Recent content by Sistine

Yes I read the edit part of your post, but in my lecture notes there was nothing said about approximation by the Yukawa potential, the integral was just given as I stated it above. Is the integral an approximation? It partially works out when integrated. (also is it related to the first born...

Thanks for your reply, I tried integrating over \mathbb{R}^3 but my integral does not converge for 0\leq r<\infty i.e. integrating \int_0^{2\pi}\int_0^{\pi}\int_0^{\infty}r e^{ikr \cos\theta} \sin\theta dr d\theta d\phi Doing the \theta integration first I get...

Homework Statement In calculating the quantum mechanical amplitude for the Coulomb potential (scattering of say \alpha particle off a massive particle of charge Ze), I came across a Fourier transform which I could not calculate. If U(r)=\frac{2Ze^2}{4\pi\epsilon_0 r} then...

Homework Statement Prove that if f is a meromorphic function f:\mathbb{C}\rightarrow\mathbb{C} with |f(z)|^5\leq |z|^6\quad\textrm{for all}\quad z\in\mathbb{C} Then f(z)=0 for all z\in\mathbb{C} Homework Equations Liouville's Theorem A bounded entire function is constant. The...

Perhaps your right. The image of the Gauss map at a point is perpendicular to the tangent space at that point , so that no linear combination of \partial_1\sigma, \partial_2\sigma could ever represent N at that point. However is it possible to represent the map N:R2->R3 as a matrix? I'll try to...

Homework Statement Consider the following parametrization of a Torus: \sigma(u,v)=((R+r\cos u)\cos v, (R+r\cos u)\sin v, r\sin u) R>r,\quad (u,v)\in [0,2\pi)^2 1. Compute the Gauss map at a given point. 2. What are the eigenvalues of that map in the base...

I have thought again about the problem. I know that all the charge within the solid will work to repel itself so that the all the charge tends towards the surface. So that by Gauss' law the electric field within the surface is zero. But I still don't know what the electric field will be on the...

Homework Statement What is the electric field inside a a solid of uniform chage density i.e. \mathbf{E}(\mathbf{r})=\frac{1}{4\pi\varepsilon_0}\int_V\rho(\mathbf{r}')\frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|}dV' What is the electric field at \mathbf{r}'=\mathbf{r} if...

Homework Statement A tank is traveling in a straight line we look at the side on view of the tank and consider its continuous track in contact with the x-axis. Its wheels have radius R and the distance between he centers of the wheels is L (The continuous track is wrapped around the wheels)...

I tried letting the radius of the contour go to infinity but I got 0 as my final answer. Numerical methods seem to point to the conclusion that the integral is non-zero, so I don't think this is correct. I know that \int_{C_{30}}f=\int_{C_R}f\quad R>30 but is it true in the limit as R\to\infty?

Homework Statement Compute the integral \oint_{|z|=30}\frac{dz}{z^9+30z+1} Homework Equations Residue theorem for a regular closed curve C \onit_C f(z)dz=2\pi i\sum_k\textrm{Res}(f,z_k) z_k a singularity of f inside C The Attempt at a Solution I'd rather not compute the...

I tried applying schwarz lemma to |f(z)|\leq |e^z| i.e. \left|\frac{f(z)}{e^z}\right|\leq 1 But this did not give me much information about f. What other Theorems from Complex Analysis could I use to gain information about f?

Homework Statement Find all entire functions f such that |f(z)|\leq e^{\textrm{Re}(z)}\quad\forall z\in\mathbb{C} Homework Equations \textrm{Re}(u+iv)=u The Attempt at a Solution I tried using Nachbin's theorem for functions of exponential type. I also tried using the Cauchy...

Homework Statement Find a function g analytic in |z|\leq 2, with g(2/3)=0 and |g(z)|= 1 on |z|=2 Homework Equations Bilinear maps B_{\alpha}(z)=\frac{z-\alpha}{1-\overline{\alpha}z} |B_{\alpha}(z)|=1 on |z|=1 The Attempt at a Solution I tried using the maximum...

Homework Statement I'm trying to prove the following inequality for 0\leq \theta\leq\frac{\pi}{2} \frac{2}{\pi} \theta \leq\sin\theta Homework Equations 0 \leq \theta\leq\frac{\pi}{2} 0\leq \frac{2}{\pi}\theta\leq 1 The Attempt at a Solution I've looked at the limit...