegative, sqrtw2 = w; if w is negative, sqrtw2 = -w. In other words, sqrt(w^2) equals w when w ≥ 0 and equals -w when w < 0.

In complex analysis, the situation is more subtle. The square root is a multivalued operation, and sqrt(w^2) is not generally equal to w. The principal square root function introduces a branch cut, so sqrt(z^2) can differ from z depending on z’s position in the complex plane. Consequently, algebraic simplifications like sqrt(w^2) → w are not universally valid in complex arithmetic.

Computational notes: many computer algebra systems simplify sqrt(w^2) to Abs(w) (the absolute value) rather than w, reflecting its real-number interpretation. In programming, expressions like sqrt(w*w) typically yield a nonnegative result for real w. When w represents a matrix or higher-dimensional object, sqrt(w^2) becomes ambiguous and a separate concept—the matrix square root—must be used, with different properties from the scalar case.

Relation to norms: for a real scalar, sqrt(w^2) is the Euclidean norm of a one-dimensional vector and equals |w|. More generally, sqrt(x^T x) represents the L2 norm, ||x||_2.