# ramanujan_dirac

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I am a physics undergrad with interests in theoretical physics and mathematics.

 Feb22 comment Proof that Legendre Polynomials are Complete @Anonymous: Isn't the other answer complete and correct? Feb10 comment Has Prof. Otelbaev shown existence of strong solutions for Navier-Stokes equations? @StephenMontgomery-Smith: What is the current status/conclusion? Has the proof been fixed? Has it shown to be completely wrong? Jan26 comment Example of infinite field of characteristic $p\neq 0$ @ZevChonoles: I wonder if there are more examples, this is the only example that I know of. Jul30 comment Advanced undergraduate(?) Real Analysis book which is concise and lots of interesting problems @Potato: Ohh. I didn't have a good look at Rudin, but I assumed it overlaps too much with the last Real Analysis text I read, Bartle and Sherbert. I will have a look. Jun3 comment Uniqueness of the vector in $\mathbb{R}^n$ specified by the curl, divergence and the normal component @Murphid: Please could you help me prove the uniqueness of the vector field? Jun2 comment Riemann, Ricci curvature tensor and Ricci scalar of the n dimensional sphere @celtschk: Its linear in the second order derivative of the metric. The expression for the 2 sphere I got is, (I am 100% sure this is correct) for the Ricci tensor is $R_{ij}=\frac{1}{R^2}g_{ij}$. So, I should be getting a same expression in the n dimensional case, with a different constant. For the 2 sphere. $R_{ijk}=\frac{1}{R^2}(\delta^i_j g_{jm}-\delta^i_m g_{jk})$ May28 comment $\sum_{m=-l, …,l; l=0,1,2,..} e^{\frac{-i E_l (t_f-t_i)}{\hbar}} Y_{lm}(\phi_f,\theta_f)Y_{lm}(\phi_i, \theta_i)$ I want to compute the $l$ sum, the $m$ sum has been done using the addition theorem for spherical harmonics. May28 comment $\sum_{m=-l, …,l; l=0,1,2,..} e^{\frac{-i E_l (t_f-t_i)}{\hbar}} Y_{lm}(\phi_f,\theta_f)Y_{lm}(\phi_i, \theta_i)$ @RonGordon: No. It is actually the angle between the two position vectors: i.e. if $(\theta_f, \phi_f)$ and $(\theta_i, \phi_i)$ correspond to $\hat{n_f}$ and $\hat{n_i}$, then $\cos{\theta}=\hat{n_f} \cdot \hat{n_i}$ May27 comment Diagonalizing/eigenvalues of a particular infinite dimensional matrix Ok. How was this related to the work done on Toeplitz matrices? May25 comment Diagonalizing/eigenvalues of a particular infinite dimensional matrix @Raskolnikov: If it is not too much trouble, please could you outline the method/properties of the characteristics. I am unable to find a suitable reference on the internet May6 comment Prove there is no element of order 6 in a simple group of order 168 Why does $n_2=21$, mean there is no element of order 6? Apr30 comment Problem evaluating an improper integral $\int_0^{\infty} \frac{(\sin{2x}-2x\cos{2x})^2}{x^6}$ using fourier transform Thanks! I missed a term in the second integral. I am really clumsy in my calculations. Sorry for the trouble. Apr30 comment Problem evaluating an improper integral $\int_0^{\infty} \frac{(\sin{2x}-2x\cos{2x})^2}{x^6}$ using fourier transform At first glance, I am flabbergasted. The first term $\int_o^{\infty}a^2 \cos{\xi x} dx=a^2\frac{\sin{a \xi}}{\xi}=\frac{a^2 \xi ^2 \sin{a \xi} }{\xi ^3}$, which is clearly missing from your answer, and clearly present in mine. The second part done by parts, should't cancel it out. Thanks a lot for the trouble though. :) Apr30 comment Problem evaluating an improper integral $\int_0^{\infty} \frac{(\sin{2x}-2x\cos{2x})^2}{x^6}$ using fourier transform Hi. Should I have taken the sine fourier transform? My sine and cos are interchanged. But the function is even, so the sine part would be zero right? Apr30 comment Problem evaluating an improper integral $\int_0^{\infty} \frac{(\sin{2x}-2x\cos{2x})^2}{x^6}$ using fourier transform @BarackObama: I didn't know the plancherel's formula, so I looked it up on the internet, which correct me in I am wrong is $\int_{\mathbb{R}}||f(x)||^2 \, dx = \int_{\mathbb{R}}||\hat{f(\xi)}||^2 \, d {\xi}$. But in my problem, the integrand is not the square of the entire fourier transform. There is an extra term, which I won't be able to integrate after squaring. How do I do it? Apr30 comment Evaluating improper integrals using laplace transform Awesome! Thanks. Apr30 comment Condition for the inverse laplace transform of a function to exist and bromwich integral Assuming your $t$ is my $x$, I don't see why $\frac{2e^{iR}}{x^3}$ would be zero, if $x$ is non zero. Have I dont the contour integral right? Here there are no singularities so my $a$ can be anything, and my contour is the straight line parallel to the imaginary axis. Also, do you have a answer to my first query. Is there any theorem, which places some restriction on my $F(s)$ so that it has a inverse laplace transform. Apr29 comment Solving an integral equations using fourier transform Thanks! Please could you tell me how you got the sign function from the integral containing sin? Apr29 comment Solving an integral equations using fourier transform I had done uptil this point, and was asking about the inverse fourier transform of my RHS, which I didn't know is a famous function called the sinc function, and who transform is the box function. BTW, should't the $\pi$ cancel out as $f(x)=\frac{1}{\pi}\int_0^{\infty}A(\xi)\cos{\xi x}$, where $A$ is the even part. The definitions vary, but I thought one of the integrals of the pair is defined with a $2 \pi$ Apr29 comment Solving an initial value ODE problem using fourier transform Thanks a lot for the answer. Would it work, if I solve the homogeneous differential equation separately, and then add it to inverse fourier transform that has been found out to form the general solution? So in the example that I have mentioned in the last sentence: I add $e^x(A \cos{2x}+B\sin{2x})$ to the solution obtained by the inverse fourier transform and then substitute in the initial conditions? Just want to make sure if this would work.