Is the infinite series ##\sum_{n=1,3,5,...}^\infty \frac {1} {n^6}## somewhat related to the Riemann zeta function?The attached image suggest the value to be inverse of the co-efficient of the series.Is there any integral representation of the series from where the series can be evaluated?
Hi guys it's me again. I need help with this exercise which reads: a particle of mass m, placed in an infinite rectangular one-dimensional potential well that confines it in the segment between ##x = -\frac{a}{2} and x=\frac{a}{2}##, is at instant ##t=0## in the state:
##|\psi \rangle =...
Since ##\left|3x+2\right|=0\rightarrow\ x=-\frac{2}{3}##, we know the vertical asymptote is at ##x=-\frac{2}{3}##.
Looking at the limit at that point, and also looking at the left- and right-sided limit, I cannot simplify it any further...
What do you guys have to say about this Mathoverflow post?
Do you have any interesting ideas about this?
https://mathoverflow.net/questions/432396/extending-reals-with-logarithm-of-zero-properties-and-reference-request
So I've thought of an admittedly crude proof that the process of pattern recognition i.e. finding patterns, be they linguistic, mathematical, artistic, whatever, is a process that can never end.
It goes like this: Imagine we find all patterns, and I mean ALL of them, and we list them on a...
Hi. I’m trying to solve an optics problem and really struggling. The problem is best described as follows…
Imagine you have a section of a wall that you want to look like a window on a spaceship. So you want to look at this “window” and see through it some “stars” (i.e. pinpoints of light) that...
Problem:
I have done part a) in spherical polar coordinates.
For part b) I thought it would be just:
$$\sigma = -\epsilon_0 \frac{\partial V}{\partial r}$$
But I got confused by "You may want to use different coordinate systems .." So I assume partial derivative w.r.t to r is the spherical...
An infinite product representation of Bessel's function of the first kind is:
$$J_\alpha(z) =\frac{(z/2)^\alpha}{\Gamma(\alpha+1)}\prod_{n=1}^\infty(1-\frac{z^2}{j_{n,\alpha}^2})$$
Here, the ##j_{n,\alpha}## are the various roots of the Bessel functions of the first kind. I found this...
I have the following problem and am almost sure of the answer but can't quite prove it:
##f(y)## is nonnegative, and I know that ##\int_0^{\infty } f(y) \, dy## is finite.
I now need to calculate (or simplify) the double integral:
$$\int_0^{\infty } \left(\int_x^{\infty } f(y) \, dy\right) \...
So I was thinking about arithmetic, geometric and harmonic means when I had a thought. Let's say we have a curve y = x^2. We want to find the AM of the points on the curve between x=1 and x=2 i.e. y = 1 and y = 4. To make thing easier, we'll start with just the endpoints and keep adding...
Hi
For an infinite well , solving the Schrodinger equation gives wavefunctions of the form sin(nπx/L). These are not eigenfunctions of the momentum operator which means there are no eigenvalues of the momentum operator. Does this mean momentum cannot be measured ?
Inside the infinite well the...
For part (b),
The solution is
However, is there really an infinite number of pairs physically speaking? It would be very hard, say, vary the force applied by ##0.0000001N## for example.
Many thanks!
The natural numbers are the smallest infinite set, aleph_0.
By taking out an infinite subset to the natural numbers (the odd naturals), we get an infinite subset, the even numbers, which has the same size, aleph_0 (e.g. the map n->2n).
We can take an even "sparser" subset of the natural...
For 6(b),
The solution is,
However, for ##a = 1## they could have also said that f is not continuous since f(1) is not defined (vertical asymptote) correct?
Many thanks!
Kurzesagt in a Nutshell said that the number of possible protein combinations the human body can have is 6.8 x 10^495. I asked GPT to multiple it by 20 million (which is the hypothetical number of possible alien civilizations in the Milky Way galaxy give or take). The chatbot gave me 1.36 x...
##\sum_{n=1}^\infty n^{-a}## converge s for ##a\gt 1## - otherwise diverges. Is there any theory for ##a_n##? For example ##a_n\gt 1## and ##\lim_{n\to \infty} a_n =1##. How about non-convergent with ##\liminf a_n=1##?
I'm reviewing some subjects that I long forgot and now I'm studying synchronous machines.
So, when you change the field current (of the rotor) of s synchronous generator, the result is a change in the magnetic flux, which change the internal voltage (Ea). That will result in a change in the...
I believe the universe could not possibly be infinite. I am not denying it is of extremely large scale but it still must be finite. No true infinity is found in nature but only in mathematics. why should it differ here ?
I wrote down the equation of motion for In(t) and I'm trying to match it with infinite spring mass system equation solution. In the spring mass system, we consider A to be the equilibrium length of the springs, and we can thus write Xn(t) = X(nA,t) and put it back into the equation of motion...
How many whole numbers are there?
infinity.
How many tenths of whole numbers are there?
ten times infinity.
How many hundredths of whole numbers are there?
100 times infinity.
How many millionths of whole numbers are there?
1,000,000 times infinity
How many decimal numbers are there?
infinity...
From my physical problem, I ended up having a sum that looks like the following.
S_N(\omega) = \sum_{q = 1}^{N-1} \left(1 - \frac{q}{N}\right) \exp{\left(-\frac{q^2\sigma^2}{2}\right)} \cos{\left(\left(\mu - \omega\right)q\right)}
I want to know what is the sum when N \to \infty. Here...
For a state to be stationary it must be time independent.
Naively, I tried to find the values of c where I don't have any time dependency.
##e^{c \cdot L_z} \psi (r,t) = e^{c L_z} \sqrt{\frac{8}{l^3}} sin(\frac{2 \pi x}{l}) sin(\frac{2 \pi u}{l}) sin(\frac{2 \pi z}{l}) e^{-iEt/\hbar}##...
I have a nanoparticle of cadmium selenide with a diameter d. When it emits a photon with a wavelenght lambda, it happens because an electron jumps from the conduction band to the occupied band across a forbidden band. I can suppose that jump as a jump from a higher energy level (the conduction...
Hi Physics Forum, I want to ask if there is an "appropriate" notation for the infinite self-iteraction of an analytic function ##f(x)##, that is ##f(f(f(...)))##. For example I know ##f^{(+\infty)}(x)## can be a way, but there is an operator notation as for the infinite sum...
Prove the existence of a strictly increasing sequence ##m_1 < m_2 < m_3 < \cdots## of integers satisfying the property that for all positive integers ##\ell##, the sequence ##\sin(\ell m_1), \sin(\ell m_2), \sin (\ell m_3),\ldots## converges.
##s_1=2##
##s_2=4##
##s_3=5.333##
##s_4=5.9999##
##(s_n)## is increasing, but unable to guess a bound. Let's see if Cauchy criterion can do something.
For n>2,
$$
s_{n+k} - s_n = \frac{2^{n+1} }{(n+1)!} + \frac{ 2^{n+2} }{(n+2)!} + \cdots \frac{2^{n+k} }{(n+k)!}
$$
$$
s_{n+k} - s_n <...
I am reading an abstract algebra textbook and enjoying it. I am working through preliminaries some more to refine my knowledge on proofs with sets before really digging in. I understand that if
$$X \subseteq Y$$
and
$$ Y \subseteq X$$
Then
$$ X = Y$$
This makes sense to me. However, the...
This is not homework, just a question of curiosity. Something that came to my mind last night.
Say I have a magnet and a piece of iron. Say I use it to do weight training exercise. I stick the magnet on the wall and stick the iron on the magnet. Say it takes 100lbs of force to pull the piece of...
The implication seems to be that from the beginning of the post expansion era, there was everywhere an average velocity of a large volume of matter which was (very near) zero everywhere with respect to a common fixed coordinate system (with a spacially uniform time expansion of distances)...
I have solved c), but don’t know how to solve the integral in d.
It looks like an integral to get c_n (photo below), but I still can’t figure out what to make of c) in the integral of d).
I also thought maybe you can rewrite c) into an initial wave function (photo below) with A,x,a but don’t...
If the Universe could somehow reach a state of infinite entropy (or at least a state of extremely high entropy), would all fundamental symmetries of the physical laws (gauge symmetries, Lorentz symmetry, CPT symmetry, symmetries linked to conservation principles...etc) fail to hold or be...
After reading up on some of the discussion in the Quantum Interpretations forums, I became interested in learning more about POVMs.
However, most of the examples are from the finite dimensional setting. If I wanted to model a POVM that approximately measures position and momentum...
I have the equations for all three regions but usually for region 3, which is Ce^ikx+De-ikx, the C term would be zero since there is no reflection, but with the infinite wall would it reflect? Would the whole wavefunction go to zero like when working with the infinite square wall? I'm stuck on...
The solution says that the tension in the string in the negative x direction is balanced by the force of the plate on the ball (red). Why is the repulsive force of the ball on the plate (in blue) not included in this calculation?
What I don't understand is how come the electric field of the negative plane isn't pointing towards the positive plane (in blue) and cancelling out the electric field of the positive plane (in red). See image
So I think I use the right approach and I get uncertainty like this:
And it's interval irrelevant(ofc),
So what kind of wave function gives us \h_bar / 2 ? I guess a normal curve? if so, why is normal curve could be? if not then what's kind of wave function can reach the lower bound
Consider a Markov chain with state space {1, 2, 3, 4} and transition matrix P given below:
Now, I have already figured out the solutions for parts a,b and c. However, I don't know how to go about solving part d? I mean the question says we can't use higher powers of matrices to justify our...
1. Individual students’ scores on a national test have a normal distribution with a mean of 18.5 and a standard deviation of 7.8. At a Trade School, 84 students took the test. If the scores at this school have the same distribution as national scores, what is the mean, standard deviation and...
3 cm is inside the cylinder. We can use a gaussian cylinder to enclose the inside of the cylinder up to 3 cm. Because the outer cylinder is infinite there is no flux out of the end caps with the inner cylinder. There is also no charge enclosed in the cylinder. So the electric field 3cm away from...
Draw a Gaussian pill box that starts from 0 (half way between the slab) and extends towards 2 cm.$$A \times \int_{0}^{0.02} \rho dz$$
I'm not sure if I should multiply the integral by A (area) or V (volume)
And if area would I multiply by 0.02^2?
I'm confused here. Thanks for your help.
Is anything in the physical universe unlimited or infinite or does Infinity only exist to make some math work? Is Infinity a real thing outside of math?
Does setting up the problem symmetrically on this axis and the boundary conditions applied make sense? I don't believe I will have a problem solving for the potential inside, but i just want to make sure I have my B.C and axis correct before proceeding.
EDIT:
Or should this be a 2-D lapace...