Thank you for commenting.
Someone already explained to me why the integral is zero. He showed me lots of graphs and it can be seen that the indefinite integral vanishes over the entire domain. However, near the origin, when the limits of integration are narrow and symmetric, the definite...
As shown in the image below, I tried to integrate a large integral. However, the result is strange. According to the result, the integral is always zero whatever the values of w, h, L, P, S and k. However, when I try to put some "test values", the result is not zero.
test values...
Consider
$$ \int_{y_0}^{y_1} cos (ay^2) dy = \sqrt{\frac{\pi}{2a}} [C(\frac{2ay_1}{\sqrt{2a\pi}})-C(\frac{2ay_0}{\sqrt{2a\pi}})]$$
where
C(x) is the C Fresnel integral
This is maximum when (for a given a=a_0)
$$\frac{2ay_1}{\sqrt{2a\pi}} =1$$
and
$$\frac{2ay_0}{\sqrt{2a\pi}} =-1$$
So...
I solved the original maximization problem with a lot of simplifications. I tried to generalize it and ended up in this problem.
In my original calculations, I found that there is a ratio between for example a and B. For instance, a/B = 2.0 which will always produce a K with a max value. The K...
Consider a double integral
$$K= \int_{-a}^a \int_{-b}^b \frac{B}{r_1(y,z)r_2^2(y,z)} \sin(kr_1+kr_2) \,dy\,dz$$
where
$$r_1 =\sqrt{A^2+y^2+z^2}$$
$$r_2=\sqrt{B^2+(C-y)^2+z^2} $$
Now consider a function:
$$C = C(a,b,k,A,B)$$
I want to find the function C such that K is maximized. In other...
I would like to clarify this particular question.
According to the answer key, the answer is D.
However, I argued to my friend that there may be an error in this question. The arrow is pointing toward a carbon atom so the question might have been " The indicated carbon atom is:". But if...
How do you find the order of convergence? Do you mean use convergence tests? Most of the time f1 > f2 so I think the series converges. Does this mean that I'm allowed to use my first expansion? Thanks.
I'm trying to expand the following using Newton's Generalized Binomial Theorem.
$$[f_1(x)+f_2(x)]^\delta = (f_1(x))^\delta + \delta (f_1(x))^{\delta-1}f_2(x) + \frac{\delta(\delta-1)}{2!}(f_1(x))^{\delta-2}(f_2(x))^2 + ...$$
where $$0<\delta<<1$$
But the condition for this formula is that...
Sir, in the generalization part, I have found some sources where the terms are infinite and not finite as written here. Can you post a link where I can find this formula? Thanks
Fresnel Number is defined as $$N_f(a) = a^2 / \lambda L$$
where a is the size of slit, lambda is wavelength and L is distance from slit to screen
In the paper, the slit was size 2a and distance from slit to screen is L + D, but D = L so
$$N_f(a) = 4a^2 / \lambda 2L = 2a^2/ \lambda L$$
I don't...
I used cos (A-B) = cos A cos B + sin A sin B (I feel bad for not noticing this. I was too focused on applying the N_f(a) ≪1 and the other condition.)
$$P(x; a)= \frac{\gamma}{2\lambda L \eta} [\frac{1}{π^2N_F(a)\eta(1 - \frac{x}{a\eta})^2} + \frac{1}{π^2N_F(a)\eta(1 + \frac{x}{a\eta})^2}...
Thanks for helping. I did it differently and arrived at the final form but I had some assumptions that I don't know if allowed or not. Like for example, $$N_F(a) =\frac{2a^2}{\lambda L}$$ but since $$N_F(a) ≪1$$, I replaced it with $$\frac{a^2}{\lambda L}$$. Is this reasonable? If not, I would...