What is Phi: Definition and 130 Discussions

Phi (; uppercase Φ, lowercase φ or ϕ; Ancient Greek: ϕεῖ pheî [pʰé͜e]; Modern Greek: φι fi [fi]) is the 21st letter of the Greek alphabet.
In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voiceless bilabial plosive ([pʰ]), which was the origin of its usual romanization as ⟨ph⟩. During the later part of Classical Antiquity, in Koine Greek (c. 4th century BC to 4th century AD), its pronunciation shifted to that of a voiceless bilabial fricative ([ɸ]), and by the Byzantine Greek period (c. 4th century AD to 15th century AD) it developed its modern pronunciation as a voiceless labiodental fricative ([f]).
The romanization of the Modern Greek phoneme is therefore usually ⟨f⟩.
It may be that phi originated as the letter qoppa, and initially represented the sound /kʷʰ/ before shifting to Classical Greek [pʰ]. In traditional Greek numerals, phi has a value of 500 (φʹ) or 500,000 (͵φ). The Cyrillic letter Ef (Ф, ф) descends from phi.
As with other Greek letters, lowercase phi (encoded as the Unicode character U+03C6 φ GREEK SMALL LETTER PHI) is used as a mathematical or scientific symbol. Some uses, such as the golden ratio, require the old-fashioned 'closed' glyph, which is separately encoded as the Unicode character U+03D5 ϕ GREEK PHI SYMBOL.

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  1. C

    What does the phase angle phi mean in the harmonic oscillation function?

    The function for simple harmonic oscillation is: Acos(ωT)+\phi Why is there an angle phi added to the function acos(ωT)?
  2. jfy4

    4-point correlation in phi 4 to first order

    Hi, Look at \begin{align} & \int d^4 x d^4 x_1 d^4 x_2 d^4 x_3 d^4 x_4 \exp [i(p_1 x_1 + p_2 x_2 -k_1 x_3 -k_2 x_4)] \\ & \times (-i\lambda)D(x_1-x)D(x_2-x)D(x_3-x)D(x_4-x) \end{align} for first order in lambda for 2-2 scattering. In Maggiore I am told to substitute y_i=x_i-x as a...
  3. andrewkirk

    Lightlike radial null geodesic - how do we know it has constant theta and phi?

    Consider a light ray emanating from the origin of a FLRW coordinate system in a homogeneous, isotropic universe. The initial velocity of that ray will have only x0 (t) and x1 (r) components. In papers I have seen it is assumed that its velocity will continue to have zero circumferential...
  4. T

    Where does the phi come from in the charge of a capacitor in an LC circuit?

    I'm reading University Physics 13e by Young and Freedman and we're given this equation: -L(di/dt)-q/C=0 So we know that i, current, is, i=dq/dt. So the first equation can be read, after some simplification, as q''+(1/LC)q=0 where q prime means the derivative of q (charge) with...
  5. J

    RThe canonical representation phi (measure theory) (Royden)

    RThe "canonical representation phi" (measure theory) (Royden) Homework Statement I need some help understanding the canonical representation of phi as described on p. 77 of Rodyen's 3rd edition. I've transcribed it below for those of you who don't own the book. Homework Equations The...
  6. O

    Maxwell's, integrals, current, elements, delta phi and confusion

    I'm working on an online EECS course, and to be frank some of it is going straight over my head - but at the same time parts of it are far below my current knowledge, so I want to work and stick with it. The speaker is working through proving current and voltage - to arrive at Kirchoff's...
  7. J

    Calculating Phi: A Sin Wave Explained

    A question I have is how is one supposed to calculate phi when simply looking at a sin wave. The equation is of the form x(t) = A*cos(wt + phi). Once again, the only thing i am given is the graph. I appreciate any help you can give!
  8. J

    Help with Euler's Phi function

    Homework Statement Find x such that phi(x) = 5,000,000, where phi(x) is Euler's function. Homework Equations I know that if x is prime, then phi(x) = x-1. Also, phi(pk) = pk - pk-1 = pk * (1 - 1/p). The Attempt at a Solution Since 5,000,001 is not a prime number...
  9. G

    Treating phi and phi-dagger as independent

    In treating the complex scalar field in QFT, \phi and \phi^\dagger are treated as independent variables. I'd like to make sure I understand what this actually means: Is this the same idea as treating \phi and \partial \phi as independent variables in the Lagrangian? Formally, we view the...
  10. V

    Tau and phi (conjugates?) fibonacci sequence

    Homework Statement Would tau and phi be considered conjugates? Homework Equations \tau = \frac{1-\sqrt{5}}{2} \phi = \frac{1+\sqrt{5}}{2} The Attempt at a Solution I know that a complex number such as 1+2i would have 1-2i as a conjugate. However, for fractions, I can't quite remember if the...
  11. U

    Volume bound by rho=2+2cos phi

    Homework Statement Find the volume bounded rho=5+2cosphi Homework Equations dV=rho squared drho d phi d theta The Attempt at a Solution I am guessing this is some cylindrical shape. Theta should be 0-2pi and phi=0 pi/2
  12. T

    Importance of pi and phi for canonical quantisation

    Homework Statement Explain in words the importance of pi(x) and phi(x) for the canonical quantisation programme. Homework Equations none The Attempt at a Solution This is a past paper exam question, which I would appreciate some clarity on. As a proposed suggestion: pi(x) and...
  13. S

    Finding Connected Graphs That Do Not Model a V-Sentence Phi

    how to define a V-sentence phi such that phi has aebitrarily large finite models and for any finite model G , G is a connected graph. after that to find a connected graph that does not model the sentence phi. please explain it to me.
  14. M

    Proving the Euler Phi Equation Divisible by 2

    Prove that the Euler phi equation is always divisible by 2. If n > 2.? I don't understand how this proof works: I think I need to show that the inverse of a and a are both generated by the same group. Therefore, there are at least 2 elements that generate all other elements in the group...
  15. M

    Inverse image of phi (totient)

    So upon introduction to Euler's phi function, we can see that \phi (1) = 1 and \phi (2) = 1, where it turns out that these are in fact the only numbers in N that map to 1. Now what I'm wondering is if there is some general way to find the inverse image of numbers in the image of phi? Also...
  16. F

    Phi Numbers in Time and Price: Exploring Consciousness

    In the market one can find a lot of Phi related numbers in time and price. But also many cubic. http://img836.imageshack.us/img836/2079/image1czh.jpg This is the stock Google. Why would you say this occur? It´s in calendar days. +- 2-3 days. 0,382 and 0,786 - I guess you know the relation...
  17. C

    How do you determine angle phi in spherical coordinates?

    In general, how do I determine the angle phi when you have to use spherical coordinates for integration?
  18. K

    Number theory - euler's phi function

    Homework Statement Let p = a prime. Show {x}^{2} ≡ a (mod {p}^{2}[/tex]) has 0 solutions if {x}^{2} ≡ a (mod p) has 0 solutions, or 2 solutions if {x}^{2} ≡ a (mod p) has 2. The Attempt at a Solution OK, my mistake, I don't think this has anything to do with the phi function. But I don't...
  19. E

    Plotting I/Io vs Phi: 3-Slit System w/650 nm Wavelength

    Homework Statement A three slit system is illuminated by light with a wavelength of 650 nm. The slits are evenly spaced with a slit space of 1.28 mm and a slit width of 0.16 mm A) Using Excel, plot the modulated interference pattern for +or - 0.5° or so. Plot I/Io. b) Are there any missing...
  20. K

    Does it make sense to define operators like r,theta, phi

    I'm pondering, since we've introduced formalism, all operators are either scalars or vector components, does it make sense to define operators like r, theta, phi (as in spherical coordinates) which are neither? In classical mechanics we can easily transform observables fro cartesian to...
  21. F

    LRFD phi for overturning, sliding, and soil bearing?

    Does anyone know the phi factors for overturning, sliding, and soil bearing in a small reinforced footing? If you could site the precise source too, that'd be great. I don't have access to many up to date specs and codes :/
  22. C

    Practical use of the phi function

    I am not sure if this is the right section but I thought you physicists may be able to give me an example. I am demonstrating the usefulness of the phi function and finding the last two digits of 107^{999999999} I am just wonder where in physics or astronomy such a massive number would come...
  23. L

    Superrenormalizable phi cubed theory

    Hi, I'm trying to show that phi^3 theory in d=4 is superrenormalisable (only finite no of terms are power counting divergent). In the following I use, d=#dimensions, I=#internal props, E=external legs, V=#number of vertices (of phi^3 type, i.e. three valent) The Superficial degree of...
  24. T

    Need Euler phi function proof help

    I am having trouble with this proof: show that if d|n then phi(d)|phi(n). I know that if d|n, then ad=n and that phi(ad)=(a,d)*phi(a)&phi(d)/phi((a,d)), but I can't seem to get anywhere with this info. Thanks for your help.
  25. K

    Proving Euler's Phi Function: Any Help Appreciated!

    "Let m and k be positive integers and φ(m) is the Euler's phi function. Then the number of integers n such that 1≤n≤mk and (n,m)=1 is kφ(m)." I can't figure out why this is true. How can we prove it? Can someone explain this, please? Any help is appreciated!
  26. A

    Electric Field at Ro, Phi, and Z Help

    A 2[\muC] point charge is located at A(4,3,5) in free space. Find E\rho, E\phi, and Ez at P(8,12,2). I know I have to start off in rectangular, convert to cylindrical coordinates. I know how to find the R vector and |R|. Since it says there is a point charge, do I need to integrate...
  27. X

    Trying solve the phi function again

    This is getting on my nerves now. I am stumped on these: Homework Statement Obtain an expression for: phi(p^3) and phi(P^n) Homework Equations phi(p^2)=p(p-1) The Attempt at a Solution The factors of P^n are 1 and P. Any number less than P^n that shares a factor must have p as a...
  28. X

    Proofing Euler's Phi Function.

    Generally, I am stumped by the Phi function. I have found out the pattern but I am having difficulty proving it. Homework Statement i) When p is a prime number, obtain an expression in terms of p for:- ϕ(p²)Homework Equations ϕ(p)=p-1 The Attempt at a Solution ϕ(p)=p-1 so ϕ(5)=4 therefore...
  29. R

    Units of Scalar Field \phi & Lagrangian Density

    What are the dimensions of a scalar field \phi ? The Lagrangian density is: \mathcal L= \partial_\mu \phi \partial^\mu \phi - m^2 \phi \phi So in order to make all the terms have the same units, you can try either: \mathcal L=\frac{\hbar^2}{c^2} \partial_\mu \phi \partial^\mu \phi -...
  30. T

    How to Determine Time Ordering in Phi-3 Theory for a 2-to-3 Particle Process?

    i have been given a problem for writing s matrix in second order perturbative theory for an interaction hamiltonian with phi 4 and phi 3 contributions. it is also given that our initial state is of 2 particles and final state is of three particles. now in solving that i have to take time...
  31. E

    What is Phi: Definition & Meaning

    what does phi mean
  32. G

    Solving for Theta and Phi in Direction of Propagation | Homework Statement

    Homework Statement The direction of propagation is defined as ex=sin(theta)cos(phi), ey=sin(theta)sin(phi), ez=cos(theta). What are the values of theta and phi characterizing the direction of propagation? Homework Equations We have a point in space with the coordinates...
  33. M

    Normalising phi for the Hydrogen atom.

    This is most likely very simple, but I can't figure it out. http://www.sussex.ac.uk/physics/teaching/btv/Lect02_2006.pdf Step 5 they've got an equation for \Phi. They then normalise it to get A = \frac{1}{\sqrt{2\pi}}. Every time I do the integral I get: A^2.^{2\pi}_{0}[...
  34. P

    Writing sin^2 theta * Sin 2 phi in terms of spherical Harmonics

    I am trying to write the term "Sin^2 theta * Sin 2 phi" in terms of spherical Harmonics (they form a combination of Y(2,-2) and Y(2,2)) but the term I get contains the imaginary number 'i'. Am I doing something wrong.. In fact this term is a part of a Hamiltonian and when I get the eigenvalues I...
  35. H

    Understanding Phi Function and Multiplicativity

    phi(20^100) = phi(4^100 * 5^100) = phi(2^200 * 5^100) = (2^200 - 2^199)(5^100 - 5^99) = 2^199(2-1) * 5^99(5-1) = 2^199 * 5^99 * 4 = 2^201 * 5^99. I don't understand line 4-7. Can anyone explain?
  36. S

    Analysis of A-Module Endomorphism \phi: Understanding Kronecker Delta Function

    Let A, M be a commutative ring and a finitely generated A-module respectively. Let \phi be an A-module endomorphism of M such that \phi (M)\subseteq \alpha\ M where \alpha is an ideal of A. Let x_1,\dots,x_n be the generators of M. Then we know that \displaystyle{\phi(x_i)=\sum_{j=1}^{n}...
  37. P

    Calculating Euler phi function

    How do i comput the euler phi function of a large interger? i know that if p is prime then phi(p)=p-1 and I've found a formula for computing non primes but i don't know how to implement in something like Matlab. Does anyone know how?
  38. P

    MATLAB Compute Euler Phi Function of Large Integer | Matlab Solution

    How do i comput the euler phi function of a large interger? i know that if p is prime then phi(p)=p-1 and I've found a formula for computing non primes but i don't know how to implement in something like Matlab. Does anyone know how?
  39. A

    Phi (the golden ratio) in prime quadruplets

    Phi exists at the center of prime quadruplets, along with its square root, and cube root! http://www.code144.com/zphithrice.png The 'pos' numbers come from the position of the prime numbers in the sequence itself, i.e. 193 is the 44th prime number, and 197 is the 45th prime number...
  40. C

    Ker(phi) = {0}, then phi injective?

    Hi all, Can anyone point to an explanation of why if ker(phi) = {0}, then phi is injective?
  41. K

    Need help on Maths essay, relationship between Phi and Fibonacci

    hey there! its my first post, and quite an urgent one. i am into my last year of high school, and i am required to write a 4000 word maths essay. i chose to write on maths, because i like maths! but i am quite stucked at this stage... i decided to writing on the mathematical relationship on...
  42. H

    Phi Physics Help: Solving for N

    \foralln\inN\varphi(n)/mid/n
  43. S

    Is Phi One-to-One and What is Its Image for a Given Rectangle?

    Let [phi](u,v)=(u^2,v). Is phi one-to-one? If not, determine a domain on which phi is one-to-one. Find the image under phi of: - The rectangle R=[-1,1]X[-1,1] The Attempt at a Solution - I'm not sure at all how to determine whether phi is one-to-one or not, so if somebody can...
  44. H

    Iteration of Euler's phi function

    Iterating Euler's totient function \varphi(n), it eventually arrives at 1. Let h(n) denote the least number of iterations to arrive at 1. For instance, h(18)=3, ie \varphi^{3}(18)=1. There is a conjecture (or is there a proof?) that the largest n for which h(n)=k is 2\times 3^{k-1}. The only...
  45. A

    Questions about phi and infinity?

    Guys How do you think our reality would differ if the phi ratio were exactly 3 ? Is infinity a possibility? To me It must be, but in he same breath it can't be! "It is impossible to imagine an impossibility" By the way I am a retired Mechanical Engineer (Electricity power...
  46. S

    Where does phi go in the probability density for hydrogen?

    The analytical solution for the wavefunction of a hydrogenic electron with quantum numbers n, l and m has a spherical harmonic part that involves theta and phi (in spherical coordinates). I was looking in Griffiths, and the spherical harmonics part only has phi as exp(i m phi) where i is the...
  47. D

    Can the Infinite Monkey Theorem Predict Pi's Digits in Phi's Expansion?

    I guess one could use any irrational numbers here, but phi and pi are favorites. I am sure that most people are aware of the infinite monkey theorem. If not use http://en.wikipedia.org/wiki/Infinite_monkey_theorem as a reference. By using this theorem, could one say that the the first...
  48. B

    Is Joining Phi Kappa Phi Worth It? A Scientist's Perspective

    So I have been invited to join Phi Kappa Phi. I'm short on funds, but that isn't the primary reason why I'm considering not joining. The people part of the honors society tell me that it's worthwhile, which comes to no surprise by me, and poking around the internet makes me think that it might...
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