Recent content by nickthequick

Hi, Given a holomorphic function u(x,y) defined in the half plane ( x\in (-\infty,\infty), y\in (-\infty,0)), with boundary value u(x,0) = f(x) , the solution to this equation (known as the Poisson integral formula) is u(x,y) = \int_{-\infty}^{\infty} \frac{y\ f(t) }{(t-x)^2 +y^2} \...

AlephZero, Thanks, this is very helpful!Nick

Hi, First off, apologies if this is in the wrong place - any directions on where this is more appropriate are appreciated. I am trying to figure out how to convert a smoothing operation in physical space into one in Fourier space. In physical space, with equally spaced points x_j, (j...

Hi, I'm trying to solve an algebraic-integral equation (I don't know if this is the proper description of this class of equations, it just seems like the least wrong way to describe them) and have run into several issues that I'll describe below, but first I'll outline the problem. I'm...

Hm. I suspected as much. I'm going to end up differentiating this term w.r.t \alpha_n as it is part of the potential term in a Lagrangian. I don't see how that would help solve this, but I note it for completeness.

Hi, I am trying to make progress on the following integral I = \int_0^{2\pi} \sqrt{(1+\sum_{n=1}^N \alpha_n e^{-inx})(1+\sum_{n=1}^N \alpha_n^* e^{inx})} \ dx where * denotes complex conjugate and the Fourier coefficients \alpha_n are constant complex coefficients, and unspecified...

Got it! Thanks

Hi, I'm trying to find \iint_S \sqrt{1-\left(\frac{x}{a}\right)^2 -\left(\frac{y}{b}\right)^2} dS where S is the surface of an ellipse with boundary given by \left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 = 1 . Any suggestions are appreciated! Thanks, Nick

On second thought, the Joukowski map seems inappropriate here. I think the map I want is z\to a \cosh(\xi + i \eta) so that x=a\sinh (\xi) \cos(\eta) and y = a\cosh (\xi) \sin(\eta). This will effectively give me the change in functional form that I expect; however, I still don't see...

Hi, Given that the flow normal to a thin disk or radius r is given by \phi = -\frac{2rU}{\pi}\sqrt{1-\frac{x^2+y^2}{r^2}} where U is the speed of the flow normal to the disk, find the flow normal to an ellipse of major axis a and minor axis b. I can only find the answer in the...

Hi, I'm trying to make headway on the following ghastly integral: \int_0^{\infty} x^{\frac{3}{2}}e^{-xd} J_o(rx) \frac{\sin (\gamma \sqrt{x}\sqrt{x^2+\alpha^2}t)}{\sqrt{x^2+\alpha^2}}\ dx where d,r, \alpha, \gamma ,t \in \mathbb{R}^+ and J_o is the zeroth order Bessel function of...

Hm. Thanks for the suggestions. Here's what I've come up with. Rewrite I as (I've changed d to h here for clarity of presentation) I = \frac{i}{3}\partial_k \partial_h\iint \frac{1}{(x^2+y^2+h^2)^{\frac{3}{2}}}e^{-i(kx+\ell y)} \ dx \ dy =\frac{i}{3}\partial_k...

Hi all, I've been trying to solve the following I = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{x}{(x^2+y^2+d^2)^{\frac{5}{2}}} e^{-i(kx+\ell y)} \ dx \ dy where d,k,\ell are constants. I haven't been able to put this into a tractable analytic form and I figured I'd consult all...

The problem I was having was happening when I was dealing with integration of surfaces at infinity. This confuses me for a variety of reasons and I'm not sure that I even need to address this issue. Instead, let's discuss what I am really interested: Assume \vec{P} is nonzero, is in a finite...

Hi, Consider P = \boldsymbol{\nabla} f +\boldsymbol{\nabla}\times \bold{A} where f and A are scalar and vector potentials, respectively, and P is strictly positive and well behaved, and only nonzero in a domain \mathcal{D}. I want to find how the magnitude of \int \boldsymbol{\nabla} f dV...