overlinephi

Overline φ, written as φ̄ or \overline{φ}, is a notation used in mathematics and related fields to denote the conjugate of a quantity named φ. Depending on context, φ̄ may indicate a complex conjugate, an algebraic conjugate, or other related notions.

In complex analysis, if φ is a complex-valued quantity, its conjugate φ̄ is obtained by changing the sign of the imaginary part. For a complex number z = a + bi, z̄ = a − bi. If φ is real-valued, φ̄ equals φ.

In algebraic number theory, φ and φ̄ may denote a pair of algebraic conjugates of a quadratic irrational. For example, the golden ratio φ = (1+√5)/2 has the conjugate φ̄ = (1−√5)/2. They satisfy φ + φ̄ = 1 and φ φ̄ = −1.

In other contexts, overline notation can denote an average or an adjoint operation. For instance, φ̄ might be used to denote the average value of φ over a region or, in field theory, the adjoint or antiparticle in some conventions. The exact meaning depends on the source.

Typography and notation: φ̄ is produced in typesetting by placing a bar over the symbol φ. Because the meaning of φ̄ varies by discipline, readers should rely on surrounding definitions to determine its interpretation.