# What is Frobenius: Definition and 105 Discussions

Frobenius is a surname. Notable people with the surname include:

Ferdinand Georg Frobenius (1849–1917), mathematician
Frobenius algebra
Frobenius endomorphism
Frobenius inner product
Frobenius norm
Frobenius method
Frobenius group
Frobenius theorem (differential topology)
Georg Ludwig Frobenius (1566–1645), German publisher
Johannes Frobenius (1460–1527), publisher and printer in Basel
Hieronymus Frobenius (1501–1563), publisher and printer in Basel, son of Johannes
Ambrosius Frobenius (1537–1602), publisher and printer in Basel, son of Hieronymus
Leo Frobenius (1873–1938), ethnographer
Nikolaj Frobenius (born 1965), Norwegian writer and screenwriter
August Sigmund Frobenius (died 1741), German chemist

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1. ### I Frobenius theorem for differential one forms

Hi, starting from this old PF thread I've some doubts about the Frobenius condition for a differential 1-form ##\omega##, namely that ##d\omega = \omega \wedge \alpha## is actually equivalent to the existence of smooth maps ##f## and ##g## such that ##\omega = fdg##. I found this About...
2. ### Solution verification of ODE using Frobenius' method

I have no problems with solving this exercise, but my solution disagrees slightly with that given in the answers in the back of the book, and I do not know who's correct. First, we rewrite the equation as $$x''+\frac{3}{2t}x'-\frac{(1+t)}{2t^2}x=0.\tag1$$ We recognize that this is so-called...
3. ### I About global inertial frame in GR - revisited

Hi, reading this old thread I'd like a clarification about the following: Fermi Normal hypersurface at an event on a comoving FLRW worldline is defined by the collection of spacetime orthogonal geodesics. Such geodesics should be spacelike since they are orthogonal to the timelike comoving...
4. ### I Derivation for the indicial exponent in the Frobenius method

I'm reading a book called Asymptotic Methods and Perturbation Theory, and I came across a derivation that I just couldn't follow. Maybe its simple and I am missing something. Equation 3.3.3b below. y(x) takes the form A(x)*(x-x0)^α and A(x) is expanded in a taylor series.

14. J

### Finding the singular points for this differential equation

Homework Statement If d^2/dx^2 + ln(x)y = 0[/B]Homework Equations included in attempt The Attempt at a Solution I was confused as to whether I include the power series for ln(x) in the solution. It makes comparing coefficients very nasty though. Whenever I expand for m=0 for the a0 I end...

31. ### Frobenius series without recurrence relation

Homework Statement Consider x^2y''-xy'+n^2y=0 where n is a constant. a) find two linearly independent solutions in the form of a Frobenius series, initially keeping at least the first 3 terms. Can you find the solution to all orders? b) for n=1 you shouild find only one linearly...
32. ### Solve Using Frobenius Method

Homework Statement Solve x(1-x)\frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+2y=0 using the Frobenius Method. Homework Equations R(x)\frac{d^{2}y}{dx^{2}}+\frac{1}{x}P(x)\frac{dy}{dx}+\frac{1}{x^{2}}V(x)y=0 R_{0}s(s-1)+P_{0}s+V_{0}=0 y=\sum^{∞}_{m=0}a_{m}x^{m+s}...
33. ### Solve Legendre Polynomial using Method of Frobenius

Not sure how this can be done. can anyone help?
34. ### Proving the Frobenius Norm Identity for Matrices: A Step-by-Step Guide

Homework Statement Prove ∥A∥F =√trace(ATA), for all A ∈ R m×n Where T= transpose Homework Equations The Attempt at a Solution I tried and i just can prove it by using numerical method. Is there anyway to prove the equation in a correct way?
35. ### Solving the O.D.E using Frobenius Method about x=1

Homework Statement The task is to find an analytic solution to the O.D.E 4(1-x^2)y''-y=0 \hspace{20mm} y'(1)=1 by using an appropriate series solution about x=1. The Attempt at a Solution The singularity at x=1 is regular, which makes me think the Frobenius method is what's meant by...
36. ### Proof of Frobenius' Theorem: Directly Showing ##\omega \wedge d\omega = 0##

Hi guys. Most of my texts have the standard proof of Frobenius' theorem (both the vector field and differential forms versions) and through multiple indirect equivalences conclude that ##\omega \wedge d\omega = 0## implies (locally) that ##\omega = \alpha d\beta## where ##\omega## is a 1-form...
37. ### Does limit exist as x approaches zero? Frobenius Method DEQ

Homework Statement what is the limit of (4x^2-1)/(4x^2) when x→0 Homework Equations In order to find the Indicial Equation, do I need to take the limit of p(x) and q(x), the non-constant coefficients? If so, can the limit of this function be found using LH Rule? The Attempt at a...
38. ### MATLAB help, code for Frobenius norm

Hello, I am trying to write a mtlab code to compute Frobenius norm of an mxn matrix A. defined by ||A||_{F} = \sqrt{ \sum_{i=1}^m \sum_{j=1}^n a^{2}_{i,j}} I have so far written this code, but it does not work, if anyone can help /guide me to the right path, would be greatly...
39. ### Frobenius method for a differential equations

Homework Statement The function satisfies the differential equation f''(x) = xf(x) and has boundary conditions f(0) = 1 and f'(0) = 1 Use Frobenius method to solve for f(x) with a taylor expansion of f(x) up to the quartic term a4x4 Homework Equations f(x) = a0 + a1x + a2x2 + a3x3 + a4x4...
40. ### MHB How Can the Frobenius Method Be Used to Solve Complex ODEs?

Ok here's a funny ODE to solve: xy'' + (1-2x)y' + (x-1)y = 0 clearly a straight forward power series substitution won't work here since we have a regular singularity at x = 0 so try the frobenius method by expanding around x = 0. Assume y = \sum_{m=0}^{\infty} a_mx^{m+r} is a solution where...
41. ### MHB Solving DE using Frobenius series method

Solve xy'= y using frobenius method The explanation given in the book is very confusing can somebody explain in simple method. Thanks
42. ### Differential eqations and frobenius method

Homework Statement y''+4xy'+(4x^2+2)y=0 find the basis of solutions using the frobenius method. can anyone solve this please...
43. ### Proving the Frobenius Norm as a Matrix Norm

Homework Statement Prove that the Frobenius norm is indeed a matrix norm. Homework Equations The definition of the the Frobenius norm is as follows: ||A||_F = sqrt{Ʃ(i=1..m)Ʃ(j=1..n)|A_ij|^2} The Attempt at a Solution I know that in order to prove that the Frobenius norm is indeed...
44. ### Using Frobenius theorem

Homework Statement In Problems 25–30, x=0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x=0...
45. ### Frobenius Method - Roots differ by integer

I'm reading up on some methods to solve differential equations. My textbook states the following: "y_{1} and y_{2} are linearly independent ... since \sigma_{1}-\sigma_2 is not an integer." Where y_{1} and y_{2} are the standard Frobenius series and \sigma_1 and \sigma_2 are the roots of...
46. ### Solving Schrodinger DE using Frobenius

My DE is \frac{h^2}{2m} \frac{d^2\psi}{dx^2} + \left(E - \frac{Ae^{-ax}}{x} \right) \psi = 0 where h, m, A < 0 and a and E are constants. I need to construct the following series solution (using the larger root of my indicial equation): \psi(x) = a_0 \left[x + \frac{Am}{h^2}x^2 +...
47. ### When is the Frobenius norm of a matrix equal to the 2-norm of a matrix?

What conditions most be true for these two norms to be equal? Or are they always equal?
48. ### Theoretical/non-tedious question about Frobenius method

When using the Frobenius method of solving differential equations using power series solutions, I get a solution y = (indicial_stuff) + (infinite_summation_stuff) = 0 for a differential equation differential_stuff = 0. WHY is it that I can say (indicial_stuff) = 0? If y =...
49. ### Variations of the Frobenius coin problem

I wasn't sure if this should go in the number theory section, but here goes: Is there a formula for solving problems such as: If there are n coin denominations x_{1},x_{2}...x_{n} that total p cents, how many combinations are possible? n and p are positive real numbers, of course. On a...
50. ### Second order diff. eq. Frobenius

Hi there. I have this exercise, which says: Demonstrate that: xy''+(1-x)y'+\lambda y=0 has a polynomial solution for some λ values. Indicate the orthogonality relation between polynomials, the fundamental interval, and the weight function. So I thought I should solve this using Frobenius...