Recent content by duc

I've found the solution which is of the following form ## a_n = \sum_{m=1}^{l} \frac{1}{m} \sum_{i=0}^{n} a_i^{(m-1)} a_{n-i}^{(l-m)} ## where a_i^(m-1) is the coefficient corresponding to the power ## x^i ## of the polynomial ## P_{m-1}(x) ## (the same convention for ## P_{l-m}(x) ##).

Homework Statement The polynomial of order ##(l-1)## denoted ## W_{l-1}(x) ## is defined by ## W_{l-1}(x) = \sum_{m=1}^{l} \frac{1}{m} P_{m-1}(x) P_{l-m}(x) ## where ## P_m(x) ## is the Legendre polynomial of first kind. In addition, one can also write ## W_{l-1}(x) = \sum_{n=0}^{l-1} a_n \cdot...

Yes i know. What matters is the factor in front of each derivative, e.g ## \frac{1}{r \sin\theta} ##. Hence, only if the derivatives ## \frac{\partial}{\partial r}, \frac{\partial}{\partial \phi} ## involves, it follows that: ## e^{\frac{\partial}{\partial r} + \frac{\partial}{\partial \phi}} =...

The radial derivative ## \frac{\partial}{\partial r} ## will commute with ## \frac{\partial}{\partial \theta}## and ## \frac{\partial}{\partial \phi}## when the function to be derived is factorizable. Ps: Sorry, I'm wrong about the commutations of ## \frac{\partial}{\partial x_i}, (x_i =...

Thanks for your reply, unfortunately, it is not that simple. Remembering that when you decompose the gradient operator into ## \overrightarrow{\nabla} = \underbrace{\frac{\partial}{\partial r}}_{\hat{A}} + \underbrace{\frac{1}{r}\frac{\partial}{\partial \theta}}_{\hat{B}} +...

Thank you very much DEvens. I've thought of it but had some doubts and didn't go further :D. It is indeed "trivial" as you said :">. Thanks again. :)

Homework Statement Find the action of the operator ## e^{\vec{a} . \vec{\nabla}} \big( f(\theta,\phi) . g(r) \big) ## where \nabla is the gradient operator given in spherical coordinates, f and g are respectively scalar functions of the angular part ## ( \theta, \phi) ## and the radial part ##...

Hi DEvens, Thanks for your reply. Maybe i should be more specific. The system of matrix equations I've mentioned is not linear system of equations that can be solved by using matrix technique. A, B, C and D here are not number but square matrices of dimension N x N (where N is integer and...

Hello everyone, I'm struggling with a coupled of matrix equations of the general form: AX + CY = cX BY + DX = cY where A, B, C and D are hermitics square matrices. X, Y and c are the eigenvector and eigenvalue to be found. I'm looking for a method or an algorithm to solve this system by using...