Phi (/ˈfaɪ, ˈfiː/ FY, FEE; uppercase Φ, lowercase φ or ϕ; Ancient Greek: ϕεῖ pheî [pʰéî̯]; Modern Greek: φι fi [fi]) is the twenty-first letter of the...

ᴺ ᴻ ᴼ ᴽ ᴾ ᴿ ᵀ ᵁ ᵂ ᵃ ᵄ ᵅ ᵆ ᵇ ᵈ ᵉ ᵊ ᵋ ᵌ ᵍ ᵏ ᵐ ᵑ ᵒ ᵓ ᵖ ᵗ ᵘ ᵚ ᵛ, Greek ᵝ ᵞ ᵟ ᵠ ᵡ, Cyrillic ᵸ, other ᵎ ᵔ ᵕ ᵙ ᵜ. These are intended to indicate secondary articulation...

a linear expression, as in: 3 φ 3 − 5 φ 2 + 4 = 3 ( φ 2 + φ ) − 5 φ 2 + 4 = 3 ( ( φ + 1 ) + φ ) − 5 ( φ + 1 ) + 4 = φ + 2 ≈ 3.618033. {\displaystyle...

prime to n. It is written using the Greek letter phi as φ ( n ) {\displaystyle \varphi (n)} or ϕ ( n ) {\displaystyle \phi (n)} , and may also be called...

cos ⁡ φ x ^ + sin ⁡ θ sin ⁡ φ y ^ + cos ⁡ θ z ^ , θ ^ = cos ⁡ θ cos ⁡ φ x ^ + cos ⁡ θ sin ⁡ φ y ^ − sin ⁡ θ z ^ , φ ^ = − sin ⁡ φ x ^ + cos ⁡ φ y ^ {\displaystyle...

latitude ϕ to the equator is given by (ϕ in radians) m ( ϕ ) = ∫ 0 ϕ M ( ϕ ′ ) d ϕ ′ = a ( 1 − e 2 ) ∫ 0 ϕ ( 1 − e 2 sin 2 ⁡ ϕ ′ ) − 3 2 d ϕ ′ {\displaystyle...

2nd projection map) and for any x in X: Φ ( 0 , x ) = x {\displaystyle \Phi (0,x)=x} Φ ( t 2 , Φ ( t 1 , x ) ) = Φ ( t 2 + t 1 , x ) , {\displaystyle \Phi...

{ ϕ ( x 1 ) ⋯ ϕ ( x n ) } | Ω ⟩ = ∫ D ϕ ϕ ( x 1 ) ⋯ ϕ ( x n ) e i ∫ d 4 x ( 1 2 ∂ μ ϕ ∂ μ ϕ − m 2 2 ϕ 2 − λ 4 ! ϕ 4 ) ∫ D ϕ e i ∫ d 4 x ( 1 2 ∂ μ ϕ ∂...

induction on n ≥ 1: φ n + 1 = ( F n φ + F n − 1 ) φ = F n φ 2 + F n − 1 φ = F n ( φ + 1 ) + F n − 1 φ = ( F n + F n − 1 ) φ + F n = F n + 1 φ + F n . {\displaystyle...

φ {\displaystyle \varphi } ⁠ is called Binet's formula: F n = φ n − φ ¯ n φ − φ ¯ = φ n − φ ¯ n 5 = T r ( φ n 5   ) 5 , L n = φ n + φ ¯ n φ + φ ¯ = φ...

when at latitude ϕ), is about 111132.92 − 559.82 cos ⁡ 2 ϕ + 1.175 cos ⁡ 4 ϕ − 0.0023 cos ⁡ 6 ϕ {\displaystyle 111132.92-559.82\,\cos 2\phi +1.175\,\cos...

Φ ( 0 ) ( x 0 ) = 1 2 π ∫ − ∞ x 0 e − t 2 / 2 d t Φ ( 1 ) ( x 0 ) = 1 2 π e − x 0 2 / 2 Φ ( n ) ( x 0 ) = − ( x 0 Φ ( n − 1 ) ( x 0 ) + ( n − 2 ) Φ (...

the following form: if X is a random variable and φ is a convex function, then φ ( E ⁡ [ X ] ) ≤ E ⁡ [ φ ( X ) ] . {\displaystyle \varphi (\operatorname...

cos ⁡ φ ,   sin ⁡ φ ) + 2 r ˙ φ ˙ ( − sin ⁡ φ ,   cos ⁡ φ ) + r φ ¨ ( − sin ⁡ φ ,   cos ⁡ φ ) − r φ ˙ 2 ( cos ⁡ φ ,   sin ⁡ φ ) = ( r ¨ − r φ ˙ 2 ) r...

{5}}}{2}}}  ≈ 1.61803399 symbolized by the Greek letter φ) as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, phinary...

whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it...

spaces. Suppose that φ : M → N {\displaystyle \varphi \colon M\to N} is a smooth map between smooth manifolds; then the differential of φ {\displaystyle \varphi...

A ϕ ( z ) ( ρ + d ρ ) d ϕ − A ϕ ( z + d z ) ( ρ + d ρ ) d ϕ + A z ( ϕ + d ϕ ) d z − A z ( ϕ ) d z ( ρ + d ρ ) d ϕ d z = − ∂ A ϕ ∂ z + 1 ρ ∂ A z ∂ ϕ {\displaystyle...

, x μ ) − L ( φ A , φ A , ν , x μ ) ] = ∂ ∂ x σ ( ∂ L ∂ φ A , σ ) δ ¯ φ A + ∂ L ∂ φ A , σ δ ¯ φ A , σ = ∂ ∂ x σ ( ∂ L ∂ φ A , σ δ ¯ φ A ) . {\displaystyle...

In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function F {\displaystyle F} of some real variable t {\displaystyle t}...

ϕ F | e − i H T | ϕ I ⟩ = ∫ d ϕ 1 ∫ d ϕ 2 ⋯ ∫ d ϕ N − 1 ⟨ ϕ F | e − i H T / N | ϕ N − 1 ⟩ ⋯ ⟨ ϕ 2 | e − i H T / N | ϕ 1 ⟩ ⟨ ϕ 1 | e − i H T / N | ϕ I...

and division can be computed as z 1 z 2 = r 1 r 2 ( cos ⁡ ( φ 1 + φ 2 ) + i sin ⁡ ( φ 1 + φ 2 ) ) . {\displaystyle z_{1}z_{2}=r_{1}r_{2}(\cos(\varphi _{1}+\varphi...

spiral r = a φ n {\displaystyle r=a\varphi ^{n}\;} the length is L = ∫ φ 1 φ 2 n 2 r 2 φ 2 + r 2 d φ = a ∫ φ 1 φ 2 φ n − 1 n 2 + φ 2 d φ   . {\displaystyle...

φ 1 − sin ⁡ φ ] = R ln ⁡ [ 1 + sin ⁡ φ cos ⁡ φ ] = R ln ⁡ ( sec ⁡ φ + tan ⁡ φ ) = R tanh − 1 ⁡ ( sin ⁡ φ ) = R sinh − 1 ⁡ ( tan ⁡ φ ) = R sgn ⁡ ( φ )...

the free dictionary. Phi (Φ, φ, or ϕ) is the 21st letter of the Greek alphabet. Phi or PHI may also refer to: Golden ratio (φ) Phi coefficient, a measure...

δ ′ , φ ⟩ = ⟨ δ ′ , x φ ⟩ = − ⟨ δ , ( x φ ) ′ ⟩ = − ⟨ δ , x ′ φ + x φ ′ ⟩ = − ⟨ δ , x ′ φ ⟩ − ⟨ δ , x φ ′ ⟩ = − x ′ ( 0 ) φ ( 0 ) − x ( 0 ) φ ′ ( 0 )...

function φ {\textstyle \varphi } whose value depends only on the distance between the input and some fixed point, either the origin, so that φ ( x ) = φ ^ (...

Y(θ, φ) = Θ(θ) Φ(φ). Applying separation of variables again to the second equation gives way to the pair of differential equations 1 Φ d 2 Φ d φ 2 = −...

cos ⁡ ( φ 1 ) cos ⁡ ( φ 2 ) + sin ⁡ ( φ 1 ) sin ⁡ ( φ 2 ) = sin ⁡ ( φ 2 ) sin ⁡ ( φ 1 ) + cos ⁡ ( φ 2 ) cos ⁡ ( φ 1 ) − cos ⁡ ( φ 2 ) cos ⁡ ( φ 1 ) + cos...

multiplication: | ϕ ⟩ ⟨ ψ | ≐ ( ϕ 1 ϕ 2 ⋮ ϕ N ) ( ψ 1 ∗ ψ 2 ∗ ⋯ ψ N ∗ ) = ( ϕ 1 ψ 1 ∗ ϕ 1 ψ 2 ∗ ⋯ ϕ 1 ψ N ∗ ϕ 2 ψ 1 ∗ ϕ 2 ψ 2 ∗ ⋯ ϕ 2 ψ N ∗ ⋮ ⋮ ⋱ ⋮ ϕ N ψ 1 ∗ ϕ N ψ...