Function Definition and 1000 Threads

I learned about Bessel functions and steady-state temperature distributions in the past. Recently, I was searching online for some example problems on the topic and found the "original question" along with the solution online as a PDF file. While I am unsure will it be appropriate for me to...

Here N, a, and b are integer constants. M is also an integer but changes for every value of x, which makes the index of the second summation dependent on the first. The problem is the relationship M(x) is analytically difficult to define. Is there a way to solve/simplify this expression?

In QM by virtue of wave function we calculate any things. But in QFT it seems that there is a lacking of notion of wave function.I do not understand why QFT still goes well(it is a good theory to calculate any things)?

My intuition for this problem is to use divergence theorem: ## \int_V \nabla^2 u dV = \int_S \nabla u \cdot \vec{n} dS## But note that ##\vec{n}## is perpendicular to x-y plane, and makes ##\int_S \nabla \ln s \cdot \vec{n} dS = 0## If we take laplacian in polar coordinate directly, then...

My textbook for Advanced Electomagnetics, by Balinas has this identity. cos θ = se^(jξ) = s( cos ξ + j sin ξ ). I have no idea what they are saying. Is there an S funtion I'm not aware of? I've looked back and forth, and he doesn't seem to explain it's use. I've inserted a picture of the...

##f(x)=-x^2 + 3## so ##f^{-1} (x)=- \sqrt{3-x} ~, x \leq 3## ##ff^{-1} = - (- \sqrt{3-x})^2 + 3 = x## and the domain will be ##x \leq 3## ##f^{-1} f = - \sqrt{3-(-x^2+3)} = -x ## and the domain will be ##x \leq 0## My question: ##ff^{-1} (x)## or ##f^{-1} f(x) ## is not always equal to...

I have this function that I need to reverse. I can't figure out how to reverse it though, I should be able to take the resulting int, put it through a function with the height value and get back distance. Normally, when I get stuck, I just ask wolfram alpha to solve it for me, but that...

E=hf-W where W is a work function. However we know that electrons in an atom will be excited only when radiated with photons of n*f0 discrete number of frequencies. where E=hf-W is a continuous function. Is this because energy level is continuous within a conductor? If we think of only...

As a part of a bigger problem, I was trying to evaluate the D'Alambertian of ##\epsilon(t)\delta(t^2-x^2-y^2-z^2)##, where ##\epsilon(t)## is a sign function. This term appears in covariant commutator function, so I was checking whether I can prove it solves Klein-Gordon equation. Since there's...

A conducting sphere of radius a is made up of 3 different shells, upper part a/2 z -a/2, the middle part -a/2 z a, the lower part -a z -a/2 where the center of the sphere is at the origin. The upper part V and the lower part has -V potentials, while the mid part is grounded. Find the...

Hi all:) In my recent exploration of Elliptic Function, Curves and Motion I have come upon a handy equation for creating orbital motion. Essentially I construct a trigonometric function and use the max distance to foci as the boundary for my motion on the x-plane. When I plot a point rotating...

1. a)I have plotted the graph on desmos and attached an image here. b i. The threshold frequency is equal to the x-intercept ~ 5.6*10^14 Hz ii. The work function is equal to the y-intercept ~ -3.75*10^19 J (would it be correct to state that this value is negative?) c. Convert to eV; 3.75*10^19...

Summary:: i) Set up a differential equation that describes how the pressure ##p## varies with the distance r from the center of the planet. Hint: You can base your reasoning on static equilibrium and Archimedes' principle. ii)Calculate how the atmospheric pressure p and the density of the...

I tested the first function with the Cauchy Riemann equations and it seemed to fail that test, so I don't believe that function is analytic. However, I'm really not sure how to show that it is or is not analytic using the definition of the complex derivative.

If my partition function is for a continuous distribution of energy, can I simply say that the probability of my ensemble being in a state with energy ##cU## is ##e^{-\beta cU} /Z##? I believe that isn't right as my energy distribution is continuous, and I need to be integrating over small...

The domain and range of this function will be the same. We can let ##𝑓(𝑥)=\sqrt{x},𝑥≥0## However, ##𝑦=𝑓(𝑥)≥0##, so the domain and range of ##f## are ##[0,+∞)## And since ##f## is a function, ##f^{-1}s## domain is the range of ##f## and ##f^{-1}s## range is ##f’s## domain. In other words...

https://phys.org/news/2020-09-function-collapse-gravity.html An interesting article I saw yesterday. However, both Nature articles (the summary [https://www.nature.com/articles/s41567-020-1026-2] and the actual technical paper [https://www.nature.com/articles/s41567-020-1008-4]) are behind a...

Let $a$ be an integer. Consider the function $y=\dfrac{12x^2-12ax}{x^2+36}$. For what integral values of $a$ the maximum and the minimum of the function $y=f(x)$ are integers?

I know that a tensor can be seen as a linear function. I know that the tensor product of three spaces can be seen as a multilinear map satisfying distributivity by addition and associativity in multiplication by a scalar.

Hey there, I am a little confused about the way most textbooks and notes I've read find the beta function for QED. They find it by looking at how the photon propagator varies with momentum ##q##, in particular in the context of a ##2\rightarrow2## scattering process which is proportional to...

## \int_0 ^ {2 \pi} \frac {dx} {3 + cos (x)} ## las únicas formas que probé fueron, multiplicar por ## \frac{3-cos (x)}{3-cos (x)} ## pero no me gusta esto porque obtengo una expresión muy complicada. También recurrí a la sustitución ## t = tan (\frac {x} {2}) ## que me gusta bastante, pero...

Hi there. I have the following function: $$f(x)=x+\frac{1}{(x+1)}$$ I've caculated the derivative to: $$f'(x)=1-\frac{1}{(1+x)^2}$$ And the domain to: $$(-\infty, -1)\cup(-1, \infty)$$ I've also found two extreme point: $$x=0, x=-2$$ I know that a function is strictly increasing if...

Not sure how to go about this. Would relying on a hole or asymptote work?

Hi, just wanting to know the answer to this. What parts of the brain do playing these games stimulate? What effects do playing these games have on the brain? what brain functions do they improve or strengthen? more specifically I mean the games call to power 1 and/or empire Earth 1. Note that i...

\[ \int_{0}^{\inf} \frac{e^{-\frac{(x-a)^2}{b}}}{x^2-c^2} dx\] or \[ \int_{0}^{constant} \frac{e^{-\frac{(x-a)^2}{b}}}{x^2-c^2} dx\] maybe application Residue theorem integral ? because this problem same the kramers kronig relation?

This is a long post with limited amount of physics in it (but it is a physics question, so hopefully it is allowed). I am a scientist but trained as an ecologist (PhD) and despite my long interest in physics, my knowledge of it remains rather rudimentary. Apologies for that, in advance. I am...

Hello, While using the IDLE editor, I noticed that it is possible to simply type the variable name, list name, dict name and view its content without using the print() function...For example, if the variable a=5, then I would expect print(a) to output 5. And it does. But I can also simply type...

If we have a set of variables ##x_1, x_2, ...x_n ## what does it mean to say that "##G## is only a function of ##x_1,x_2,x_3##"? My thoughts: Context 1: The function ##G## has been previously defined. In Context 1, saying "##G## is only a function of ##x_1,x_2,x_3##" means the same thing as...

The graph's turning point of a quadratic function $$f(x)=ax^2+bx+c$$ is over the X-axis. If the coordinate of the turning point is (p, q) and a > 0, the correct statement is ... A. c is less than zero B. c is more than zero C. q is less than zero D. q equals zero Since the point (p, q) is over...

the explanation about the question I got from internet is, A very small change in area divided by the dx will give the function of graph so anti-derivative of function of graph should be equal to the area of the function. It also seem quite obvious to me but I am not satisfied by it, It seems to...

Hi, I was hoping to gain more insight into these window questions when looking at frequency spectra questions. I don't really know what makes windows better than one another. My attempt: In the question, we have f(t) = cos(\omega_0 t) and therefore its F.T is F(\omega ) = \pi \left(...

Consider the following function: $$f(x) = \begin{cases} 1 & \text{when} & -\pi<x<0\\ 0 & \text{when} & 0<x<\pi \end{cases}$$ Beyond ##-\pi## and ##\pi##, the function just repeats itself; it is periodic. I want to plot this function for values beyond ##-\pi## and ##\pi##. The graph should look...

Here is the figure: The answer is $$Q_A<Q_B$$ which I can show by calculation using the above equations. What's confusing to me is I thought that the change in internal energy was a state function. Which would mean since the initial and final points are the same, $$\Delta E_A=\Delta E_B$$ or by...

I never took any physics courses nor don't have a background in mathematics never the less I became very interested in quantum physics after reading Sean Carroll's book Something deeply hidden. One of the difficult things for me to wrap my head around was the concept of superposition and...

I have a strange problem. I pass an Array[20] to a function. When I put cout << sizeof(Ar); (line 29 in the program) it only shows the length of one element ( 4bytes). BUT if I write cout << Ar[10]; (line 30 in the program) It will give me the correct number of the original idNum[10]. I tried a...

I have the characteristic function of the Cauchy distribution ##C(t)= e^{-(\mid t \mid)}##. Now, how would I show that the Cauchy distribution has no moments using this? I think you have to show it has no Taylor expansion around the origin. I am not sure how to do this.

Honestly not sure how to go about this. Again this is one equation of 4 that I have. I considered using Laplace transforms but taking the Laplace transform of a step function whose argument is one of the variables being solved for doesn't seem possible. Also, if there is an alternative way to...

Homework Statement:: Why is the heaviside function in the inverse laplace transform of 1? Relevant Equations:: N/A This is a small segment of a larger problem I've been working on, and in my book it gives the transform of 1 as 1/s and vice versa. But as I've looked online for help in figuring...

For a single particle, $$Z=\frac{1}{h^2}\int_{-\infty}^{\infty} e^{-\beta \frac{P^2}{2m}}d^2p \int e^{-U(r)}drd\theta= \frac{1}{h^2}(\frac{2\pi m}{\beta}) 2\pi [\int_{0}^{r_0}e^{U_0}dr+\int_{r_0}^{R}dr]$$ $$ =\frac{1}{h^2}(\frac{2\pi m}{\beta}) 2\pi [e^{U_0}(r_0)+(R-r_0)]=\frac{\pi...

The beginning is straight forward and I found f=x^2-2yz, which satisfies grad(f)=F. Then I calculated W= f(x,y,z)-f(0,1,1) since it's conservative. I get stuck when trying to find the max and mins. Given grad(f)=0 at extrema, we can see (0,0,0) is a point. On the boundary, I have to...

I am not sure, but since the partition function Z is just the sum of all Boltzmann Factor We can just add: (some terms don't appear in the image, by the way, the estimative is nice, the result is above ANS) But i didn't understand what the author did: While i didn't even care about the...

I would express the pendulum motion in form of polar coordinates with corresponding unit vectors in the x-z-plane (view images, ignore the german, sorry). How would you now bring in the circular movement, which is constantly changing its plane? Is it enough to simply add the representation of a...

Hey, please tell me if the following is correct. We have a continuous, increasing and strictly monotonic function on ##[a, b]##, and ##x_0\in[a,b]##. Let ##g(y)## be its inverse, and ##f(x_0)=y_0##. I want to show that ##|y-y_0|<\delta\implies|g(y)-g(y_0)|<\epsilon##. \begin{align*}...

In this paper, on quantum Ising model dynamics, they consider the Hamiltonian $$\mathcal{H} = \sum_{j < k} J_{jk} \hat{\sigma}_{j}^{z}\hat{\sigma}_{k}^{z}$$ and the correlation function $$\mathcal{G} = \langle \mathcal{T}_C(\hat{\sigma}^{a_n}_{j_n}(t_n^*)\cdot\cdot\cdot...

Obviously, a priori it is not possible tu use the Taylor series because the derivative ##\sim (x-1)^{1/n-1}## is not well defined in x=1. Is there any mathematical trick? or, other approximation?

could anyone explain why in the page of book this figure is related to hartree-fock? I mean why if t1>t2 we have these possibilities? and why not particle propagate from x2t2 to x3t3 instead x3t3+?

h(x) = 0 for x ≤ 0 h(x) = x^2 for x>0 But my book says h(x) = 0 for x<0 h(x) = x^2 for x≥0 Can my solution (the first one) work as well? Because the actual function value at x = 0 is zero. I feel like my solution is more elegant.

I have to show that $\forall z\in B(0,0.4)$, there exists an $x\in B(0,1)$ such that $f(x)=z$ but I am not sure how to show this. From the reverse triangle inequality $$-|f(x)-f(y)|+|x-y|\leq 0.1|x-y|\implies |f(x)-f(y)|\geq 0.9|x-y|$$ im not sure if this helps.

Is it possible to find three quadratic polynomials $f(x),\,g(x)$ and $h(x)$ such that the equation $f(g(h(x)))=0$ has the eight roots 1, 2, 3, 4, 5, 6, 7 and 8?