Prove that the Legendre polynomials satisfy the orthogonality condition, given by ♫_₁ P₁(x)P, (x) dx = 22²-18c -Sel 2l+1

Q: 3) Let f(T) e R[T] be a polynomial of degree > 3. Show that there exists polynomials g(T) and h(T)…

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Q: Consider the polynomial p(z) = z 101 + 7· z² + 3 · i · z – 1. Using Rouche's Theorem, determine how…

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Q: Find all homogeneous polynomials of degree 3 in R² 3 p = Ax² + Bx²³y + Cxy² + Dy ³ that are…

A: Let, the homogeneous polynomial beThen, the function p is called harmonic if

Q: Explain why the fundamental theorem of algebra does not apply to f(x) = Vx + 3. That is, no complex…

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Q: (a) Calculate the resultant of A = 3x² +3 and B = (x-2)(x+5) by hand. (b) Let A, B, C be…

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Q: – x²y? x2 + y? f(x, y)

A: fx, y=-x2y2x2+y2 Now, z0=fa, b=f1, -2=-12-2212+-22=-45 Now partially differentiating fx,…

Q: Verify that U(x,y) = x3y - xy3 is harmonic on the complex plane.

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Q: Express K (x) in terms of the orthonormal Legendre's polynomials, with the inner product. Here, K(x)…

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Q: 6. Let 1 f (x) - cos x + sin(2x), x E [0, 1] 2 Find the least squares polynomial approximation of…

A: Let fx=12cos x-sin2x, x∈0,1. To find quadratic fitting  fx≈P2x=a0+a1x+a2x2 Consider, s0=∫011dx=1

Q: Define T:P,-R as shown to the right. P(- 1) a. Find the image under T of p(t) = - 2-t b. Show that…

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Q: For the strong form: d (1– 2x²) du + и 3 х3, for 0 <x <1 du u(0) = 1, = 2 |x=1 Obtain (i) primary…

A: Since, you have post multiple sub parts. I am answering first three parts. Please post the question…

Q: Construct the Lagrange interpolation Polynomial for the function Y= Sin Tx, the ponts =0, 2, =…

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Q: Mümkün olan en yüksek dereceden polinomlar için aşağıdaki kareleme formülünü kesin eşitliğe…

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Q: Find the best least squares approximation to f (x) = |x| on [−π, π] by a trigonometric polynomial of…

A: Introduction: To find the best least squares approximation to f(x) = |x| on [-π, π] by a…

Q: equational-style proof to prove the theorem schema + (B+C) = (((Bv C) → C)) = (((BAC) V T)) (3) Use…

A: Let's prove the given theorem schema using equational-style proof. Want to show:1. Prove :   Will…

Q: Consider the function f(x) = -x³+2x²+x-2. Generate four evenly-space values of f(x) from x = -1 to…

A: Given function is, fx=-x3+2x2+x-2 The four evenly spaced values of fx from x=-1 to x=2 Then, the…

Q: Let p: R → R be a polynomial of odd degrees. Prove that the equation p(x) = 0 has a solution in real…

A: Given p:R→R be a polynomial of odd degree. We have to prove that the equation px=0 has a solution in…

Q: a) Find a Lagrange interpolating polynomial in the form of P, (x) = ax³ + bx² + cx+d befitting the…

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Q: Determine the variation of the functional (b) J(x)=√ [x7(1) + x₁(1)x₂(1) + x3(1) + 2x₁(1)×₂(1)] dt.

A: To determine the variation of the functional J(x), we need to find the variation δJ for an arbitrary…

Q: Consider the polynomials a(x) = r' + x³ + x + 4 and b(x) = 2r³ + 4.x² + x + 2 from Z5[r]. a) Find…

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Q: Consider the quadratic function f: R3 →R defined by f(X) = xỉ – x1x2 + x – x2x3 + x3 + 1 A: Express…

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Q: that the Lagrange Polynomial of values Know follows b) F a) T as 0 1 2 = (x²-19x + 6) :- 3 7

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Q: Prove that for any nonzero real number m, the functions f(x) = sin(mx) and g(x) = cos(mx) are…

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Q: Show that u = √3+ √2i is algebraic over Q and type its minimal polynomial p(x) = irr(u, Q) below.

A: This question is about group theory.

Q: Show that u = √3+ √2i is algebraic over Q and type its minimal polynomial p(x) = irr(u, Q) below

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Q: 3. Given the functions f, (x) = x and f2(x) = sin2x on the interval - %3D (a) Find the norm of f, on…

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Q: Verify that the real-valued function u(x, y) = x^3 − 3xy^2 − 2y defined on the complex plane C is…

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Q: 1: On the polynomial space P₁ John chooses the scalar product (flg)₁ = ₁f(x)g(x) dx, while Jim uses…

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Q: 2. Consider the Newton polynomial Pn(x) = ao+a1(xr-xo) + a₂(x-xo)(x − x₁) +... +an(x-xo)(x -…

A: Given that   that interpolate then we have Now  ,       putting

Q: Q[V2 ]is Principal Ideal Domain..

A: Q[√2] is Principal Ideal Domain. True.

Q: f(x) = x² − 3 is the minimal polynomial of √3 over Q,

A: We have that              f(x) = x2-3  be a polynomial of two degree.

Q: a) Factor a +x³ + x into irreducible polynomials in Z2[r];

A: i) Linear factors are irreducible. ii) Just substitute values in Zp and see the expression becomes…

Q: here the spacing is suc

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