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Rodrigues Formula for Hermite polynomials and Orthogonality

10:21

Recursion Relations and Generating Function of Hermite Polynomials

10:49

The Hermite Differential Equation and Hermite Polynomials

11:28

The Generating Function for Laguerre Polynomials

10:00

Recursion Relation for Laguerre Polynomials

7:39

Rodrigues Formula and Orthogonality of the Laguerre Polynomials

9:10

Schrodinger Equation for a Quantum Mechanical Harmonic Oscillator

We consider the Schrodinger equation for the wave function on the real line with a harmonic potential. This is one of the rare cases when we have find exact forms of the solutions. When we separate variables we obtain two differential equations for which the eigenvalues (energy levels) have solutions in the form of the product of Gaussians and Hermite polynomials.
#mikethemathematician, #mikedabkowski, #profdabkowski

Відео

Rodrigues Formula for Hermite polynomials and Orthogonality

10:21

Rodrigues Formula for Hermite polynomials and Orthogonality

Переглядів 632 години тому

We use the generating function for the Hermite polynomials (ua-cam.com/video/aixs-OoixAM/v-deo.html) to prove the Rodrigues formula for the Hermite polynomials. Once we have the Rodrigues formula we can easily prove that the Hermite polynomials are orthogonal with respect to the Gaussian measure on the real line. We also find the L^2 norm of the Hermite polynomials with respect to this measure....

Recursion Relations and Generating Function of Hermite Polynomials

10:49

Recursion Relations and Generating Function of Hermite Polynomials

Переглядів 1084 години тому

We use the exact formula for the Hermite polynomials (ua-cam.com/video/X2_yu-nlo_U/v-deo.html) to find a three term recursion formula for the Hermite polynomials. Once we obtain the recursion formula, we use the standard technique to find the generating function for the Hermite polynomials. We will use this generating function to find the Rodrigues formula and find eigenfunctions of the Fourier...

The Hermite Differential Equation and Hermite Polynomials

11:28

The Hermite Differential Equation and Hermite Polynomials

Переглядів 2017 годин тому

We consider the Hermite differential equation, which is a second order linear nonconstant coefficient equation. The origin is an ordinary point, so we can write down the series solution of this differential equation. For certain values of the parameter, there will be polynomial solutions: the Hermite polynomials. This polynomials arise in the study of the quantum mechanical harmonic oscillator ...

The Generating Function for Laguerre Polynomials

10:00

The Generating Function for Laguerre Polynomials

Переглядів 779 годин тому

We use the recursion relation for Laguerre polynomials (ua-cam.com/video/v_qjpPSSz4U/v-deo.html) to find a closed form expression for the generating function of the Laguerre polynomials. Generating functions play an essential role in discovering values of these polynomials at certain points and relationships among the derivatives of these functions. The generating function, and its generalizati...

7:39

Recursion Relation for Laguerre Polynomials

Переглядів 5212 годин тому

We use the Rodrigues Formula for the Laguerre polynomials (ua-cam.com/video/YFdPDbaVFqo/v-deo.html) to find a recursion relation for the Laguerre polynomials. As we have seen, these polynomials are orthogonal with respect to the weight exp(-x), so there must be a three term recursion formula for them. We will use this recursion to derive the generating function for the Laguerre polynomials whic...

Rodrigues Formula and Orthogonality of the Laguerre Polynomials

9:10

Rodrigues Formula and Orthogonality of the Laguerre Polynomials

Переглядів 10714 годин тому

We continue our study of the Laguerre polynomials (ua-cam.com/video/iDONsJKd9pQ/v-deo.html) by proving the Rodrigues formula. This formula expresses the Laguerre polynomials as an nth derivative of a product of x^n and exp(-x) then scaled by exp(x) and 1/n!. This formula is then used to prove that these polynomials are orthogonal on the positive real axis with respect to the measure exp(-x)dx (...

The Laguerre Differential Equation and Laguerre Polynomials

8:00

The Laguerre Differential Equation and Laguerre Polynomials

Переглядів 17116 годин тому

We consider the Laguerre Differential Equation which arises when seperating variables in the Schrodinger Equation for the hydrogen atom. The fact that the Laguerre equation admits polynomial solutions allow for radial solutions which have finite second moment. We study these polynomials solutions and assemble a collection of properties of these polynomials which will help us understand the solu...

10:19

Hypergeometric Functions

Переглядів 20419 годин тому

We consider the hypergeometric ordinary differential equation. This equation has regular singular points at 0, 1, and infinity. Every linear differential equation with exactly three regular singular points can be transformed into this equation. We find the series representation of the hypergeometric function using the Method of Frobenius. #mikethemathematician, #profdabkowski, #mikedabkowski

8:52

L^2 Norms of the Legendre Polynomials

Переглядів 11921 годину тому

We use the generating function of the Legendre polynomials to compute the L^2 norm of the nth Legendre polynomial. These values are important when used in conjunction with the orthogonality of these polynomials to construct an orthonormal basis of functions for which we can find the Fourier-Legendre expansions of functions which will be useful in solving boundary value problems. #mikethemathema...

The Generating Function for the Legendre Polynomials

11:59

The Generating Function for the Legendre Polynomials

Переглядів 91День тому

We derive the formula for the generating function of the Legendre polynomials. Starting with the Bonnet Recursion Formula (ua-cam.com/video/2oWIzqm6x8c/v-deo.html), we write down a Maclaurin series whose coefficients are the Legendre polynomials. Shifting the series by one term and using the Bonnet recursion formula, we arrive at an integral equation for the series. Differentiating that equatio...

Bonnet's Recursion Formula for the Legendre Polynomials

10:02

Bonnet's Recursion Formula for the Legendre Polynomials

Переглядів 83День тому

We use the Rodrigues formula for the Legendre polynomials (ua-cam.com/video/TF3x4YXFMME/v-deo.html) to prove the Bonnet recursion formula for the Legendre polynomials. This recursion relates one Legendre polynomial to the previous two. It will be used in the proof of the generating function for the Legendre polynomials. It is also very useful for quickly computing the first several Legendre pol...

Rodrigues Formula for the Legendre Polynomials and Orthogonality

7:53

Rodrigues Formula for the Legendre Polynomials and Orthogonality

Переглядів 167День тому

We consider the Legendre polynomials in closed form (ua-cam.com/video/Ow-K3BzIpgs/v-deo.html), we show that these polynomials can be expressed as the nth derivative of a polynomial of twice the degree that has roots at -1 and 1 of order n. This formula is one instance of the Rodrigues formula for orthogonal polynomials. The Rodrigues formula has a wide number of applications, in particular to t...

12:03

The Legendre Polynomials in Closed Form

Переглядів 142День тому

We use the recursion relationship for the the coefficients of the power series solution to the Legendre differential equation (ua-cam.com/video/KRZCjY_J1So/v-deo.html) to find a closed form for the Legendre polynomials. While the closed form may appear slightly complex, it will be important to know the exact coefficients when dealing with problems in electrostatics or gravitation. We express th...

7:38

The Legendre Differential Equation

Переглядів 8914 днів тому

We study the Legendre differential equation which arises in the study of spherical harmonics and electrostatics. This second order linear differential equation corresponds to Sturm-Liouville eigenvalue problem. We will find a solution to this differential equation around the ordinary point of zero (it has three regular singular points at 1, -1 and infinity). We show that there are polynomial so...