φ

In lowercase

• The letter phi is commonly used in physics to represent wave functions in quantum mechanics, such as in the Schrödinger equation and bra–ket notation: ${\displaystyle \langle \phi |\psi \rangle }$.
• The golden ratio${\displaystyle {\tfrac {1+{\sqrt {5}}}{2}}\approx }$1.618033988749894848204586834... in mathematics, art, and architecture. (Its reciprocal, 1/φ, is ${\displaystyle {\tfrac {{\sqrt {5}}-1}{2}}}$ and is equal to φ − 1.)
• Euler's totient functionφ(n) in number theory; also called Euler's phi function.
• The cyclotomic polynomial functions Φ_n(x) of algebra.
• The number of electrical phases in a power system in electrical engineering, for example 1ϕ for single phase, 3ϕ for three phase.
• In algebra, group or ringhomomorphisms
• In probability theory, ${\textstyle \varphi (x)=(2\pi )^{-{\frac {1}{2}}}e^{-{\frac {x^{2}}{2}}}}$ is the probability density function of the standard normal distribution.
• In probability theory, φ_X(t) = E[e] is the characteristic function of a random variable X.
• In statistics, the phi coefficient is a measure of association between two binary variables.
• An angle, typically the second angle mentioned, after θ (theta). It can denote:
  • The argument of a complex number.
  • The phase of a wave in signal processing.
  • In spherical coordinates, mathematicians usually refer to ϕ as the polar angle (from the z-axis). The convention in physics is to use ϕ as the azimuthal angle (from the x-axis).
  • One of the dihedral angles in the backbones of proteins in a Ramachandran plot
  • Internal or effective angle of friction.
  • In cartography and navigation, ɸ denotes latitude; likewise, in geodesy it denotes geodetic latitude.
  • In ship stability, ɸ represents the angle of list.
  • In aircraft flight mechanics as the symbol for bank angle (sometimes represented with the letter theta, which is also used for pitch angle).