Recent content by eschiesser

But then, isn't my contour running over the pole in each integral?

Homework Statement I need to solve this integral for a>0: \int _0^{\infty }\frac{\text{Sin}[x]}{x}\frac{1}{x^2+a^2}dx The Attempt at a Solution Using wolfram mathematica, I get that this integral is: \frac{\pi -e^{-x} \pi }{2x^2}=\frac{\pi (1-\text{Cosh}[a]+\text{Sinh}[a])}{2...

Here's a quote from my notes: "For a given 'Illumination Aperture Stop', the degree of coherence depends on the size of the Pupil Plane Aperture (i.e. resolution)." and COHERENT ILLUMINATION (in context of a microscope with Koehler Illumination) Remember the size of the Pupil determines the...

Yes Andy, I suppose it does. It is essentially a confirmation of my intuition, but I am still uncertain about what my professors mean when they ask me about the coherence of a system. Perhaps this is a question for them. So, what role does my imaging system play in the coherence of my image, in...

This is a very vague question, but I hope by asking it, we can pin down an answer eventually. In optics, there are such things as "Coherent Imaging" and "Incoherent Imaging." From what I understand, the degree of coherence has to do with the size of your Point Spread Function (PSF), a.k.a...

if I am understanding the problem correctly, which is not an assumption I would bet any substantial amount of money on, I thought that the point was to show that f_m is the best substitution for \gamma_m, e.g. show that f_m=\gamma_m reduces E better than any other substitution for \gamma_m...

Homework Statement So I'm supposed to show that a finite Fourier approximation is the optimal approximation for a given function. I am to suppose we have a given set of functions \phi _k(x),k=1,2,\text{...}N defined on a\leq x\leq b. They are orthogonal \int _a^b\phi _m(x)\phi _n(x)dx=0...

wow...never even crossed my mind to use trig substitution. In fact, I completely forgot that as a method. And its so useful! Thank you! I guess I ought to review calc 2 stuff from HS... Thanks again!

Homework Statement \int \frac{1}{\left(x^2+z^2\right)^{3/2}} \, dx I have been trying various u-substitutions for about 2 hours now, but I cannot seem to find a way to solve this by hand! I used mathematica to solve the problem. I feel like it will be fairly straightforward once I figure...

This is simply in the context of the Geometrical Optics course I'm taking. I'm just getting into optics in school, and I am just trying to gain some insight into this method for evaluating optical systems. I use Excel to automate the process. The 20 lens system was just hypothetical. Moving...

I have heard those terms briefly. My professor at the institute of optics at the university of rochester probably just wanted to shorten the terms, so he calls them a and b rays. Anyways, I'm just trying to get a good grasp of how to evaluate a given optical system using those rays, why you...

An a-ray is just an axial ray that goes through the aperture stop. Generally, you find the angle in the object plane that corresponds to when the a-ray just kisses the edge of the aperture stop, AKA the Y height at that point is the radius of the aperture stop. A b-ray is a ray that goes...

Is anyone familiar with the "y n u" ray trace method for optical systems, using paraxial rays where Snell's law is approximated using: n1sinx1 ≈ n1tanx1 ≈ n1u1 = n2u2 If anyone is familiar with this, I'd be interested to discuss the finer points of "a-ray" and "b-ray" tracing, as well...

I had the same thought. Should it be defined to mean the number of "n" terms? Maybe a large N? Or are you saying there is something more fundamentally wrong with it. Thanks!

I thought of another possible solution: [1-e^(i*n*x)]/[1-e^(i*x)] Is it within the parameters of the problem to have the integer "n" included in the answer? I was typing this as you posted ha. This would be the sum of a finite geometric series, no? I now realize what the clue was...