Polynomials Definition and 784 Threads

Homework Statement Show that L11(x) and L12(x) are precisely the polynomials for 1s and 2s orbitals. What is the role of variable x in each case? Homework Equations L1n(x) = d/dx Ln(x), n = 1, 2, 3... The Attempt at a Solution Because L1(x) = 1 - x L2(x) = 2 - 4x + x2: I did: L11(x) = d/dx...

I was tutoring a student and I came across the following question. I feel like I'm missing something obvious, but it seems like there are too many variables for an answer to be determined. The attached picture contains all of the question details.

Hi! (Wave) I want to show that $\forall a,b$ with $b>0$: $$(n+a)^b=\Theta(n^b)$$ According to my notes, it is like that: $\forall n \geq n_0=\lceil |a| \rceil$ $$(1) \Rightarrow (n+a)^b \leq (2n)^b=2^b n^b=c n^b \Rightarrow (n+a)^b=O(n^b), \forall n \geq \lceil |a| \rceil \text{ and }...

Hi, I'm new to lisp and I've been set some coursework in it and I don't really know how to begin. I need to implement polynomial arithmetic so I can add, subtract and multiply polynomials. So like: $\left(x + y\right)\left(x + y\right) = \left({x}^{2} + 2xy + {y}^{2}\right)$ It also needs to...

(x2)3(2x3)3/(4x)2 I got 1/2x^13, is this correct?

I will be using /= to mean 'does not equal'. From my textbook: Division Algorithm: Let R be any ring and let f(x) and g(x) be polynomials in R[x]. Assume that f(x) /= 0 and that the leading coefficient of f(x) is a unit in R. then unique determined polynomials q(x) and r(x) exist such that 1)...

Hello! (Wave) I want to show that if $I,J$ ideals of $K[x_1, x_2, \dots , x_n]$, then $V(I \cap J)=V(I) \cup V(J)$. Do I have to show that $ V(IJ)=V(I\cap J)$ and then $V(IJ)=V(I)\cup V(J)$? (Thinking) If so, that's what I have tried: $x \in V(IJ) \leftrightarrow (f_i \cdot g_j)(x)=0$, where...

m=1 and l=1 x = cos(θ) What would be the solution to this? Thanks.

Hi, I am just curious, are Hermite and Legendre polynomials related to one another? From what I have learned so far, I understand that they are both set examples of orthogonal polynomials...so I am curious if Hermite and Legendre are related to one another, not simply as sets of orthogonal...

Hi, I need suggestions for picking up some standard textbooks for the following set of topics as given below: Ordinary and singular points of linear differential equations Series solutions of linear homogenous differential equations about ordinary and regular singular points...

Homework Statement Could someone explain how Legendre polynomials are derived, particularly first three ones? I was only given the table in the class, not steps to solving them...so I am curious. Homework Equations P0(x) = 1 P1(x) = x P2(x) = 1/2 (3x2 - 1) The Attempt at a Solution ...

I just had a few questions not directly addressed in my textbook, and they're a little odd so I thought I would ask, if you don't mind. :) -Firstly, I was just wondering, why is it that Legendre polynomials are only evaluated on a domain of {-1. 1]? In realistic applications, is this a limiting...

Homework Statement In the linear space of all real polynomials with inner product (x, y) = integral (0 to 1)(x(t)y(t))dt, let xn(t) = tn for n = 0, 1, 2,... Prove that the functions y0(t) = 1, y1(t) = sqrt(3)(2t-1), and y2 = sqrt(5)(6t2-6t+1) form an orthonormal set spanning the same subspace...

Hey! :o Let $K$ be a field. I want to show that the ring $K[x]$ has infinitely many irreducible polynomials.I have done the following: We suppose that there are finite many irreducible polynomials, $f_1(x), f_2(x), \dots, f_n(x)$ with $deg f_i(x)>0$. Let $g(x)=f_1(x) \cdot f_2(x) \cdots...

Homework Statement (3y-2/y+3) - (3y+1)/(y2+6y-9) Homework EquationsThe Attempt at a Solution Ok, I have attempted to solve in 2 steps, step 1: solve 3y-2/y+3 step 2: solve 3y+1/y2+6y-9 and then subtract the answers. This doesn't seem to work, as I get: 3y-2/y+3 = 3 remainder 11 and...

When you have a polynomial say ax^4+bx^3+cx^2+dx+e where a,b,c,d and e are constants and divide this by a polynomial say ax+b it follows that the quotient will be a cubic polynomial. Assuming that a remainder exists, then the remainder will be a constant because in my reasoning, the remainder...

Hi. I'm off to solve this integral and I'm not seeing how \int dx Hm(x)Hm(x)e^{-2x^2} Where Hm(x) is the hermite polynomial of m-th order. I know the hermite polynomials are a orthogonal set under the distribution exp(-x^2) but this is not the case here. Using Hm(x)=(-1)^m...

Proposition 5.2.1 in Artin states that: THEOREM. Let $p_k(t)\in \mathbf C[t]$ be a sequence of monic polynomials of degree $\leq n$, and let $p(t)\in \mathbf C[t]$ be another monic polynomial of degree $n$. Let $\alpha_{k,1},\ldots,\alpha_{k,n}$ and $\alpha_1,\ldots,\alpha_n$ be the roots...

Hey! In my calculus book they claim that a second degree polynomial always can be rewritten as x^2 - a^2 or as x^2 + a^2, if you use an appropriate change of variable. I was thinking about how this works. Let's say we have a second degree polynomial (on the general form?) ax^2 +bx + c = 0...

Whats up guys ! currently studying for calculas exam and could use someone going over my answers ! Homework Statement Q1. Calculate the taylor polynomial of degree 5 centred 0 for f(x) = e-x. Simply coeffcients and use the error formula to estimate the error when p5(0.1) Q.2 Q1...

I am reading Chapter 2: Commutative Rings in Joseph Rotman's book, Advanced Modern Algebra (Second Edition). I need help with Exercise 2.47 on page 114. Problem 2.47 reads as follows: I need help with showing that f(x) has a root \alpha \in \mathbb{F}_4 . My work on this part of the problem...

I came across the Legendre differential equation today and I'm curious about how to solve it. The equation has the form: $$(1 - x^2)y'' - 2xy' + \nu(\nu +1)y = 0, (1)$$ Where ##\nu## is a constant. The equation has singularities at ##x_1 = \pm 1## where both ##p## and ##q## are not analytic...

Homework Statement Trying to make sense of my notes... "A polynomial in n variables on an n-dimensional F-vector space V is a formal sum of the form: p(x)= ∑(C_i)x^β" so basically can somebody help me understand how polynomials represent vector spaces? Whatever degree the polynomial is...

[/itex]Homework Statement Find the first three coeficents c_n of the expansion of Cos(x) in Hermite Polynomials. The first three Hermite Polinomials are: H_0(x) = 1 H_1(x) = 2x H_0(x) = 4x^2-2The Attempt at a Solution I know how to solve a similar problem where the function is a polynomial of...

I was studying a article that solves the cube and quartic equation in the inverse sense: ##x = \sqrt[3]{A} + \sqrt[3]{B}## ##x = \sqrt[4]{A} + \sqrt[4]{B} + \sqrt[4]{C}## https://www.physicsforums.com/attachment.php?attachmentid=70239&stc=1&d=1401676309I found this relationship too...

Homework Statement Reason or prove whether there exist polynomials A, B and C such that the following is satisfied where y=e^{k\cdot arcsinx}: A\cdot y''+B\cdot y'+C\cdot y=0 Note that this is high school level calculus so it shouldn't be something too complicated. While I said...

I'm in the first of 3 courses in quantum mechanics, and we just started chapter 4 of Griffiths. He goes into great detail in most of the solution of the radial equation, except for one part: translating the recursion relation into a form that matches the definition of the Laguerre polynomials...

(1) P_{l}(u) is normalised such that P_{l}(1) = 1. Find P_{0}(u) and P_{2}(u) We have the recursion relation: a_{n+2} = \frac{n(n+1) - l(l+1)}{(n+2)(n+1)}a_{n} I'm going to include a second similar question, which I'm hoping is solved in a similar way, so I can relate it to the above...

Homework Statement Write ##sin(ax)## for ##a \in \mathbb{R}##. (Use generating function for appropriate ##z##) Homework Equations ##e^{2xz-z^2}=\sum _{n=0}^{\infty }\frac{H_n(x)}{n!}z^n## The Attempt at a Solution No idea what to do. My idea was that since...

(a) Use Taylor's Theorem to estimate the error in using the Taylor Polynomial of f(x)=sqrt{x} of degree 2 to approximate sqrt{8}. (The answer should be something like 1/2 * 8^{-7/2}. (b) Find a bound on the difference of sin(x) and x- x^{3}/6 + x^{5}/120 for x in [0,1]This is a problem on a...

Pl(u) is normalized such that Pl(1) = 1. Find P0(u) and P2(u) note: l, 0 and 2 are subscript recursion relation an+2 = [n(n+1) - l (l+1) / (n+2)(n+1)] an n is subscript substituted λ = l(l+1) and put n=0 for P0(u) and n=2 for P2(u), didnt get very far please could someone...

If $P(0)=3$ and $P(1)=11$ where $P$ is a polynomial of degree 3 with integer coefficients and $P$ has only 2 integer roots, find how many such polynomials $P$ exist?

i.e., does the set of functions of the form, \{ x^{\frac{n}{m}}\}_{n=0}^{\infty} for some fixed m produce a linearly independent set? Either way, can you give a brief argument why or why not? Just curious :)

What is the maximum number of roots of a multivariate polynomial over a field? Is there a multivariate version of the fundamental theorem of algebra?

Hello! :cool: I want to find the greatest common divisor of $x^4+1$ and $x^2+x+1$. I applied the Euclidean division and found that $x^4+1=(x^2+x+1) \cdot (x^2-x)+(x+1)$. So,isn't it like that: $gcd(x^4+1,x^2+x+1)=x+1$ ? But.. in my textbook,the result is $1$! Which of the both results is...

Homework Statement Suppose that u and v are real numbers for which u + iv has modulus 3. Express the imaginary part of (u + iv)^−3 in terms of a polynomial in v.Homework Equations The Attempt at a Solution |u+iv|=3 then sort(u^2+i^2) = 3 then u = 3 and v=0 or u=0 and v=3(0+3i)^-3 i swear i am...

Hello. I open this 'thread', in number theory, but he also wears "calculation". I've done a little research, I share with you. Let \ r_1, r_2, \cdots, r_n, roots of the polynomial. P(x)=p_0x^n+p_1x^{n-1}+ \cdots+p_{n-1}x+p_n Let \ Q(x)=q_0 x^n+q_1x^{n-1}+ \cdots +q_n, such that its roots...

I've been taught that with the basic form of a function's maclaurin series, complex forms of the same series can be found. For example, the first three terms for arctan(x) are x-x^3/3 + x^5/5, meaning the first three terms for arctan(x^2+1) at a=0 should be (x^2+1) - ((x^2+1)^3)/3 +...

Homework Statement I want to transform a polynomial of kind p(x)=ax³+bx²+cx+d in another like p(t)=At³+B. Is possible? Homework Equations Is possible to transform a polynomial of kind ax³+bx²+cx+d in another like t³+pt+q...

Hello. Homework Statement Basically I want to evaluate the integral as shown in this document: Homework Equations The Attempt at a Solution The integral with the complex exponentials yields a Kronecker Delta. My question is whether this Delta can be taken inside the integral...

Greetings! :biggrin: Homework Statement Starting from the Rodrigues formula, derive the orthonormality condition for the Legendre polynomials: \int^{+1}_{-1} P_l(x)P_{l'}(x)dx=(\frac{2}{2l + 1}) δ_{ll'} Hint: Use integration by parts Homework Equations P_l=...

Hey :o ! Could you help me at the following exercise? $k, n \in \mathbb{N}$ $f(x)=cos(k \pi x), x \in [0,1]$ $x_i=ih, i=0,1,2,...,n, h=\frac{1}{n}$ Let $p \in \mathbb{P}_n$ the Lagrange interpolating polynomials of $f$ at the points $x_i$. Calculate an upper bound of the maximum error...

Contemporary Abstract Algebra by Gallian This is Exercise 14 Chapter 3 Page 69 Question Let $G$ be the group of polynomials under the addition with coefficients from $Z_{10}$. Find the order of $f=7x^2+5x+4$ . Note: this is not the full question, I removed the remaining parts. Attempt...

Homework Statement Use a Taylor Polynomial about pi/4 to approximate cos(42){degrees} to an accuracy of 10^-6. *To get an accuracy of 10^-6, use the error term to determine an nth Taylor Polynomial to use. Homework Equations x = 45 or pi/4, x0 = 42 or 7pi/30 cos(x) = Pn(x) + Rn(x)...

Homework Statement I want to varify that the components of a homogenous electric field in spherical coordinates \vec{E} = E_r \vec{e}_r + E_{\theta} \vec{e}_{\theta} + E_{\varphi} \vec{e}_{\varphi} are given via: E_r = - \sum\limits_{l=0}^\infty (l+1) [a_{l+1}r^l P_{l+1}(cos \theta) - b_l...

show that the first derivative of the legendre polynomials satisfy a self-adjoint differential equation with eigenvalue λ=n(n+1)-2 The attempt at a solution: (1-x^2 ) P_n^''-2xP_n^'=λP_n λ = n(n + 1) - 2 and (1-x^2 ) P_n^''-2xP_n^'=nP_(n-1)^'-nP_n-nxP_n^' ∴nP_(n-1)^'-nP_n-nxP_n^'=(...

Hi everyone, :) Here's a question I encountered and I need your help to solve it. Question: Let \(V\) be the space of real polynomials of degree \(\leq n\). a) Check that setting \(\left(f(x),\,g(x)\right)=\int_{0}^{1}f(x)g(x)\,dx\) turns \(V\) to a Euclidean space. b) If \(n=1\), find...

I am reading R Y Sharp: Steps in Commutative Algebra. In Chapter 3 (Prime Ideals and Maximal Ideals) on page 44 we find Exercise 3.24 which reads as follows: ----------------------------------------------------------------------------- Show that the residue class ring S of the ring of...

Does anyone understand this project? I desperately need your help! Please let me know. Appreciate a lot!

Homework Statement Let T: R^6 -> R^6 be the linear operator defined by the following matrix(with respect to the standard basis of R^6): (0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 ) a) Find the T-cyclic subspace generated by each standard basis vector...