Polynomials Definition and 784 Threads

I'm now studying the application of legendre polynomials to numerical integration in the so called gaussian quadrature. There one exploits the fact that an orthogonal polynomial of degree n is orthogonal to all other polynomials of degree less than n with respect to some weight function. For...

Hello all! I am trying to work through and understand the derivation of the Legendre Polynomials from Jackson's Classical electrodynamics. I have reached a part that I cannot get through however. Jackson starts with the following orthogonality statement and jumps (as it seems) in his proof...

Homework Statement Are the following statements true or false? Explain your answers carefully, giving all necessary working. (1) p_{1}(t) = 3 + t^{2} and p_{2}(t) = -1 +5t +7t^{2} form a basis for P_{2} (2) p_{1}(t) = 1 + 2t + t^{2}, p_{2}(t) = -1 + t^{2} and p_{3}(t) = 7 + 5t -6t^{2}...

Homework Statement Two Questions: 1. Prove, by contradiction, that if a and b are integers and b is odd,, then -1 is not a root of f(x)= ax^2+bx+a. 2. Prove, by contradiction, that there are infinitely many primes as follows. Assume that there only finite primes. Let P be the largest...

Homework Statement For p(x)=x4-2x3+3x2-3x+1 and A= 1 1 1 -1 -1 0 -2 1 0 0 1 0 1 0 0 0 you can check that P(A)=0 using this find a polynomial q(x) so that q(A)=A-1. The point is A4-2A3+3A2-3A=A(-A3+2A2-3A+3I)=I a) What is q(x)? I don't really understand how to approach...

Homework Statement Thanks very much for reading. I actually have two problems, I hope it's ok to state both of the in the same thread. 1. Let Vn be the space of all functions having the n'th derivitve in the point x0. I've been given the semi-norm (holds all the norm axioms other than ||v|| =...

Please I need Visual Basic algorithm for computing Hermite polynomials. Any one with useful info? Thanks.

Homework Statement Let f(x) = 2x^3 + 5x + 3 Find the inverse at f^-1(x) = 1 Homework Equations N/AThe Attempt at a Solution The only way that I know how to solve inverses is by solving for X, then replacing it by Y. Then I supposed I would sub 1 into the inverted polynomial. However I'm...

Homework Statement determine whether the following polynomials are irreducible over Q, i)f(x) = x^5+25x^4+15x^2+20 ii)f(x) = x^3+2x^2+3x+5 iii)f(x) = x^3+4x^2+3x+2 iv)f(x) = x^4+x^3+x^2+x+1 Homework Equations The Attempt at a Solution By eisensteins criterion let...

I am trying to find a way to integrate the following expression Integral {Ylm(theta, phi) Conjugate (Yl'm'(theta, phi) LegendrePolynomial(n, Cos[theta])} dtheta dphi for definite values of l,m,n,l',m' . You normally do this in Mathematica very easily. But it happens that I need to use this...

How are Hermite Polynomials related to the solutions to the Schrodinger equation for a harmonic oscillator? Are they the solutions themselves, or are the solutions to the equation the product of a Hermite polynomial and an exponential function? Thanks!

Hi, If I'm given something like this for a problem, Approximate the given quantities using Taylor polynomials with n=3 sqrt(101) how do I know what I should set f(x) equal to? I could set it to many different things, sqrt(x), sqrt(x+100), sqrt(x+50). My answer would be very different...

Hi! Im trying to do some rather easy QM-calculations in Fortran. To do that i need a routine that calculates the generalized Laguerre polynomials. I just did the simplest implementation of the equation: L^l_n(x)=\sum_{k=0}^n\frac{(n+l)!(-x^2)^k}{(n-k)!k!} I implemented this in the...

I've been doing some work and I keep running into polynomials of the following form: P(x,y,z) = ax^2 + by^2 + cz^2 + 2(exy + fxz + gyz) \mod p where a,b,c \in \mathbb{Z}_p/ \{0\} and d , e, f \in \mathbb{Z}_p . It would be great if I knew anything about the existence of roots of P ...

I've just read a paper that references the use of student-t orthogonal polynomials. I understand how the Gauss-Hermite polynomials are derived, however applying the same process to the weight function (1 + t^2/v)^-(v+1)/2 I can't quite get an answer that looks anything like a polynomial...

Hi Could someone please explain how to best handle rational polynomials in Maple? I have matrix of rational polynomials and for some reason Maple keeps grumbling i.e. "error, (in, linearalgebra:- HermiteForm) expecting a matrix of rational polynomials" The matrix I am working with is...

while attempting to solve for radius of a circular orbit, I ended up getting P(x)=0 and dP(x)/dx=0 where P is a fourth order polynomial. I am not sure how can I solve it. Can someone shed some light on it. Thanks

Hi, yet another question regarding polynomials :). Just curious about this. Let f(x), g(x) be irreducible polynomials over the finite field GF(q) with coprime degrees n, m resp. Let \alpha , \beta be roots of f(x), g(x) resp. Then the roots of f(x), g(x), are \alpha^{q^i}, 0\leq i \leq n-1...

Hi, I'm currently doing a project and this topic has come up. Are there any known famous classes of polynomials (besides cyclotomic polynomials) that fit that description? In particular, I'm more interested in the case where the polynomials have odd degree. I know for example that the roots of...

hey i want to find out if the set s = {t2-2t , t3+8 , t3-t2 , t2-4} spans P3 For vectors, i would setup a matrix (v1 v2 v3 v4 .. vn | x) where x is a column vector (x , y ,z .. etc) and reduce the system. If a solution exists then the vectors span the space, if there are no solutions then...

given a set of orthogonal polynomials \int_{-\infty}^{\infty}dx P_{m} (x) P_{n} (x) w(x) = \delta _{m,n} the measure is EVEN and positive, so all the polynomials will be even or odd my question is if we suppose that for n-->oo \frac{ P_{2n} (x)}{P_{2n}(0)}= f(x) for a known...

Hi all, I am currently a 2nd year mathematics and physics student. I am working, for the first time, on my own research and just sort of getting my feet wet. I got in touch with a professor that studies Special Functions and he led me to the Legendre functions and associated Legendre...

Hi I was wondering since i have problems factoring any polynomial past 2nd degree i was wondering if anyone can show a way i can remember for finals ^_^. IE. let's say we have a 3rd degree polynomial. X^3 - 3X^2 +4 i tried looking it up but most don't show how they did the work so i can...

[-1]int[1]P(x)Q(x)dx P,Q\inS verify that this is an inner product.

Homework Statement There is a recursion relation between the Legendre polynomial. To see this, show that the polynomial x p_k is orthogonal to all the polynomials of degree less than or equal k-2. Homework Equations <p,q>=0 if and only if p and q are orthogonal. The Attempt at a...

1. Let \mathbb{R}[x]_n^+ and } \mathbb{R}[x]_n^- denote the vector subspaces of even and odd polynomials in \mathbb{R}[x]_n Show \mathbb{R}[x]_n=\mathbb{R}[x]_n^+ \oplus\mathbb{R}[x]_n^- 3. For every p^+(x) \in \mathbb{R}[x]_n^+ \displaystyle p^+(x)=\sum_{m=0}^n a_m x^m=p^+(-x)...

Let T:V to V be a linear operator on an n-dimensional vector space V. Let T have n distinct eigenvalues. Prove that the minimal polynomial and the characteristic polynomial are identical up to a factor of +/- 1 I'm probably over thinking this, but it seems that if you have n distinct...

In how many ways can obtain polynomial from [PLAIN]http://im3.gulfup.com/2011-05-05/1304543619801.gif notes that c any coffieceints is in{0.1} also in how many ways can obtain even ploynomials?whats the probability that we can obtain P(1,1,1)=0

Homework Statement For each n\,\in\,\mathbb{N}, let p_n(x)\,\in\,\mathbb{Z}_2[x] be the polynomial 1\,+\,x\,+\,\cdots\,x^{n\,-\,1}\,+\,x^n Use mathematical induction to prove that p_n(x)\,\cdot\,p_n(x)\,=\,1\,+\,x^2\,+\,\cdots\,+x^{2n\,-\,2}\,+\,x^{2n}Homework Equations Induction steps...

Homework Statement Find two polynomials, each of degree 2, in Z6[x] whose product has degree 0. Can you repeat the same in Z7[x]? Explain. Homework Equations In Z6[x] and Z7[x] can the only variable be x? The Attempt at a Solution I know Z6 consists of {0,1,2,3,4,5} and Z7...

Homework Statement I'm doing work finding the centroids of 2d graphs. I'm working these problems using double integrals of regions that are horizontally or vertically simple. To do this I have to be able to convert line equations from one variable to the other. Some are simple but others...

In a past exam paper at my uni I am asked to show that the hermite polynomials are solutions of the hermite diff. equation but first there is the following \Phi(x,t)=\exp (2xh-h^2)=\sum_{n=0}^{\infty} \frac{h^n}{n!}H_n(x) So I need to find the form of H_n first, and I'm stuck. I tried...

Homework Statement Factor x^{16}-x in F_8[x] Homework Equations The Attempt at a Solution I know how to do it in F_2[x] . I also feel the factorizations are the same in the two fields..but not sure how to prove it.

Find a polynomial that is orthogonal to f(x)=x2-1/2 using L2[0,1]. I have looked all in the textbook and all over the internet and have found some hints if the interval is [-1,1], but still do not even know where to start here. I think I was gone the day our professor taught this because I do...

1.Let F be an extension field of K and let u be in F. Show that K(a^2)contained in K(a) and [K(u):K(a^2)]=1 or 2. 2.Let F be an extension field of K and let a be in F be algebraic over K with minimal polynomial m(x). Show that if degm(x) is odd then K(u)=K(a^2). 1. I was thinking of...

Homework Statement Show that x^2\,+\,x can be factored in two ways in \mathbb{Z}_6[x] as the product of nonconstant polynomials that are not units.Homework Equations Theorem 4.8 Let R be an integral domain. then f(x) is a unit in R[x] if and only if f(x) is a constant polynomial that is a...

Homework Statement Hi, I need to show that \alpha+1=[x] is a primitive element of GF(9)= \mathbb{Z}_3[x]/<x^{2}+x+2> I have already worked out that the function in the < > is irreducible but I do not know where to go from this. Homework Equations there are 8 elements in the...

Hello, I'm currently doing an undergrad project on this topic and I was wondering if any of you guys know what is the fastest algorithm (asymptotically) that has been discovered so far, for such purpose. Here is paper by Shoup (1993) which gave the fastest algorithm up to then...

Hello, I'm currently doing an undergrad project on this topic and I was wondering if any of you guys know what is the fastest algorithm (asymptotically) that has been discovered so far, for such purpose. Here is paper by Shoup (1993) which gave the fastest algorithm up to then...

I've recently been working with Legendre polynomials, particularly in the context of Spherical Harmonics. For the moment, it's enough to consider the regular L. polynomials which solve the differential equation [(1-x^2) P_n']'+\lambda P=0 However, I've run into a problem. Why in the...

Homework Statement If the characteristic polynomial of an operator T is (-1)^n*t^n, is T nilpotent? Homework Equations The Attempt at a Solution My first instinct for this question is that the answer is yes, because the matrix form of T must have 0's on the diagonal and must...

If the characteristic polynomial of an operator T is (-1)^n*t^n, is T nilpotent? My first instinct for this question is that the answer is yes, because the matrix form of T must have 0's on the diagonal and must be either upper triangular or lower triangular. This is what I found when I tried...

I would like to know if the following correct. Suppose I start with a connection on a real vector bundle and extend it to the complexification of the bundle. The curvature forms of the complexification seem to be the same as curvature 2 forms of the real bundle. From this it seems that the...

Homework Statement Prove that a polynomial f of degree n with coefficients in a field F has at most n roots in F. Homework Equations The Attempt at a Solution So we could prove this by induction by using a is a root of f if and only if x-a divides f. My question is: why do...

Why should the exponents of polynomials just be whole numbers ?

Hi everybody! I have this problem: Either P = (X+2)m+(X+3)n and Q = x2+5x+7; Determine m, n such that Q | P;( m, n = ? (Q divide P)); May you help me please? Thank You!

Homework Statement attached Homework Equations The Attempt at a Solution what is x_i? is it the coefficient of x or simply add up 1-5? i found the notation different from http://mathworld.wolfram.com/PolynomialNorm.html so i am confused. Thx!

Homework Statement 1. Use the definition of "f(x) is O(g(x))" to show that 2^x + 17 is O(3^x)'). 2. Determine whether the function x^4/2 is O(x^2) 2. The attempt at a solution 1. From my understanding, I would say of course 2^x + 17 is O(3^x) because the constant is of such low...

Homework Statement How to prove that the only irreducible polys over the reals are the linear ones and the quadratic ones no real roots? What about the ones with higher degree? I feel that I'm missing something that's really obvious. Homework Equations The Attempt at a Solution

Anyone who knows the limits of orthogonality for Bessel polynomials? Been searching the Internet for a while now and I can't find a single source which explicitly states these limits (wiki, wolfram, articles, etc). One thought: since the Bessel polynomials can be expressed as a generalized...