Show that the eigenfunctions un(x) are orthogonal .

Expert Solution

Colby Messinger answered 2 years ago

Based on Euler's identity, show the linear superposition of the eigenfunctions

Show that the Legendre polynomials P1 and P2 are orthogonal by explicit integration. Also show that...

Show that the Legendre polynomials P1 and P2 are orthogonal by explicit integration. Also show that when (P2)^ 2 is integrated over the full range of integration, the result is 2 /(2l+1) , where l is the order of the polynomial.

Show that the following are eigenfunctions of the Laplacian operator and determine the eigenvalues: a. (r^-1)sinkr...

Show that the following are eigenfunctions of the Laplacian operator and determine the eigenvalues: a. (r^-1)sinkr b.( r^-3)[(sin^2)θ]sin2φ

Find the linear space of eigenfunctions for the problem with periodic boundary conditions u′′(x) = λu(x)...

Find the linear space of eigenfunctions for the problem with periodic boundary conditions u′′(x) = λu(x) u(0) = u(2π) u′(0) = u′(2π) for (a) λ = −1 (b) λ = 0 (c) λ = 1. Note that you should look for nontrivial eigenfunctions

Determine the eigenvalues and the corresponding normalized eigenfunctions of the following Sturm–Liouville problem: y''(x) + λy(x)...

Determine the eigenvalues and the corresponding normalized eigenfunctions of the following Sturm–Liouville problem: y''(x) + λy(x) = 0, x ∈ [0;L], y(0) = 0, y(L) = 0,

Show that the bound state wave function of the δ function's well is well orthogonal to...

Show that the bound state wave function of the δ function's well is well orthogonal to all scattering state wave functions.

Show that in 2D, the general orthogonal transformation as matrix A given by {{cos, sin}, {-sin,...

Show that in 2D, the general orthogonal transformation as matrix A given by {{cos, sin}, {-sin, cos}}. Verify that det[A] = 1 and that the transpose of A equals its inverse. Let Tij be a tensor in this space. Write down in full the transformation equations for all its components and deduce that Tii is an invariant.

Find the family of curves orthogonal to: y4 = k(x − 2)3.

Un = {x ∈ Zn* | x & n are relatively prime}; w/ operator multiplication modulo(n)...

Un = {x ∈ Zn* | x & n are relatively prime}; w/ operator multiplication modulo(n) show: Un is a commutative group.

Find the eigenvalues λn and eigenfunctions yn(x) for the given boundary-value problem. (Give your answers in...

Find the eigenvalues λn and eigenfunctions yn(x) for the given boundary-value problem. (Give your answers in terms of k, making sure that each value of k corresponds to two unique eigenvalues.) y'' + λy = 0, y(−π) = 0, y(π) = 0 λ2k − 1 =, k=1,2,3,... y2k − 1(x) =, k=1,2,3,... λ2k =, k=1,2,3,... y2k(x) =, k=1,2,3,...

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